3D rotation group - Wikipedia

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</ul> </li> <li id="toc-Infinitesimal_rotations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Infinitesimal_rotations"> <div class="vector-toc-text"> 12 Infinitesimal rotations </div> </a> <ul id="toc-Infinitesimal_rotations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Realizations_of_rotations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Realizations_of_rotations"> <div class="vector-toc-text"> 13 Realizations of rotations </div> </a> <ul id="toc-Realizations_of_rotations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spherical_harmonics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Spherical_harmonics"> <div class="vector-toc-text"> 14 Spherical harmonics </div> </a> <ul id="toc-Spherical_harmonics-sublist" class="vector-toc-list"> </ul> </li> <li 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v trojrozměrném prostoru" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Drehgruppe" title="Drehgruppe – German" lang="de" hreflang="de" data-title="Drehgruppe" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Grupo_de_rotaci%C3%B3n_SO(3)" title="Grupo de rotación SO(3) – Spanish" lang="es" hreflang="es" data-title="Grupo de rotación SO(3)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/3-dimensia_turnada_grupo" title="3-dimensia turnada grupo – Esperanto" lang="eo" hreflang="eo" data-title="3-dimensia turnada grupo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/3%EC%B0%A8%EC%9B%90_%EC%A7%81%EA%B5%90%EA%B5%B0" title="3차원 직교군 – Korean" lang="ko" hreflang="ko" data-title="3차원 직교군" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%AA_%D7%94%D7%A1%D7%99%D7%91%D7%95%D7%91_(3)SO" title="חבורת הסיבוב (3)SO – Hebrew" lang="he" hreflang="he" data-title="חבורת הסיבוב (3)SO" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Rotatiegroep" title="Rotatiegroep – Dutch" lang="nl" hreflang="nl" data-title="Rotatiegroep" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Grupa_obrot%C3%B3w" title="Grupa obrotów – Polish" lang="pl" hreflang="pl" data-title="Grupa obrotów" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grupo_de_rota%C3%A7%C3%A3o" title="Grupo de rotação – Portuguese" lang="pt" hreflang="pt" data-title="Grupo de rotação" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%BF%D0%B0_%D0%B2%D1%80%D0%B0%D1%89%D0%B5%D0%BD%D0%B8%D0%B9" title="Группа 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Group of rotations in 3 dimensions</div> In <a href="/wiki/Classical_mechanics" title="Classical mechanics">mechanics</a> and <a href="/wiki/Geometry" title="Geometry">geometry</a>, the 3D rotation group, often denoted <a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">SO</a>(3), is the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> of all <a href="/wiki/Rotation" title="Rotation">rotations</a> about the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a> of <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional</a> <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> under the operation of <a href="/wiki/Function_composition" title="Function composition">composition</a>.<a href="#cite_note-1">[1]</a> By definition, a rotation about the origin is a transformation that preserves the origin, <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a> (so it is an <a href="/wiki/Isometry" title="Isometry">isometry</a>), and <a href="/wiki/Orientation_(vector_space)" title="Orientation (vector space)">orientation</a> (i.e., handedness of space). Composing two rotations results in another rotation, every rotation has a unique <a href="/wiki/Inverse_function" title="Inverse function">inverse</a> rotation, and the <a href="/wiki/Identity_map" class="mw-redirect" title="Identity map">identity map</a> satisfies the definition of a rotation. Owing to the above properties (along composite rotations' <a href="/wiki/Associative_property" title="Associative property">associative property</a>), the set of all rotations is a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> under composition. Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. Rotations are not commutative (for example, rotating R 90° in the x-y plane followed by S 90° in the y-z plane is not the same as S followed by R), making the 3D rotation group a <a href="/wiki/Non-abelian_group" title="Non-abelian group">nonabelian group</a>. Moreover, the rotation group has a natural structure as a <a href="/wiki/Manifold" title="Manifold">manifold</a> for which the group operations are <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smoothly differentiable</a>, so it is in fact a <a href="/wiki/Lie_group" title="Lie group">Lie group</a>. It is <a href="/wiki/Compact_space" title="Compact space">compact</a> and has dimension 3. Rotations are <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformations</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> and can therefore be represented by <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> once a <a href="/wiki/Basis_of_a_vector_space" class="mw-redirect" title="Basis of a vector space">basis</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> has been chosen. Specifically, if we choose an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}">, every rotation is described by an <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal 3 × 3 matrix</a> (i.e., a 3 × 3 matrix with real entries which, when multiplied by its <a href="/wiki/Transpose" title="Transpose">transpose</a>, results in the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>) with <a href="/wiki/Determinant" title="Determinant">determinant</a> 1. The group SO(3) can therefore be identified with the group of these matrices under <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>. These matrices are known as "special orthogonal matrices", explaining the notation SO(3). The group SO(3) is used to describe the possible rotational symmetries of an object, as well as the possible orientations of an object in space. Its <a href="/wiki/Group_representation" title="Group representation">representations</a> are important in physics, where they give rise to the <a href="/wiki/Elementary_particle" title="Elementary particle">elementary particles</a> of integer <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Length_and_angle">Length and angle</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=1" title="Edit section: Length and angle">edit</a>]</div> Besides just preserving length, rotations also preserve the <a href="/wiki/Angle" title="Angle">angles</a> between vectors. This follows from the fact that the standard <a href="/wiki/Dot_product" title="Dot product">dot product</a> between two vectors u and v can be written purely in terms of length (see the <a href="/wiki/Law_of_cosines" title="Law of cosines">law of cosines</a>): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} \cdot \mathbf {v} ={\frac {1}{2}}\left(\|\mathbf {u} +\mathbf {v} \|^{2}-\|\mathbf {u} \|^{2}-\|\mathbf {v} \|^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \cdot \mathbf {v} ={\frac {1}{2}}\left(\|\mathbf {u} +\mathbf {v} \|^{2}-\|\mathbf {u} \|^{2}-\|\mathbf {v} \|^{2}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2be32bc3925318102e0a1d5f45a696051b5198d2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.674ex; height:5.176ex;" alt="{\displaystyle \mathbf {u} \cdot \mathbf {v} ={\frac {1}{2}}\left(\|\mathbf {u} +\mathbf {v} \|^{2}-\|\mathbf {u} \|^{2}-\|\mathbf {v} \|^{2}\right).}"> It follows that every length-preserving linear transformation in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> preserves the dot product, and thus the angle between vectors. Rotations are often defined as linear transformations that preserve the inner product on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}">, which is equivalent to requiring them to preserve length. See <a href="/wiki/Classical_group" title="Classical group">classical group</a> for a treatment of this more general approach, where SO(3) appears as a special case. <div class="mw-heading mw-heading2"><h2 id="Orthogonal_and_rotation_matrices">Orthogonal and rotation matrices</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=2" title="Edit section: Orthogonal and rotation matrices">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">Orthogonal matrix</a> and <a href="/wiki/Rotation_matrix" title="Rotation matrix">Rotation matrix</a></div> Every rotation maps an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> to another orthonormal basis. Like any linear transformation of <a href="/wiki/Finite-dimensional" class="mw-redirect" title="Finite-dimensional">finite-dimensional</a> vector spaces, a rotation can always be represented by a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>. Let R be a given rotation. With respect to the <a href="/wiki/Standard_basis" title="Standard basis">standard basis</a> e1, e2, e3 of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> the columns of R are given by (Re1, Re2, Re3). Since the standard basis is orthonormal, and since R preserves angles and length, the columns of R form another orthonormal basis. This orthonormality condition can be expressed in the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{\mathsf {T}}R=RR^{\mathsf {T}}=I,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mi>R</mi> <mo>=</mo> <mi>R</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <mi>I</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{\mathsf {T}}R=RR^{\mathsf {T}}=I,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f00b1d1fa4a159fc813f017112c4b3bc45b7635" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.775ex; height:3.009ex;" alt="{\displaystyle R^{\mathsf {T}}R=RR^{\mathsf {T}}=I,}"></dd></dl> where RT denotes the <a href="/wiki/Transpose" title="Transpose">transpose</a> of R and I is the 3 × 3 <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>. Matrices for which this property holds are called <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrices</a>. The group of all 3 × 3 orthogonal matrices is denoted O(3), and consists of all proper and improper rotations. In addition to preserving length, proper rotations must also preserve orientation. A matrix will preserve or reverse orientation according to whether the <a href="/wiki/Determinant" title="Determinant">determinant</a> of the matrix is positive or negative. For an orthogonal matrix R, note that det RT = det R implies (det R)2 = 1, so that det R = ±1. The <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of orthogonal matrices with determinant +1 is called the special <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a>, denoted SO(3). Thus every rotation can be represented uniquely by an orthogonal matrix with unit determinant. Moreover, since composition of rotations corresponds to <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>, the rotation group is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to the special orthogonal group SO(3). <a href="/wiki/Improper_rotation" title="Improper rotation">Improper rotations</a> correspond to orthogonal matrices with determinant −1, and they do not form a group because the product of two improper rotations is a proper rotation. <div class="mw-heading mw-heading2"><h2 id="Group_structure">Group structure</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=3" title="Edit section: Group structure">edit</a>]</div> The rotation group is a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> under <a href="/wiki/Function_composition" title="Function composition">function composition</a> (or equivalently the <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">product of linear transformations</a>). It is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a> consisting of all <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible</a> linear transformations of the <a href="/wiki/Real_coordinate_space" title="Real coordinate space">real 3-space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}">.<a href="#cite_note-2">[2]</a> Furthermore, the rotation group is <a href="/wiki/Nonabelian_group" class="mw-redirect" title="Nonabelian group">nonabelian</a>. That is, the order in which rotations are composed makes a difference. For example, a quarter turn around the positive x-axis followed by a quarter turn around the positive y-axis is a different rotation than the one obtained by first rotating around y and then x. The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the <a href="/wiki/Cartan%E2%80%93Dieudonn%C3%A9_theorem" title="Cartan–Dieudonné theorem">Cartan–Dieudonné theorem</a>. <div class="mw-heading mw-heading3"><h3 id="Complete_classification_of_finite_subgroups">Complete classification of finite subgroups</h3>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=4" title="Edit section: Complete classification of finite subgroups">edit</a>]</div> The finite subgroups of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} (3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8366fc6e92660ba077b87b745b305a4176b1d1ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle \mathrm {SO} (3)}"> are completely <a href="/wiki/Classification_theorem" title="Classification theorem">classified</a>.<a href="#cite_note-coxeter-3">[3]</a> Every finite subgroup is isomorphic to either an element of one of two <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a> families of planar isometries: the <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0301812adb392070d834ca2df4ed97f1cf132f33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle C_{n}}"> or the <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{2n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/317005e1541e49becbb38da665de9d667b8863b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.965ex; height:2.509ex;" alt="{\displaystyle D_{2n}}">, or to one of three other groups: the <a href="/wiki/Tetrahedral_group" class="mw-redirect" title="Tetrahedral group">tetrahedral group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cong A_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cong A_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c787947b23c62f7360e3fdd060af896e32c6bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.251ex; height:2.509ex;" alt="{\displaystyle \cong A_{4}}">, the <a href="/wiki/Octahedral_group" class="mw-redirect" title="Octahedral group">octahedral group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cong S_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≅</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cong S_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9037c415dceccb174f6bd8e2cb6d777053b5122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.932ex; height:2.509ex;" alt="{\displaystyle \cong S_{4}}">, or the <a href="/wiki/Icosahedral_group" class="mw-redirect" title="Icosahedral group">icosahedral group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cong A_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cong A_{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dea51f976c1fc6e55ca425cbf639c25f1530a1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.251ex; height:2.509ex;" alt="{\displaystyle \cong A_{5}}">. <div class="mw-heading mw-heading2"><h2 id="Axis_of_rotation">Axis of rotation</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=5" title="Edit section: Axis of rotation">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Axis%E2%80%93angle_representation" title="Axis–angle representation">Axis–angle representation</a></div> Every nontrivial proper rotation in 3 dimensions fixes a unique 1-dimensional <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> which is called the axis of rotation (this is <a href="/wiki/Euler%27s_rotation_theorem" title="Euler's rotation theorem">Euler's rotation theorem</a>). Each such rotation acts as an ordinary 2-dimensional rotation in the plane <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> to this axis. Since every 2-dimensional rotation can be represented by an angle φ, an arbitrary 3-dimensional rotation can be specified by an axis of rotation together with an <a href="/wiki/Angle_of_rotation" class="mw-redirect" title="Angle of rotation">angle of rotation</a> about this axis. (Technically, one needs to specify an orientation for the axis and whether the rotation is taken to be <a href="/wiki/Clockwise_and_counterclockwise" class="mw-redirect" title="Clockwise and counterclockwise">clockwise</a> or <a href="/wiki/Counterclockwise" class="mw-redirect" title="Counterclockwise">counterclockwise</a> with respect to this orientation). For example, counterclockwise rotation about the positive z-axis by angle φ is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{z}(\phi )={\begin{bmatrix}\cos \phi &-\sin \phi &0\\\sin \phi &\cos \phi &0\\0&0&1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{z}(\phi )={\begin{bmatrix}\cos \phi &-\sin \phi &0\\\sin \phi &\cos \phi &0\\0&0&1\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f62c196069d2c1a2ce655d79fa0e48e7fc977255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:31.072ex; height:9.509ex;" alt="{\displaystyle R_{z}(\phi )={\begin{bmatrix}\cos \phi &-\sin \phi &0\\\sin \phi &\cos \phi &0\\0&0&1\end{bmatrix}}.}"></dd></dl> Given a <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> n in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> and an angle φ, let R(φ, n) represent a counterclockwise rotation about the axis through n (with orientation determined by n). Then <ul><li>R(0, n) is the identity transformation for any n</li> <li>R(φ, n) = R(−φ, −n)</li> <li>R(π + φ, n) = R(π − φ, −n).</li></ul> Using these properties one can show that any rotation can be represented by a unique angle φ in the range 0 ≤ φ ≤ π and a unit vector n such that <ul><li>n is arbitrary if φ = 0</li> <li>n is unique if 0 < φ < π</li> <li>n is unique up to a <a href="/wiki/Sign_(mathematics)" title="Sign (mathematics)">sign</a> if φ = π (that is, the rotations R(π, ±n) are identical).</li></ul> In the next section, this representation of rotations is used to identify SO(3) topologically with three-dimensional real projective space. <div class="mw-heading mw-heading2"><h2 id="Topology">Topology</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=6" title="Edit section: Topology">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hypersphere_of_rotations" class="mw-redirect" title="Hypersphere of rotations">Hypersphere of rotations</a></div> The Lie group SO(3) is <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphic</a> to the <a href="/wiki/Real_projective_space" title="Real projective space">real projective space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{3}(\mathbb {R} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{3}(\mathbb {R} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f5ea9bdc7df9ead8e852c208b16ebdbb1a8f5ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.609ex; height:3.176ex;" alt="{\displaystyle \mathbb {P} ^{3}(\mathbb {R} ).}"><a href="#cite_note-4">[4]</a> Consider the solid ball in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> of radius π (that is, all points of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> of distance π or less from the origin). Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin. The identity rotation corresponds to the point at the center of the ball. Rotations through an angle 𝜃 between 0 and π (not including either) are on the same axis at the same distance. Rotation through angles between 0 and −π correspond to the point on the same axis and distance from the origin but on the opposite side of the origin. The one remaining issue is that the two rotations through π and through −π are the same. So we <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">identify</a> (or "glue together") <a href="/wiki/Antipodal_point" title="Antipodal point">antipodal points</a> on the surface of the ball. After this identification, we arrive at a <a href="/wiki/Topological_space" title="Topological space">topological space</a> <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to the rotation group. Indeed, the ball with antipodal surface points identified is a <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a>, and this manifold is <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphic</a> to the rotation group. It is also diffeomorphic to the <a href="/wiki/Real_projective_space" title="Real projective space">real 3-dimensional projective space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{3}(\mathbb {R} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{3}(\mathbb {R} ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eacc73ad667eb9da5d2c9532fc4287673081c21a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.609ex; height:3.176ex;" alt="{\displaystyle \mathbb {P} ^{3}(\mathbb {R} ),}"> so the latter can also serve as a topological model for the rotation group. These identifications illustrate that SO(3) is <a href="/wiki/Connected_space" title="Connected space">connected</a> but not <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a>. As to the latter, in the ball with antipodal surface points identified, consider the path running from the "north pole" straight through the interior down to the south pole. This is a closed loop, since the north pole and the south pole are identified. This loop cannot be shrunk to a point, since no matter how it is deformed, the start and end point have to remain antipodal, or else the loop will "break open". In terms of rotations, this loop represents a continuous sequence of rotations about the z-axis starting (by example) at the identity (center of the ball), through the south pole, jumping to the north pole and ending again at the identity rotation (i.e., a series of rotation through an angle φ where φ runs from 0 to <a href="/wiki/Turn_(geometry)" class="mw-redirect" title="Turn (geometry)">2π</a>). Surprisingly, running through the path twice, i.e., running from the north pole down to the south pole, jumping back to the north pole (using the fact that north and south poles are identified), and then again running from the north pole down to the south pole, so that φ runs from 0 to 4π, gives a closed loop which can be shrunk to a single point: first move the paths continuously to the ball's surface, still connecting north pole to south pole twice. The second path can then be mirrored over to the antipodal side without changing the path at all. Now we have an ordinary closed loop on the surface of the ball, connecting the north pole to itself along a great circle. This circle can be shrunk to the north pole without problems. The <a href="/wiki/Plate_trick" title="Plate trick">plate trick</a> and similar tricks demonstrate this practically. The same argument can be performed in general, and it shows that the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of SO(3) is the <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> of order 2 (a fundamental group with two elements). In <a href="/wiki/Physics" title="Physics">physics</a> applications, the non-triviality (more than one element) of the fundamental group allows for the existence of objects known as <a href="/wiki/Spinor" title="Spinor">spinors</a>, and is an important tool in the development of the <a href="/wiki/Spin%E2%80%93statistics_theorem" title="Spin–statistics theorem">spin–statistics theorem</a>. The <a href="/wiki/Universal_cover" class="mw-redirect" title="Universal cover">universal cover</a> of SO(3) is a <a href="/wiki/Lie_group" title="Lie group">Lie group</a> called <a href="/wiki/Spin(3)" class="mw-redirect" title="Spin(3)">Spin(3)</a>. The group Spin(3) is isomorphic to the <a href="/wiki/Special_unitary_group" title="Special unitary group">special unitary group</a> SU(2); it is also diffeomorphic to the unit <a href="/wiki/3-sphere" title="3-sphere">3-sphere</a> S3 and can be understood as the group of <a href="/wiki/Versor" title="Versor">versors</a> (<a href="/wiki/Quaternion" title="Quaternion">quaternions</a> with <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> 1). The connection between quaternions and rotations, commonly exploited in <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>, is explained in <a href="/wiki/Quaternions_and_spatial_rotation" title="Quaternions and spatial rotation">quaternions and spatial rotations</a>. The map from S3 onto SO(3) that identifies antipodal points of S3 is a <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a> <a href="/wiki/Homomorphism" title="Homomorphism">homomorphism</a> of Lie groups, with <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> {±1}. Topologically, this map is a two-to-one <a href="/wiki/Covering_map" class="mw-redirect" title="Covering map">covering map</a>. (See the <a href="/wiki/Plate_trick" title="Plate trick">plate trick</a>.) <div class="mw-heading mw-heading2"><h2 id="Connection_between_SO(3)_and_SU(2)">Connection between SO(3) and SU(2)</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=7" title="Edit section: Connection between SO(3) and SU(2)">edit</a>]</div> In this section, we give two different constructions of a two-to-one and <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a> <a href="/wiki/Homomorphism" title="Homomorphism">homomorphism</a> of SU(2) onto SO(3). <div class="mw-heading mw-heading3"><h3 id="Using_quaternions_of_unit_norm">Using quaternions of unit norm</h3>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=8" title="Edit section: Using quaternions of unit norm">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quaternions_and_spatial_rotation" title="Quaternions and spatial rotation">Quaternions and spatial rotation</a></div> The group SU(2) is <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a> to the <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> of unit norm via a map given by<a href="#cite_note-5">[5]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=a\mathbf {1} +b\mathbf {i} +c\mathbf {j} +d\mathbf {k} =\alpha +\beta \mathbf {j} \leftrightarrow {\begin{bmatrix}\alpha &\beta \\-{\overline {\beta }}&{\overline {\alpha }}\end{bmatrix}}=U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>=</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo stretchy="false">↔</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo accent="false">¯</mo> </mover> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=a\mathbf {1} +b\mathbf {i} +c\mathbf {j} +d\mathbf {k} =\alpha +\beta \mathbf {j} \leftrightarrow {\begin{bmatrix}\alpha &\beta \\-{\overline {\beta }}&{\overline {\alpha }}\end{bmatrix}}=U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0abb62269bdfd9ffdc6fe1b0b19809ae520fd3e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.913ex; height:6.509ex;" alt="{\displaystyle q=a\mathbf {1} +b\mathbf {i} +c\mathbf {j} +d\mathbf {k} =\alpha +\beta \mathbf {j} \leftrightarrow {\begin{bmatrix}\alpha &\beta \\-{\overline {\beta }}&{\overline {\alpha }}\end{bmatrix}}=U}"> restricted to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a^{2}+b^{2}+c^{2}+d^{2}=|\alpha |^{2}+|\beta |^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a^{2}+b^{2}+c^{2}+d^{2}=|\alpha |^{2}+|\beta |^{2}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2c98f0ac4153f86226e446fd89db4987b49305" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.906ex; height:3.343ex;" alt="{\textstyle a^{2}+b^{2}+c^{2}+d^{2}=|\alpha |^{2}+|\beta |^{2}=1}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle q\in \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>q</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle q\in \mathbb {H} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e618d5daa1be1c10782e6ba5c231e71041b16397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.718ex; height:2.509ex;" alt="{\textstyle q\in \mathbb {H} }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a,b,c,d\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a,b,c,d\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/771c12a66ce5c339701694d32ee8a1b556b2c81b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.071ex; height:2.509ex;" alt="{\textstyle a,b,c,d\in \mathbb {R} }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle U\in \operatorname {SU} (2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>U</mi> <mo>∈</mo> <mi>SU</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle U\in \operatorname {SU} (2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca7ea747ca20bede37094d1c2032e4444dc5a0a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.631ex; height:2.843ex;" alt="{\textstyle U\in \operatorname {SU} (2)}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =a+bi\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =a+bi\in \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bf4b2b879c14129fbfc446d06f3a6610c9871d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.975ex; height:2.343ex;" alt="{\displaystyle \alpha =a+bi\in \mathbb {C} }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =c+di\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =c+di\in \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac599d8c272460331eaf2bb2f4975c446b22eff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.815ex; height:2.509ex;" alt="{\displaystyle \beta =c+di\in \mathbb {C} }">. Let us now identify <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> with the span of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {i} ,\mathbf {j} ,\mathbf {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {i} ,\mathbf {j} ,\mathbf {k} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d0ff20e01dd78f8f2149bcd2193013bd4aa8035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.037ex; height:2.509ex;" alt="{\displaystyle \mathbf {i} ,\mathbf {j} ,\mathbf {k} }">. One can then verify that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> is in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> is a unit quaternion, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle qvq^{-1}\in \mathbb {R} ^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mi>v</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle qvq^{-1}\in \mathbb {R} ^{3}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8c9e28a565e979fc227ea796af35b5e9493d2ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.829ex; height:3.009ex;" alt="{\displaystyle qvq^{-1}\in \mathbb {R} ^{3}.}"> Furthermore, the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\mapsto qvq^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">↦</mo> <mi>q</mi> <mi>v</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\mapsto qvq^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b4be07ce2ccd2a07afd7de386589bc28046c3fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.351ex; height:3.009ex;" alt="{\displaystyle v\mapsto qvq^{-1}}"> is a rotation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b00b2b4fd27c2cbffa02df568472f77b194a6db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}.}"> Moreover, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-q)v(-q)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-q)v(-q)^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd8bca5eeb2c21ffdcf5d010927c01d700693997" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.834ex; height:3.176ex;" alt="{\displaystyle (-q)v(-q)^{-1}}"> is the same as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle qvq^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mi>v</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle qvq^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6321214bc52695b7e410628dfa92da50eee04ea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.609ex; height:3.009ex;" alt="{\displaystyle qvq^{-1}}">. This means that there is a 2:1 homomorphism from quaternions of unit norm to the 3D rotation group SO(3). One can work this homomorphism out explicitly: the unit quaternion, q, with <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}q&=w+x\mathbf {i} +y\mathbf {j} +z\mathbf {k} ,\\1&=w^{2}+x^{2}+y^{2}+z^{2},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>q</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>w</mi> <mo>+</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>+</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}q&=w+x\mathbf {i} +y\mathbf {j} +z\mathbf {k} ,\\1&=w^{2}+x^{2}+y^{2}+z^{2},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369445259e51e13f2c8faa1224933f37f6bbdfe7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.642ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}q&=w+x\mathbf {i} +y\mathbf {j} +z\mathbf {k} ,\\1&=w^{2}+x^{2}+y^{2}+z^{2},\end{aligned}}}"> is mapped to the rotation matrix <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q={\begin{bmatrix}1-2y^{2}-2z^{2}&2xy-2zw&2xz+2yw\\2xy+2zw&1-2x^{2}-2z^{2}&2yz-2xw\\2xz-2yw&2yz+2xw&1-2x^{2}-2y^{2}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mn>2</mn> <mi>z</mi> <mi>w</mi> </mtd> <mtd> <mn>2</mn> <mi>x</mi> <mi>z</mi> <mo>+</mo> <mn>2</mn> <mi>y</mi> <mi>w</mi> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>2</mn> <mi>z</mi> <mi>w</mi> </mtd> <mtd> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>2</mn> <mi>y</mi> <mi>z</mi> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mi>w</mi> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> <mi>z</mi> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mi>w</mi> </mtd> <mtd> <mn>2</mn> <mi>y</mi> <mi>z</mi> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>w</mi> </mtd> <mtd> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q={\begin{bmatrix}1-2y^{2}-2z^{2}&2xy-2zw&2xz+2yw\\2xy+2zw&1-2x^{2}-2z^{2}&2yz-2xw\\2xz-2yw&2yz+2xw&1-2x^{2}-2y^{2}\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e462d99341f03df03fe2faabecb908cd60a21860" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.973ex; margin-bottom: -0.198ex; width:55.072ex; height:9.509ex;" alt="{\displaystyle Q={\begin{bmatrix}1-2y^{2}-2z^{2}&2xy-2zw&2xz+2yw\\2xy+2zw&1-2x^{2}-2z^{2}&2yz-2xw\\2xz-2yw&2yz+2xw&1-2x^{2}-2y^{2}\end{bmatrix}}.}"> This is a rotation around the vector (x, y, z) by an angle 2θ, where cos θ = w and |sin θ| = ‖(x, y, z)‖. The proper sign for sin θ is implied, once the signs of the axis components are fixed. The 2:1-nature is apparent since both q and −q map to the same Q. <div class="mw-heading mw-heading3"><h3 id="Using_Möbius_transformations">Using Möbius transformations</h3>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=9" title="Edit section: Using Möbius transformations">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Stereoprojnegone.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Stereoprojnegone.svg/300px-Stereoprojnegone.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Stereoprojnegone.svg/450px-Stereoprojnegone.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Stereoprojnegone.svg/600px-Stereoprojnegone.svg.png 2x" data-file-width="232" data-file-height="232" /></a><figcaption>Stereographic projection from the sphere of radius <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠1/2⁠ from the north pole (x, y, z) = (0, 0, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠) onto the plane M given by z = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ coordinatized by (ξ, η), here shown in cross section.</figcaption></figure> The general reference for this section is <a href="#CITEREFGelfandMinlosShapiro1963">Gelfand, Minlos & Shapiro (1963)</a>. The points P on the sphere <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {S} =\left\{(x,y,z)\in \mathbb {R} ^{3}:x^{2}+y^{2}+z^{2}={\frac {1}{4}}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>:</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} =\left\{(x,y,z)\in \mathbb {R} ^{3}:x^{2}+y^{2}+z^{2}={\frac {1}{4}}\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39623077830169fb70dc52dfb7e86bdd4dbc7cb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.552ex; height:6.176ex;" alt="{\displaystyle \mathbf {S} =\left\{(x,y,z)\in \mathbb {R} ^{3}:x^{2}+y^{2}+z^{2}={\frac {1}{4}}\right\}}"></dd></dl> can, barring the north pole N, be put into one-to-one bijection with points S(P) = P' on the plane M defined by z = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, see figure. The map S is called <a href="/wiki/Stereographic_projection" title="Stereographic projection">stereographic projection</a>. Let the coordinates on M be (ξ, η). The line L passing through N and P can be parametrized as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(t)=N+t(N-P)=\left(0,0,{\frac {1}{2}}\right)+t\left(\left(0,0,{\frac {1}{2}}\right)-(x,y,z)\right),\quad t\in \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>N</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(t)=N+t(N-P)=\left(0,0,{\frac {1}{2}}\right)+t\left(\left(0,0,{\frac {1}{2}}\right)-(x,y,z)\right),\quad t\in \mathbb {R} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f744cc4126c0fa13bcf3abfb45bc787698225cb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:71.784ex; height:6.176ex;" alt="{\displaystyle L(t)=N+t(N-P)=\left(0,0,{\frac {1}{2}}\right)+t\left(\left(0,0,{\frac {1}{2}}\right)-(x,y,z)\right),\quad t\in \mathbb {R} .}"></dd></dl> Demanding that the z-coordinate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(t_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(t_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27777fe9cd5e27daa7fc2bcefe88d8b79936db42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.286ex; height:2.843ex;" alt="{\displaystyle L(t_{0})}"> equals −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, one finds <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}={\frac {1}{z-{\frac {1}{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}={\frac {1}{z-{\frac {1}{2}}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc4fab98a5bb129545ef548a3faaa8f13fe286c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:12.062ex; height:6.676ex;" alt="{\displaystyle t_{0}={\frac {1}{z-{\frac {1}{2}}}}.}"></dd></dl> We have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(t_{0})=(\xi ,\eta ,-1/2).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>,</mo> <mi>η</mi> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(t_{0})=(\xi ,\eta ,-1/2).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/153133882839c421d9d9d2fef1ba344efc40a1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.403ex; height:2.843ex;" alt="{\displaystyle L(t_{0})=(\xi ,\eta ,-1/2).}"> Hence the map <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}S:\mathbf {S} \to M\\P=(x,y,z)\longmapsto P'=(\xi ,\eta )=\left({\frac {x}{{\frac {1}{2}}-z}},{\frac {y}{{\frac {1}{2}}-z}}\right)\equiv \zeta =\xi +i\eta \end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>S</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo stretchy="false">→</mo> <mi>M</mi> </mtd> </mtr> <mtr> <mtd> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⟼</mo> <msup> <mi>P</mi> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>,</mo> <mi>η</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>≡</mo> <mi>ζ</mi> <mo>=</mo> <mi>ξ</mi> <mo>+</mo> <mi>i</mi> <mi>η</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}S:\mathbf {S} \to M\\P=(x,y,z)\longmapsto P'=(\xi ,\eta )=\left({\frac {x}{{\frac {1}{2}}-z}},{\frac {y}{{\frac {1}{2}}-z}}\right)\equiv \zeta =\xi +i\eta \end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22160cbbded3475d2cf441400470fead286e3802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:60.578ex; height:10.176ex;" alt="{\displaystyle {\begin{cases}S:\mathbf {S} \to M\\P=(x,y,z)\longmapsto P'=(\xi ,\eta )=\left({\frac {x}{{\frac {1}{2}}-z}},{\frac {y}{{\frac {1}{2}}-z}}\right)\equiv \zeta =\xi +i\eta \end{cases}}}"></dd></dl> where, for later convenience, the plane M is identified with the complex plane <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4d5d3ec97eee8b915d3b14d3fb38579ee639d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} .}"> For the inverse, write L as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=N+s(P'-N)=\left(0,0,{\frac {1}{2}}\right)+s\left(\left(\xi ,\eta ,-{\frac {1}{2}}\right)-\left(0,0,{\frac {1}{2}}\right)\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>N</mi> <mo>+</mo> <mi>s</mi> <mo stretchy="false">(</mo> <msup> <mi>P</mi> <mo>′</mo> </msup> <mo>−</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>s</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>ξ</mi> <mo>,</mo> <mi>η</mi> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=N+s(P'-N)=\left(0,0,{\frac {1}{2}}\right)+s\left(\left(\xi ,\eta ,-{\frac {1}{2}}\right)-\left(0,0,{\frac {1}{2}}\right)\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9270b308a50b4fad11ce0746f01d9c31b4cc42f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:65.728ex; height:6.176ex;" alt="{\displaystyle L=N+s(P'-N)=\left(0,0,{\frac {1}{2}}\right)+s\left(\left(\xi ,\eta ,-{\frac {1}{2}}\right)-\left(0,0,{\frac {1}{2}}\right)\right),}"></dd></dl> and demand x2 + y2 + z2 = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠ to find s = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/1 + ξ2 + η2⁠ and thus <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}S^{-1}:M\to \mathbf {S} \\P'=(\xi ,\eta )\longmapsto P=(x,y,z)=\left({\frac {\xi }{1+\xi ^{2}+\eta ^{2}}},{\frac {\eta }{1+\xi ^{2}+\eta ^{2}}},{\frac {-1+\xi ^{2}+\eta ^{2}}{2+2\xi ^{2}+2\eta ^{2}}}\right)\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>P</mi> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>,</mo> <mi>η</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⟼</mo> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ξ</mi> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>η</mi> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo>+</mo> <mn>2</mn> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}S^{-1}:M\to \mathbf {S} \\P'=(\xi ,\eta )\longmapsto P=(x,y,z)=\left({\frac {\xi }{1+\xi ^{2}+\eta ^{2}}},{\frac {\eta }{1+\xi ^{2}+\eta ^{2}}},{\frac {-1+\xi ^{2}+\eta ^{2}}{2+2\xi ^{2}+2\eta ^{2}}}\right)\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3a4e15da4658b0d2df30b0b2447766337cb467b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:63.404ex; height:8.843ex;" alt="{\displaystyle {\begin{cases}S^{-1}:M\to \mathbf {S} \\P'=(\xi ,\eta )\longmapsto P=(x,y,z)=\left({\frac {\xi }{1+\xi ^{2}+\eta ^{2}}},{\frac {\eta }{1+\xi ^{2}+\eta ^{2}}},{\frac {-1+\xi ^{2}+\eta ^{2}}{2+2\xi ^{2}+2\eta ^{2}}}\right)\end{cases}}}"></dd></dl> If g ∈ SO(3) is a rotation, then it will take points on S to points on S by its standard action Πs(g) on the embedding space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b00b2b4fd27c2cbffa02df568472f77b194a6db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}.}"> By composing this action with S one obtains a transformation S ∘ Πs(g) ∘ S−1 of M, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta =P'\longmapsto P\longmapsto \Pi _{s}(g)P=gP\longmapsto S(gP)\equiv \Pi _{u}(g)\zeta =\zeta '.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>=</mo> <msup> <mi>P</mi> <mo>′</mo> </msup> <mo stretchy="false">⟼</mo> <mi>P</mi> <mo stretchy="false">⟼</mo> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo>=</mo> <mi>g</mi> <mi>P</mi> <mo stretchy="false">⟼</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mi>ζ</mi> <mo>=</mo> <msup> <mi>ζ</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta =P'\longmapsto P\longmapsto \Pi _{s}(g)P=gP\longmapsto S(gP)\equiv \Pi _{u}(g)\zeta =\zeta '.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f624854e9dd2fc4536050e9ce136d732bc485cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.863ex; height:3.009ex;" alt="{\displaystyle \zeta =P'\longmapsto P\longmapsto \Pi _{s}(g)P=gP\longmapsto S(gP)\equiv \Pi _{u}(g)\zeta =\zeta '.}"></dd></dl> Thus Πu(g) is a transformation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"> associated to the transformation Πs(g) of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}">. It turns out that g ∈ SO(3) represented in this way by Πu(g) can be expressed as a matrix Πu(g) ∈ SU(2) (where the notation is recycled to use the same name for the matrix as for the transformation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"> it represents). To identify this matrix, consider first a rotation gφ about the z-axis through an angle φ, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x'&=x\cos \phi -y\sin \phi ,\\y'&=x\sin \phi +y\cos \phi ,\\z'&=z.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>−</mo> <mi>y</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>+</mo> <mi>y</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x'&=x\cos \phi -y\sin \phi ,\\y'&=x\sin \phi +y\cos \phi ,\\z'&=z.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7adcf12eabf85bff8a48a92451a69a70bc2f6d2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:22.123ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}x'&=x\cos \phi -y\sin \phi ,\\y'&=x\sin \phi +y\cos \phi ,\\z'&=z.\end{aligned}}}"></dd></dl> Hence <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta '={\frac {x'+iy'}{{\frac {1}{2}}-z'}}={\frac {e^{i\phi }(x+iy)}{{\frac {1}{2}}-z}}=e^{i\phi }\zeta ={\frac {e^{\frac {i\phi }{2}}\zeta +0}{0\zeta +e^{-{\frac {i\phi }{2}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ζ</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>i</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> <mi>ζ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>ϕ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mi>ζ</mi> <mo>+</mo> <mn>0</mn> </mrow> <mrow> <mn>0</mn> <mi>ζ</mi> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>ϕ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta '={\frac {x'+iy'}{{\frac {1}{2}}-z'}}={\frac {e^{i\phi }(x+iy)}{{\frac {1}{2}}-z}}=e^{i\phi }\zeta ={\frac {e^{\frac {i\phi }{2}}\zeta +0}{0\zeta +e^{-{\frac {i\phi }{2}}}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f11eb6c0aebb6af4a7b47b24b68a4e08351f26f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:49.397ex; height:9.176ex;" alt="{\displaystyle \zeta '={\frac {x'+iy'}{{\frac {1}{2}}-z'}}={\frac {e^{i\phi }(x+iy)}{{\frac {1}{2}}-z}}=e^{i\phi }\zeta ={\frac {e^{\frac {i\phi }{2}}\zeta +0}{0\zeta +e^{-{\frac {i\phi }{2}}}}},}"></dd></dl> which, unsurprisingly, is a rotation in the complex plane. In an analogous way, if gθ is a rotation about the x-axis through an angle θ, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w'=e^{i\theta }w,\quad w={\frac {y+iz}{{\frac {1}{2}}-x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>w</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mi>w</mi> <mo>,</mo> <mspace width="1em" /> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>y</mi> <mo>+</mo> <mi>i</mi> <mi>z</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w'=e^{i\theta }w,\quad w={\frac {y+iz}{{\frac {1}{2}}-x}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c2b3dfe5bb9ce68f4bee4f35eb3fa597a885fbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.254ex; height:6.843ex;" alt="{\displaystyle w'=e^{i\theta }w,\quad w={\frac {y+iz}{{\frac {1}{2}}-x}},}"></dd></dl> which, after a little algebra, becomes <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta '={\frac {\cos {\frac {\theta }{2}}\zeta +i\sin {\frac {\theta }{2}}}{i\sin {\frac {\theta }{2}}\zeta +\cos {\frac {\theta }{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ζ</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mi>ζ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mi>ζ</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta '={\frac {\cos {\frac {\theta }{2}}\zeta +i\sin {\frac {\theta }{2}}}{i\sin {\frac {\theta }{2}}\zeta +\cos {\frac {\theta }{2}}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4570940e1529243572cebe39846e607e4c46458" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:21.567ex; height:8.176ex;" alt="{\displaystyle \zeta '={\frac {\cos {\frac {\theta }{2}}\zeta +i\sin {\frac {\theta }{2}}}{i\sin {\frac {\theta }{2}}\zeta +\cos {\frac {\theta }{2}}}}.}"></dd></dl> These two rotations, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\phi },g_{\theta },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\phi },g_{\theta },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4997463b1e060a7fbfaeaa877b1af29b5e471b38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.114ex; height:2.343ex;" alt="{\displaystyle g_{\phi },g_{\theta },}"> thus correspond to <a href="/wiki/Bilinear_transform" title="Bilinear transform">bilinear transforms</a> of R2 ≃ C ≃ M, namely, they are examples of <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a>. A general Möbius transformation is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta '={\frac {\alpha \zeta +\beta }{\gamma \zeta +\delta }},\quad \alpha \delta -\beta \gamma \neq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ζ</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mi>ζ</mi> <mo>+</mo> <mi>β</mi> </mrow> <mrow> <mi>γ</mi> <mi>ζ</mi> <mo>+</mo> <mi>δ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>α</mi> <mi>δ</mi> <mo>−</mo> <mi>β</mi> <mi>γ</mi> <mo>≠</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta '={\frac {\alpha \zeta +\beta }{\gamma \zeta +\delta }},\quad \alpha \delta -\beta \gamma \neq 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93fd5b026af4e5e8c1f07a42aaae41336f935c46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.728ex; height:6.009ex;" alt="{\displaystyle \zeta '={\frac {\alpha \zeta +\beta }{\gamma \zeta +\delta }},\quad \alpha \delta -\beta \gamma \neq 0.}"></dd></dl> The rotations, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\phi },g_{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\phi },g_{\theta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c258a3cb0f4727cc571b971767fbc99b4436e91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.467ex; height:2.343ex;" alt="{\displaystyle g_{\phi },g_{\theta }}"> generate all of SO(3) and the composition rules of the Möbius transformations show that any composition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\phi },g_{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\phi },g_{\theta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c258a3cb0f4727cc571b971767fbc99b4436e91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.467ex; height:2.343ex;" alt="{\displaystyle g_{\phi },g_{\theta }}"> translates to the corresponding composition of Möbius transformations. The Möbius transformations can be represented by matrices <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}},\qquad \alpha \delta -\beta \gamma =1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mi>γ</mi> </mtd> <mtd> <mi>δ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>α</mi> <mi>δ</mi> <mo>−</mo> <mi>β</mi> <mi>γ</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}},\qquad \alpha \delta -\beta \gamma =1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e081f99a19f12cbe9dfe44567a92e06edd587452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.873ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}},\qquad \alpha \delta -\beta \gamma =1,}"></dd></dl> since a common factor of α, β, γ, δ cancels. For the same reason, the matrix is not uniquely defined since multiplication by −I has no effect on either the determinant or the Möbius transformation. The composition law of Möbius transformations follow that of the corresponding matrices. The conclusion is that each Möbius transformation corresponds to two matrices g, −g ∈ SL(2, C). Using this correspondence one may write <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Pi _{u}(g_{\phi })&=\Pi _{u}\left[{\begin{pmatrix}\cos \phi &-\sin \phi &0\\\sin \phi &\cos \phi &0\\0&0&1\end{pmatrix}}\right]=\pm {\begin{pmatrix}e^{i{\frac {\phi }{2}}}&0\\0&e^{-i{\frac {\phi }{2}}}\end{pmatrix}},\\\Pi _{u}(g_{\theta })&=\Pi _{u}\left[{\begin{pmatrix}1&0&0\\0&\cos \theta &-\sin \theta \\0&\sin \theta &\cos \theta \end{pmatrix}}\right]=\pm {\begin{pmatrix}\cos {\frac {\theta }{2}}&i\sin {\frac {\theta }{2}}\\i\sin {\frac {\theta }{2}}&\cos {\frac {\theta }{2}}\end{pmatrix}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Pi _{u}(g_{\phi })&=\Pi _{u}\left[{\begin{pmatrix}\cos \phi &-\sin \phi &0\\\sin \phi &\cos \phi &0\\0&0&1\end{pmatrix}}\right]=\pm {\begin{pmatrix}e^{i{\frac {\phi }{2}}}&0\\0&e^{-i{\frac {\phi }{2}}}\end{pmatrix}},\\\Pi _{u}(g_{\theta })&=\Pi _{u}\left[{\begin{pmatrix}1&0&0\\0&\cos \theta &-\sin \theta \\0&\sin \theta &\cos \theta \end{pmatrix}}\right]=\pm {\begin{pmatrix}\cos {\frac {\theta }{2}}&i\sin {\frac {\theta }{2}}\\i\sin {\frac {\theta }{2}}&\cos {\frac {\theta }{2}}\end{pmatrix}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22565ba61ca4a960754f63a9332628a797792ac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:63.532ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}\Pi _{u}(g_{\phi })&=\Pi _{u}\left[{\begin{pmatrix}\cos \phi &-\sin \phi &0\\\sin \phi &\cos \phi &0\\0&0&1\end{pmatrix}}\right]=\pm {\begin{pmatrix}e^{i{\frac {\phi }{2}}}&0\\0&e^{-i{\frac {\phi }{2}}}\end{pmatrix}},\\\Pi _{u}(g_{\theta })&=\Pi _{u}\left[{\begin{pmatrix}1&0&0\\0&\cos \theta &-\sin \theta \\0&\sin \theta &\cos \theta \end{pmatrix}}\right]=\pm {\begin{pmatrix}\cos {\frac {\theta }{2}}&i\sin {\frac {\theta }{2}}\\i\sin {\frac {\theta }{2}}&\cos {\frac {\theta }{2}}\end{pmatrix}}.\end{aligned}}}"></dd></dl> These matrices are unitary and thus Πu(SO(3)) ⊂ SU(2) ⊂ SL(2, C). In terms of <a href="/wiki/Euler_angles" title="Euler angles">Euler angles</a><a href="#cite_note-6">[nb 1]</a> one finds for a general rotation <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}g(\phi ,\theta ,\psi )=g_{\phi }g_{\theta }g_{\psi }&={\begin{pmatrix}\cos \phi &-\sin \phi &0\\\sin \phi &\cos \phi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \theta &-\sin \theta \\0&\sin \theta &\cos \theta \end{pmatrix}}{\begin{pmatrix}\cos \psi &-\sin \psi &0\\\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}\\&={\begin{pmatrix}\cos \phi \cos \psi -\cos \theta \sin \phi \sin \psi &-\cos \phi \sin \psi -\cos \theta \sin \phi \cos \psi &\sin \phi \sin \theta \\\sin \phi \cos \psi +\cos \theta \cos \phi \sin \psi &-\sin \phi \sin \psi +\cos \theta \cos \phi \cos \psi &-\cos \phi \sin \theta \\\sin \psi \sin \theta &\cos \psi \sin \theta &\cos \theta \end{pmatrix}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>g</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g(\phi ,\theta ,\psi )=g_{\phi }g_{\theta }g_{\psi }&={\begin{pmatrix}\cos \phi &-\sin \phi &0\\\sin \phi &\cos \phi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \theta &-\sin \theta \\0&\sin \theta &\cos \theta \end{pmatrix}}{\begin{pmatrix}\cos \psi &-\sin \psi &0\\\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}\\&={\begin{pmatrix}\cos \phi \cos \psi -\cos \theta \sin \phi \sin \psi &-\cos \phi \sin \psi -\cos \theta \sin \phi \cos \psi &\sin \phi \sin \theta \\\sin \phi \cos \psi +\cos \theta \cos \phi \sin \psi &-\sin \phi \sin \psi +\cos \theta \cos \phi \cos \psi &-\cos \phi \sin \theta \\\sin \psi \sin \theta &\cos \psi \sin \theta &\cos \theta \end{pmatrix}},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d8390d24d69712dd7c475961c67270b072644cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:102.619ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}g(\phi ,\theta ,\psi )=g_{\phi }g_{\theta }g_{\psi }&={\begin{pmatrix}\cos \phi &-\sin \phi &0\\\sin \phi &\cos \phi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \theta &-\sin \theta \\0&\sin \theta &\cos \theta \end{pmatrix}}{\begin{pmatrix}\cos \psi &-\sin \psi &0\\\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}\\&={\begin{pmatrix}\cos \phi \cos \psi -\cos \theta \sin \phi \sin \psi &-\cos \phi \sin \psi -\cos \theta \sin \phi \cos \psi &\sin \phi \sin \theta \\\sin \phi \cos \psi +\cos \theta \cos \phi \sin \psi &-\sin \phi \sin \psi +\cos \theta \cos \phi \cos \psi &-\cos \phi \sin \theta \\\sin \psi \sin \theta &\cos \psi \sin \theta &\cos \theta \end{pmatrix}},\end{aligned}}}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(1)</td></tr></tbody></table> one has<a href="#cite_note-7">[6]</a> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Pi _{u}(g(\phi ,\theta ,\psi ))&=\pm {\begin{pmatrix}e^{i{\frac {\phi }{2}}}&0\\0&e^{-i{\frac {\phi }{2}}}\end{pmatrix}}{\begin{pmatrix}\cos {\frac {\theta }{2}}&i\sin {\frac {\theta }{2}}\\i\sin {\frac {\theta }{2}}&\cos {\frac {\theta }{2}}\end{pmatrix}}{\begin{pmatrix}e^{i{\frac {\psi }{2}}}&0\\0&e^{-i{\frac {\psi }{2}}}\end{pmatrix}}\\&=\pm {\begin{pmatrix}\cos {\frac {\theta }{2}}e^{i{\frac {\phi +\psi }{2}}}&i\sin {\frac {\theta }{2}}e^{i{\frac {\phi -\psi }{2}}}\\i\sin {\frac {\theta }{2}}e^{-i{\frac {\phi -\psi }{2}}}&\cos {\frac {\theta }{2}}e^{-i{\frac {\phi +\psi }{2}}}\end{pmatrix}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ψ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ψ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <mo>+</mo> <mi>ψ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mtd> <mtd> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <mo>−</mo> <mi>ψ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <mo>−</mo> <mi>ψ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <mo>+</mo> <mi>ψ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Pi _{u}(g(\phi ,\theta ,\psi ))&=\pm {\begin{pmatrix}e^{i{\frac {\phi }{2}}}&0\\0&e^{-i{\frac {\phi }{2}}}\end{pmatrix}}{\begin{pmatrix}\cos {\frac {\theta }{2}}&i\sin {\frac {\theta }{2}}\\i\sin {\frac {\theta }{2}}&\cos {\frac {\theta }{2}}\end{pmatrix}}{\begin{pmatrix}e^{i{\frac {\psi }{2}}}&0\\0&e^{-i{\frac {\psi }{2}}}\end{pmatrix}}\\&=\pm {\begin{pmatrix}\cos {\frac {\theta }{2}}e^{i{\frac {\phi +\psi }{2}}}&i\sin {\frac {\theta }{2}}e^{i{\frac {\phi -\psi }{2}}}\\i\sin {\frac {\theta }{2}}e^{-i{\frac {\phi -\psi }{2}}}&\cos {\frac {\theta }{2}}e^{-i{\frac {\phi +\psi }{2}}}\end{pmatrix}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6fc3558a803a13c84ca4ad60b57cd8bd5dfed0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:69.345ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}\Pi _{u}(g(\phi ,\theta ,\psi ))&=\pm {\begin{pmatrix}e^{i{\frac {\phi }{2}}}&0\\0&e^{-i{\frac {\phi }{2}}}\end{pmatrix}}{\begin{pmatrix}\cos {\frac {\theta }{2}}&i\sin {\frac {\theta }{2}}\\i\sin {\frac {\theta }{2}}&\cos {\frac {\theta }{2}}\end{pmatrix}}{\begin{pmatrix}e^{i{\frac {\psi }{2}}}&0\\0&e^{-i{\frac {\psi }{2}}}\end{pmatrix}}\\&=\pm {\begin{pmatrix}\cos {\frac {\theta }{2}}e^{i{\frac {\phi +\psi }{2}}}&i\sin {\frac {\theta }{2}}e^{i{\frac {\phi -\psi }{2}}}\\i\sin {\frac {\theta }{2}}e^{-i{\frac {\phi -\psi }{2}}}&\cos {\frac {\theta }{2}}e^{-i{\frac {\phi +\psi }{2}}}\end{pmatrix}}.\end{aligned}}}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(2)</td></tr></tbody></table> For the converse, consider a general matrix <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \Pi _{u}(g_{\alpha ,\beta })=\pm {\begin{pmatrix}\alpha &\beta \\-{\overline {\beta }}&{\overline {\alpha }}\end{pmatrix}}\in \operatorname {SU} (2).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo accent="false">¯</mo> </mover> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>∈</mo> <mi>SU</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \Pi _{u}(g_{\alpha ,\beta })=\pm {\begin{pmatrix}\alpha &\beta \\-{\overline {\beta }}&{\overline {\alpha }}\end{pmatrix}}\in \operatorname {SU} (2).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30578f2b7ed280a30053fcf681b8d9eb83de2ff4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.09ex; height:6.509ex;" alt="{\displaystyle \pm \Pi _{u}(g_{\alpha ,\beta })=\pm {\begin{pmatrix}\alpha &\beta \\-{\overline {\beta }}&{\overline {\alpha }}\end{pmatrix}}\in \operatorname {SU} (2).}"></dd></dl> Make the substitutions <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\cos {\frac {\theta }{2}}&=|\alpha |,&\sin {\frac {\theta }{2}}&=|\beta |,&(0\leq \theta \leq \pi ),\\{\frac {\phi +\psi }{2}}&=\arg \alpha ,&{\frac {\psi -\phi }{2}}&=\arg \beta .&\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>0</mn> <mo>≤</mo> <mi>θ</mi> <mo>≤</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <mo>+</mo> <mi>ψ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <mi>α</mi> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ψ</mi> <mo>−</mo> <mi>ϕ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <mi>β</mi> <mo>.</mo> </mtd> <mtd /> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos {\frac {\theta }{2}}&=|\alpha |,&\sin {\frac {\theta }{2}}&=|\beta |,&(0\leq \theta \leq \pi ),\\{\frac {\phi +\psi }{2}}&=\arg \alpha ,&{\frac {\psi -\phi }{2}}&=\arg \beta .&\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db80f4c3e21dde038386915f923871017e310c2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.619ex; margin-bottom: -0.219ex; width:52.988ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\cos {\frac {\theta }{2}}&=|\alpha |,&\sin {\frac {\theta }{2}}&=|\beta |,&(0\leq \theta \leq \pi ),\\{\frac {\phi +\psi }{2}}&=\arg \alpha ,&{\frac {\psi -\phi }{2}}&=\arg \beta .&\end{aligned}}}"></dd></dl> With the substitutions, Π(gα, β) assumes the form of the right hand side (<a href="/wiki/Right-hand_side" class="mw-redirect" title="Right-hand side">RHS</a>) of (<a href="#math_2">2</a>), which corresponds under Πu to a matrix on the form of the RHS of (<a href="#math_1">1</a>) with the same φ, θ, ψ. In terms of the complex parameters α, β, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\alpha ,\beta }={\begin{pmatrix}{\frac {1}{2}}\left(\alpha ^{2}-\beta ^{2}+{\overline {\alpha ^{2}}}-{\overline {\beta ^{2}}}\right)&{\frac {i}{2}}\left(-\alpha ^{2}-\beta ^{2}+{\overline {\alpha ^{2}}}+{\overline {\beta ^{2}}}\right)&-\alpha \beta -{\overline {\alpha }}{\overline {\beta }}\\{\frac {i}{2}}\left(\alpha ^{2}-\beta ^{2}-{\overline {\alpha ^{2}}}+{\overline {\beta ^{2}}}\right)&{\frac {1}{2}}\left(\alpha ^{2}+\beta ^{2}+{\overline {\alpha ^{2}}}+{\overline {\beta ^{2}}}\right)&-i\left(+\alpha \beta -{\overline {\alpha }}{\overline {\beta }}\right)\\\alpha {\overline {\beta }}+{\overline {\alpha }}\beta &i\left(-\alpha {\overline {\beta }}+{\overline {\alpha }}\beta \right)&\alpha {\overline {\alpha }}-\beta {\overline {\beta }}\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mo>−</mo> <mi>α</mi> <mi>β</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo accent="false">¯</mo> </mover> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mo>−</mo> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mo>+</mo> <mi>α</mi> <mi>β</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo accent="false">¯</mo> </mover> </mrow> <mi>β</mi> </mtd> <mtd> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo accent="false">¯</mo> </mover> </mrow> <mi>β</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo accent="false">¯</mo> </mover> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\alpha ,\beta }={\begin{pmatrix}{\frac {1}{2}}\left(\alpha ^{2}-\beta ^{2}+{\overline {\alpha ^{2}}}-{\overline {\beta ^{2}}}\right)&{\frac {i}{2}}\left(-\alpha ^{2}-\beta ^{2}+{\overline {\alpha ^{2}}}+{\overline {\beta ^{2}}}\right)&-\alpha \beta -{\overline {\alpha }}{\overline {\beta }}\\{\frac {i}{2}}\left(\alpha ^{2}-\beta ^{2}-{\overline {\alpha ^{2}}}+{\overline {\beta ^{2}}}\right)&{\frac {1}{2}}\left(\alpha ^{2}+\beta ^{2}+{\overline {\alpha ^{2}}}+{\overline {\beta ^{2}}}\right)&-i\left(+\alpha \beta -{\overline {\alpha }}{\overline {\beta }}\right)\\\alpha {\overline {\beta }}+{\overline {\alpha }}\beta &i\left(-\alpha {\overline {\beta }}+{\overline {\alpha }}\beta \right)&\alpha {\overline {\alpha }}-\beta {\overline {\beta }}\end{pmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea2ab88ff2215e9e7822cea22f158232ef0550f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:81.987ex; height:14.843ex;" alt="{\displaystyle g_{\alpha ,\beta }={\begin{pmatrix}{\frac {1}{2}}\left(\alpha ^{2}-\beta ^{2}+{\overline {\alpha ^{2}}}-{\overline {\beta ^{2}}}\right)&{\frac {i}{2}}\left(-\alpha ^{2}-\beta ^{2}+{\overline {\alpha ^{2}}}+{\overline {\beta ^{2}}}\right)&-\alpha \beta -{\overline {\alpha }}{\overline {\beta }}\\{\frac {i}{2}}\left(\alpha ^{2}-\beta ^{2}-{\overline {\alpha ^{2}}}+{\overline {\beta ^{2}}}\right)&{\frac {1}{2}}\left(\alpha ^{2}+\beta ^{2}+{\overline {\alpha ^{2}}}+{\overline {\beta ^{2}}}\right)&-i\left(+\alpha \beta -{\overline {\alpha }}{\overline {\beta }}\right)\\\alpha {\overline {\beta }}+{\overline {\alpha }}\beta &i\left(-\alpha {\overline {\beta }}+{\overline {\alpha }}\beta \right)&\alpha {\overline {\alpha }}-\beta {\overline {\beta }}\end{pmatrix}}.}"></dd></dl> To verify this, substitute for α. β the elements of the matrix on the RHS of (<a href="#math_2">2</a>). After some manipulation, the matrix assumes the form of the RHS of (<a href="#math_1">1</a>). It is clear from the explicit form in terms of Euler angles that the map <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}p:\operatorname {SU} (2)\to \operatorname {SO} (3)\\\pm \Pi _{u}(g_{\alpha \beta })\mapsto g_{\alpha \beta }\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>p</mi> <mo>:</mo> <mi>SU</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>±</mo> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}p:\operatorname {SU} (2)\to \operatorname {SO} (3)\\\pm \Pi _{u}(g_{\alpha \beta })\mapsto g_{\alpha \beta }\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a531860bd071aa03a8b6fddd843382d7efc4703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.295ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}p:\operatorname {SU} (2)\to \operatorname {SO} (3)\\\pm \Pi _{u}(g_{\alpha \beta })\mapsto g_{\alpha \beta }\end{cases}}}"></dd></dl> just described is a smooth, 2:1 and surjective <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a>. It is hence an explicit description of the <a href="/wiki/Universal_covering_space" class="mw-redirect" title="Universal covering space">universal covering space</a> of SO(3) from the <a href="/wiki/Universal_covering_group" class="mw-redirect" title="Universal covering group">universal covering group</a> SU(2). <div class="mw-heading mw-heading2"><h2 id="Lie_algebra">Lie algebra</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=10" title="Edit section: Lie algebra">edit</a>]</div> Associated with every <a href="/wiki/Lie_group" title="Lie group">Lie group</a> is its <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a>, a linear space of the same dimension as the Lie group, closed under a bilinear alternating product called the <a href="/wiki/Lie_bracket" class="mw-redirect" title="Lie bracket">Lie bracket</a>. The Lie algebra of SO(3) is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4f1d3d3bf3da64b92af1a1018ce00545808b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.179ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(3)}"> and consists of all <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">skew-symmetric</a> 3 × 3 matrices.<a href="#cite_note-8">[7]</a> This may be seen by differentiating the <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonality condition</a>, ATA = I, A ∈ SO(3).<a href="#cite_note-9">[nb 2]</a> The Lie bracket of two elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4f1d3d3bf3da64b92af1a1018ce00545808b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.179ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(3)}"> is, as for the Lie algebra of every matrix group, given by the matrix <a href="/wiki/Commutator" title="Commutator">commutator</a>, [A1, A2] = A1A2 − A2A1, which is again a skew-symmetric matrix. The Lie algebra bracket captures the essence of the Lie group product in a sense made precise by the <a href="/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula" title="Baker–Campbell–Hausdorff formula">Baker–Campbell–Hausdorff formula</a>. The elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4f1d3d3bf3da64b92af1a1018ce00545808b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.179ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(3)}"> are the "infinitesimal generators" of rotations, i.e., they are the elements of the <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a> of the manifold SO(3) at the identity element. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(\phi ,{\boldsymbol {n}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">n</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(\phi ,{\boldsymbol {n}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9099b4e9721d58ed9554336e0041496850706973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.65ex; height:2.843ex;" alt="{\displaystyle R(\phi ,{\boldsymbol {n}})}"> denotes a counterclockwise rotation with angle φ about the axis specified by the unit vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">n</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {n}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/336c0aeeee838a26734050e07d045b040a8bfed2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.304ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {n}},}"> then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall {\boldsymbol {u}}\in \mathbb {R} ^{3}:\qquad \left.{\frac {\operatorname {d} }{\operatorname {d} \phi }}\right|_{\phi =0}R(\phi ,{\boldsymbol {n}}){\boldsymbol {u}}={\boldsymbol {n}}\times {\boldsymbol {u}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>:</mo> <mspace width="2em" /> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">d</mi> <mrow> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mi>R</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">n</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">n</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall {\boldsymbol {u}}\in \mathbb {R} ^{3}:\qquad \left.{\frac {\operatorname {d} }{\operatorname {d} \phi }}\right|_{\phi =0}R(\phi ,{\boldsymbol {n}}){\boldsymbol {u}}={\boldsymbol {n}}\times {\boldsymbol {u}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1db89b93c6e6c1a577734788aca73da24ebdbb88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:41.95ex; height:6.343ex;" alt="{\displaystyle \forall {\boldsymbol {u}}\in \mathbb {R} ^{3}:\qquad \left.{\frac {\operatorname {d} }{\operatorname {d} \phi }}\right|_{\phi =0}R(\phi ,{\boldsymbol {n}}){\boldsymbol {u}}={\boldsymbol {n}}\times {\boldsymbol {u}}.}"></dd></dl> This can be used to show that the Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4f1d3d3bf3da64b92af1a1018ce00545808b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.179ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(3)}"> (with commutator) is isomorphic to the Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> (with <a href="/wiki/Cross_product" title="Cross product">cross product</a>). Under this isomorphism, an <a href="/wiki/Axis%E2%80%93angle_representation#Rotation_vector" title="Axis–angle representation">Euler vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}\in \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}\in \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a91a22435f9cba0a4ff8cd335d43d54991be8717" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.242ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {\omega }}\in \mathbb {R} ^{3}}"> corresponds to the linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {\boldsymbol {\omega }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ω</mi> <mo>~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widetilde {\boldsymbol {\omega }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982e8b8c2d4d591af4059190bcd8ee062b115d70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.021ex; margin-bottom: -0.317ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\widetilde {\boldsymbol {\omega }}}}"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {\boldsymbol {\omega }}}({\boldsymbol {u}})={\boldsymbol {\omega }}\times {\boldsymbol {u}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ω</mi> <mo>~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widetilde {\boldsymbol {\omega }}}({\boldsymbol {u}})={\boldsymbol {\omega }}\times {\boldsymbol {u}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e16ab14ded114a3e77f788d972116546edcca00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.898ex; height:2.843ex;" alt="{\displaystyle {\widetilde {\boldsymbol {\omega }}}({\boldsymbol {u}})={\boldsymbol {\omega }}\times {\boldsymbol {u}}.}"> In more detail, most often a suitable basis for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4f1d3d3bf3da64b92af1a1018ce00545808b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.179ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(3)}"> as a 3-dimensional vector space is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {L}}_{x}={\begin{bmatrix}0&0&0\\0&0&-1\\0&1&0\end{bmatrix}},\quad {\boldsymbol {L}}_{y}={\begin{bmatrix}0&0&1\\0&0&0\\-1&0&0\end{bmatrix}},\quad {\boldsymbol {L}}_{z}={\begin{bmatrix}0&-1&0\\1&0&0\\0&0&0\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {L}}_{x}={\begin{bmatrix}0&0&0\\0&0&-1\\0&1&0\end{bmatrix}},\quad {\boldsymbol {L}}_{y}={\begin{bmatrix}0&0&1\\0&0&0\\-1&0&0\end{bmatrix}},\quad {\boldsymbol {L}}_{z}={\begin{bmatrix}0&-1&0\\1&0&0\\0&0&0\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eb7f297e9859218d3edc956e79c948d0980c5b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:66.528ex; height:9.176ex;" alt="{\displaystyle {\boldsymbol {L}}_{x}={\begin{bmatrix}0&0&0\\0&0&-1\\0&1&0\end{bmatrix}},\quad {\boldsymbol {L}}_{y}={\begin{bmatrix}0&0&1\\0&0&0\\-1&0&0\end{bmatrix}},\quad {\boldsymbol {L}}_{z}={\begin{bmatrix}0&-1&0\\1&0&0\\0&0&0\end{bmatrix}}.}"></dd></dl> The <a href="/wiki/Commutation_relation" class="mw-redirect" title="Commutation relation">commutation relations</a> of these basis elements are, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\boldsymbol {L}}_{x},{\boldsymbol {L}}_{y}]={\boldsymbol {L}}_{z},\quad [{\boldsymbol {L}}_{z},{\boldsymbol {L}}_{x}]={\boldsymbol {L}}_{y},\quad [{\boldsymbol {L}}_{y},{\boldsymbol {L}}_{z}]={\boldsymbol {L}}_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\boldsymbol {L}}_{x},{\boldsymbol {L}}_{y}]={\boldsymbol {L}}_{z},\quad [{\boldsymbol {L}}_{z},{\boldsymbol {L}}_{x}]={\boldsymbol {L}}_{y},\quad [{\boldsymbol {L}}_{y},{\boldsymbol {L}}_{z}]={\boldsymbol {L}}_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21b43a3a337e8450064d485be4433ce497ddd0b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.475ex; height:3.009ex;" alt="{\displaystyle [{\boldsymbol {L}}_{x},{\boldsymbol {L}}_{y}]={\boldsymbol {L}}_{z},\quad [{\boldsymbol {L}}_{z},{\boldsymbol {L}}_{x}]={\boldsymbol {L}}_{y},\quad [{\boldsymbol {L}}_{y},{\boldsymbol {L}}_{z}]={\boldsymbol {L}}_{x}}"></dd></dl> which agree with the relations of the three <a href="/wiki/Standard_basis" title="Standard basis">standard unit vectors</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> under the cross product. As announced above, one can identify any matrix in this Lie algebra with an Euler vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}=(x,y,z)\in \mathbb {R} ^{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}=(x,y,z)\in \mathbb {R} ^{3},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c46b371dc7404f2cfbcf67f8819dc42775a8799e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.438ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {\omega }}=(x,y,z)\in \mathbb {R} ^{3},}"><a href="#cite_note-10">[8]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\boldsymbol {\omega }}}={\boldsymbol {\omega }}\cdot {\boldsymbol {L}}=x{\boldsymbol {L}}_{x}+y{\boldsymbol {L}}_{y}+z{\boldsymbol {L}}_{z}={\begin{bmatrix}0&-z&y\\z&0&-x\\-y&x&0\end{bmatrix}}\in {\mathfrak {so}}(3).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ω</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mo>=</mo> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>z</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>z</mi> </mtd> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>y</mi> </mtd> <mtd> <mi>x</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\boldsymbol {\omega }}}={\boldsymbol {\omega }}\cdot {\boldsymbol {L}}=x{\boldsymbol {L}}_{x}+y{\boldsymbol {L}}_{y}+z{\boldsymbol {L}}_{z}={\begin{bmatrix}0&-z&y\\z&0&-x\\-y&x&0\end{bmatrix}}\in {\mathfrak {so}}(3).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c711ca67c4c13799bf6db0f14eb9ab929939edaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:59.939ex; height:9.509ex;" alt="{\displaystyle {\widehat {\boldsymbol {\omega }}}={\boldsymbol {\omega }}\cdot {\boldsymbol {L}}=x{\boldsymbol {L}}_{x}+y{\boldsymbol {L}}_{y}+z{\boldsymbol {L}}_{z}={\begin{bmatrix}0&-z&y\\z&0&-x\\-y&x&0\end{bmatrix}}\in {\mathfrak {so}}(3).}"></dd></dl> This identification is sometimes called the hat-map.<a href="#cite_note-Engø_2001-11">[9]</a> Under this identification, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4f1d3d3bf3da64b92af1a1018ce00545808b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.179ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(3)}"> bracket corresponds in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> to the <a href="/wiki/Cross_product" title="Cross product">cross product</a>, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\widehat {\boldsymbol {u}}},{\widehat {\boldsymbol {v}}}\right]={\widehat {{\boldsymbol {u}}\times {\boldsymbol {v}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">u</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">v</mi> <mo>^</mo> </mover> </mrow> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\widehat {\boldsymbol {u}}},{\widehat {\boldsymbol {v}}}\right]={\widehat {{\boldsymbol {u}}\times {\boldsymbol {v}}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88bb32c38386e402116e3386f7600d72b0d95988" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.776ex; height:3.176ex;" alt="{\displaystyle \left[{\widehat {\boldsymbol {u}}},{\widehat {\boldsymbol {v}}}\right]={\widehat {{\boldsymbol {u}}\times {\boldsymbol {v}}}}.}"></dd></dl> The matrix identified with a vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {u}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d60e374e33b2c1d75888c0e8759f9e770e718f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {u}}}"> has the property that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\boldsymbol {u}}}{\boldsymbol {v}}={\boldsymbol {u}}\times {\boldsymbol {v}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">u</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\boldsymbol {u}}}{\boldsymbol {v}}={\boldsymbol {u}}\times {\boldsymbol {v}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8efe267986fc12390d7901e2eec8ae7e2011950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.388ex; height:2.676ex;" alt="{\displaystyle {\widehat {\boldsymbol {u}}}{\boldsymbol {v}}={\boldsymbol {u}}\times {\boldsymbol {v}},}"></dd></dl> where the left-hand side we have ordinary matrix multiplication. This implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {u}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d60e374e33b2c1d75888c0e8759f9e770e718f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {u}}}"> is in the <a href="/wiki/Null_space" class="mw-redirect" title="Null space">null space</a> of the skew-symmetric matrix with which it is identified, because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {u}}\times {\boldsymbol {u}}={\boldsymbol {0}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {u}}\times {\boldsymbol {u}}={\boldsymbol {0}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df60f2b23bdfa2af5e16677db7feaf1b99287e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.088ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {u}}\times {\boldsymbol {u}}={\boldsymbol {0}}.}"> <div class="mw-heading mw-heading3"><h3 id="A_note_on_Lie_algebras">A note on Lie algebras</h3>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=11" title="Edit section: A note on Lie algebras">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">Angular momentum operator</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Representation_theory_of_SU(2)" title="Representation theory of SU(2)">Representation theory of SU(2)</a> and <a href="/wiki/Jordan_map" title="Jordan map">Jordan map</a></div> In <a href="/wiki/Lie_algebra_representation" title="Lie algebra representation">Lie algebra representations</a>, the group SO(3) is compact and simple of rank 1, and so it has a single independent <a href="/wiki/Casimir_element" title="Casimir element">Casimir element</a>, a quadratic invariant function of the three generators which commutes with all of them. The Killing form for the rotation group is just the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>, and so this Casimir invariant is simply the sum of the squares of the generators, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {J}}_{x},{\boldsymbol {J}}_{y},{\boldsymbol {J}}_{z},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {J}}_{x},{\boldsymbol {J}}_{y},{\boldsymbol {J}}_{z},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a70f7cc60947545d505aa58dc0d48603d5240abc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.338ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {J}}_{x},{\boldsymbol {J}}_{y},{\boldsymbol {J}}_{z},}"> of the algebra <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\boldsymbol {J}}_{x},{\boldsymbol {J}}_{y}]={\boldsymbol {J}}_{z},\quad [{\boldsymbol {J}}_{z},{\boldsymbol {J}}_{x}]={\boldsymbol {J}}_{y},\quad [{\boldsymbol {J}}_{y},{\boldsymbol {J}}_{z}]={\boldsymbol {J}}_{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\boldsymbol {J}}_{x},{\boldsymbol {J}}_{y}]={\boldsymbol {J}}_{z},\quad [{\boldsymbol {J}}_{z},{\boldsymbol {J}}_{x}]={\boldsymbol {J}}_{y},\quad [{\boldsymbol {J}}_{y},{\boldsymbol {J}}_{z}]={\boldsymbol {J}}_{x}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb35b6ad6e2c39742ca6423fac8bf75ad4a2c9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.509ex; height:3.009ex;" alt="{\displaystyle [{\boldsymbol {J}}_{x},{\boldsymbol {J}}_{y}]={\boldsymbol {J}}_{z},\quad [{\boldsymbol {J}}_{z},{\boldsymbol {J}}_{x}]={\boldsymbol {J}}_{y},\quad [{\boldsymbol {J}}_{y},{\boldsymbol {J}}_{z}]={\boldsymbol {J}}_{x}.}"></dd></dl> That is, the Casimir invariant is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {J}}^{2}\equiv {\boldsymbol {J}}\cdot {\boldsymbol {J}}={\boldsymbol {J}}_{x}^{2}+{\boldsymbol {J}}_{y}^{2}+{\boldsymbol {J}}_{z}^{2}\propto {\boldsymbol {I}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">I</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {J}}^{2}\equiv {\boldsymbol {J}}\cdot {\boldsymbol {J}}={\boldsymbol {J}}_{x}^{2}+{\boldsymbol {J}}_{y}^{2}+{\boldsymbol {J}}_{z}^{2}\propto {\boldsymbol {I}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dec64044c017e6a267c63737f0bc1d4a38042dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.705ex; height:3.509ex;" alt="{\displaystyle {\boldsymbol {J}}^{2}\equiv {\boldsymbol {J}}\cdot {\boldsymbol {J}}={\boldsymbol {J}}_{x}^{2}+{\boldsymbol {J}}_{y}^{2}+{\boldsymbol {J}}_{z}^{2}\propto {\boldsymbol {I}}.}"></dd></dl> For unitary irreducible <a href="/wiki/Lie_algebra_representation" title="Lie algebra representation">representations</a> Dj, the eigenvalues of this invariant are real and discrete, and characterize each representation, which is finite dimensional, of dimensionality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2j+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2j+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0a258cc33d54429124aedc154d9b3a968c4d99b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.123ex; height:2.509ex;" alt="{\displaystyle 2j+1}">. That is, the eigenvalues of this Casimir operator are <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {J}}^{2}=-j(j+1){\boldsymbol {I}}_{2j+1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>j</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {J}}^{2}=-j(j+1){\boldsymbol {I}}_{2j+1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/955a6240a4be70673e13d7b7492d72ea95df4fb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.013ex; height:3.343ex;" alt="{\displaystyle {\boldsymbol {J}}^{2}=-j(j+1){\boldsymbol {I}}_{2j+1},}"></dd></dl> where j is integer or half-integer, and referred to as the <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> or <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>. So, the 3 × 3 generators L displayed above act on the triplet (spin 1) representation, while the 2 × 2 generators below, t, act on the <a href="/wiki/Spinor" title="Spinor">doublet</a> (<a href="/wiki/Spin-1/2" title="Spin-1/2">spin-1/2</a>) representation. By taking <a href="/wiki/Kronecker_product" title="Kronecker product">Kronecker products</a> of D1/2 with itself repeatedly, one may construct all higher irreducible representations Dj. That is, the resulting generators for higher spin systems in three spatial dimensions, for arbitrarily large j, can be calculated using these <a href="/wiki/Spin_operator" class="mw-redirect" title="Spin operator">spin operators</a> and <a href="/wiki/Ladder_operator" title="Ladder operator">ladder operators</a>. For every unitary irreducible representations Dj there is an equivalent one, D−j−1. All infinite-dimensional irreducible representations must be non-unitary, since the group is compact. In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, the Casimir invariant is the "angular-momentum-squared" operator; integer values of spin j characterize <a href="/wiki/Boson" title="Boson">bosonic representations</a>, while half-integer values <a href="/wiki/Fermion" title="Fermion">fermionic representations</a>. The <a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">antihermitian</a> matrices used above are utilized as <a href="/wiki/Spin_operator" class="mw-redirect" title="Spin operator">spin operators</a>, after they are multiplied by i, so they are now <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">hermitian</a> (like the Pauli matrices). Thus, in this language, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\boldsymbol {J}}_{x},{\boldsymbol {J}}_{y}]=i{\boldsymbol {J}}_{z},\quad [{\boldsymbol {J}}_{z},{\boldsymbol {J}}_{x}]=i{\boldsymbol {J}}_{y},\quad [{\boldsymbol {J}}_{y},{\boldsymbol {J}}_{z}]=i{\boldsymbol {J}}_{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>i</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>i</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>i</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\boldsymbol {J}}_{x},{\boldsymbol {J}}_{y}]=i{\boldsymbol {J}}_{z},\quad [{\boldsymbol {J}}_{z},{\boldsymbol {J}}_{x}]=i{\boldsymbol {J}}_{y},\quad [{\boldsymbol {J}}_{y},{\boldsymbol {J}}_{z}]=i{\boldsymbol {J}}_{x}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77cbe4e368f4f50472c0d68f8375936e9d6a055b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.916ex; height:3.009ex;" alt="{\displaystyle [{\boldsymbol {J}}_{x},{\boldsymbol {J}}_{y}]=i{\boldsymbol {J}}_{z},\quad [{\boldsymbol {J}}_{z},{\boldsymbol {J}}_{x}]=i{\boldsymbol {J}}_{y},\quad [{\boldsymbol {J}}_{y},{\boldsymbol {J}}_{z}]=i{\boldsymbol {J}}_{x}.}"></dd></dl> and hence <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {J}}^{2}=j(j+1){\boldsymbol {I}}_{2j+1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>j</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {J}}^{2}=j(j+1){\boldsymbol {I}}_{2j+1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87d1634e5cc6655f8fdec7ce15ef6472d285c525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.205ex; height:3.343ex;" alt="{\displaystyle {\boldsymbol {J}}^{2}=j(j+1){\boldsymbol {I}}_{2j+1}.}"></dd></dl> Explicit expressions for these Dj are, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left({\boldsymbol {J}}_{z}^{(j)}\right)_{ba}&=(j+1-a)\delta _{b,a}\\\left({\boldsymbol {J}}_{x}^{(j)}\right)_{ba}&={\frac {1}{2}}\left(\delta _{b,a+1}+\delta _{b+1,a}\right){\sqrt {(j+1)(a+b-1)-ab}}\\\left({\boldsymbol {J}}_{y}^{(j)}\right)_{ba}&={\frac {1}{2i}}\left(\delta _{b,a+1}-\delta _{b+1,a}\right){\sqrt {(j+1)(a+b-1)-ab}}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow> <mo>(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>a</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>,</mo> <mi>a</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mrow> <mo>(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>a</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>a</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>a</mi> <mi>b</mi> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow> <mo>(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>a</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>a</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>a</mi> <mi>b</mi> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left({\boldsymbol {J}}_{z}^{(j)}\right)_{ba}&=(j+1-a)\delta _{b,a}\\\left({\boldsymbol {J}}_{x}^{(j)}\right)_{ba}&={\frac {1}{2}}\left(\delta _{b,a+1}+\delta _{b+1,a}\right){\sqrt {(j+1)(a+b-1)-ab}}\\\left({\boldsymbol {J}}_{y}^{(j)}\right)_{ba}&={\frac {1}{2i}}\left(\delta _{b,a+1}-\delta _{b+1,a}\right){\sqrt {(j+1)(a+b-1)-ab}}\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fdc76b81a12cc92cc3b5c6325b4bac1c43a8d72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:56.34ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\left({\boldsymbol {J}}_{z}^{(j)}\right)_{ba}&=(j+1-a)\delta _{b,a}\\\left({\boldsymbol {J}}_{x}^{(j)}\right)_{ba}&={\frac {1}{2}}\left(\delta _{b,a+1}+\delta _{b+1,a}\right){\sqrt {(j+1)(a+b-1)-ab}}\\\left({\boldsymbol {J}}_{y}^{(j)}\right)_{ba}&={\frac {1}{2i}}\left(\delta _{b,a+1}-\delta _{b+1,a}\right){\sqrt {(j+1)(a+b-1)-ab}}\\\end{aligned}}}"></dd></dl> where j is arbitrary and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq a,b\leq 2j+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>≤</mo> <mn>2</mn> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq a,b\leq 2j+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f299d7753bb1df8cc351cd527c4aef5964c46a64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.744ex; height:2.509ex;" alt="{\displaystyle 1\leq a,b\leq 2j+1}">. For example, the resulting spin matrices for spin 1 (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91423fb032c471061948939f4a6811bf463780e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.246ex; height:2.509ex;" alt="{\displaystyle j=1}">) are <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\boldsymbol {J}}_{x}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}}\\{\boldsymbol {J}}_{y}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}0&-i&0\\i&0&-i\\0&i&0\end{pmatrix}}\\{\boldsymbol {J}}_{z}&={\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\boldsymbol {J}}_{x}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}}\\{\boldsymbol {J}}_{y}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}0&-i&0\\i&0&-i\\0&i&0\end{pmatrix}}\\{\boldsymbol {J}}_{z}&={\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe7c423226c7bdbfea35ab165e1ea786e3261ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.671ex; width:26.271ex; height:28.509ex;" alt="{\displaystyle {\begin{aligned}{\boldsymbol {J}}_{x}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}}\\{\boldsymbol {J}}_{y}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}0&-i&0\\i&0&-i\\0&i&0\end{pmatrix}}\\{\boldsymbol {J}}_{z}&={\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}}\end{aligned}}}"></dd></dl> Note, however, how these are in an equivalent, but different basis, the <a href="/wiki/Spherical_basis#Change_of_basis_matrix" title="Spherical basis">spherical basis</a>, than the above iL in the Cartesian basis.<a href="#cite_note-12">[nb 3]</a> For higher spins, such as spin <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/2⁠ (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j={\tfrac {3}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j={\tfrac {3}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/475370dbf5495aee4664cf82b0c1c8a035a303c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.027ex; width:5.741ex; height:3.509ex;" alt="{\displaystyle j={\tfrac {3}{2}}}">): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\boldsymbol {J}}_{x}&={\frac {1}{2}}{\begin{pmatrix}0&{\sqrt {3}}&0&0\\{\sqrt {3}}&0&2&0\\0&2&0&{\sqrt {3}}\\0&0&{\sqrt {3}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{y}&={\frac {1}{2}}{\begin{pmatrix}0&-i{\sqrt {3}}&0&0\\i{\sqrt {3}}&0&-2i&0\\0&2i&0&-i{\sqrt {3}}\\0&0&i{\sqrt {3}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{z}&={\frac {1}{2}}{\begin{pmatrix}3&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-3\end{pmatrix}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\boldsymbol {J}}_{x}&={\frac {1}{2}}{\begin{pmatrix}0&{\sqrt {3}}&0&0\\{\sqrt {3}}&0&2&0\\0&2&0&{\sqrt {3}}\\0&0&{\sqrt {3}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{y}&={\frac {1}{2}}{\begin{pmatrix}0&-i{\sqrt {3}}&0&0\\i{\sqrt {3}}&0&-2i&0\\0&2i&0&-i{\sqrt {3}}\\0&0&i{\sqrt {3}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{z}&={\frac {1}{2}}{\begin{pmatrix}3&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-3\end{pmatrix}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d982fbfb23786b5e4691c3fce45d9faedf9987b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.338ex; width:39.493ex; height:39.843ex;" alt="{\displaystyle {\begin{aligned}{\boldsymbol {J}}_{x}&={\frac {1}{2}}{\begin{pmatrix}0&{\sqrt {3}}&0&0\\{\sqrt {3}}&0&2&0\\0&2&0&{\sqrt {3}}\\0&0&{\sqrt {3}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{y}&={\frac {1}{2}}{\begin{pmatrix}0&-i{\sqrt {3}}&0&0\\i{\sqrt {3}}&0&-2i&0\\0&2i&0&-i{\sqrt {3}}\\0&0&i{\sqrt {3}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{z}&={\frac {1}{2}}{\begin{pmatrix}3&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-3\end{pmatrix}}.\end{aligned}}}"></dd></dl> For spin <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/2⁠ (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j={\tfrac {5}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j={\tfrac {5}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29faf5a5b97663200ccf93d8153f7671b746fcd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.027ex; width:5.741ex; height:3.509ex;" alt="{\displaystyle j={\tfrac {5}{2}}}">), <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\boldsymbol {J}}_{x}&={\frac {1}{2}}{\begin{pmatrix}0&{\sqrt {5}}&0&0&0&0\\{\sqrt {5}}&0&2{\sqrt {2}}&0&0&0\\0&2{\sqrt {2}}&0&3&0&0\\0&0&3&0&2{\sqrt {2}}&0\\0&0&0&2{\sqrt {2}}&0&{\sqrt {5}}\\0&0&0&0&{\sqrt {5}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{y}&={\frac {1}{2}}{\begin{pmatrix}0&-i{\sqrt {5}}&0&0&0&0\\i{\sqrt {5}}&0&-2i{\sqrt {2}}&0&0&0\\0&2i{\sqrt {2}}&0&-3i&0&0\\0&0&3i&0&-2i{\sqrt {2}}&0\\0&0&0&2i{\sqrt {2}}&0&-i{\sqrt {5}}\\0&0&0&0&i{\sqrt {5}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{z}&={\frac {1}{2}}{\begin{pmatrix}5&0&0&0&0&0\\0&3&0&0&0&0\\0&0&1&0&0&0\\0&0&0&-1&0&0\\0&0&0&0&-3&0\\0&0&0&0&0&-5\end{pmatrix}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>5</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\boldsymbol {J}}_{x}&={\frac {1}{2}}{\begin{pmatrix}0&{\sqrt {5}}&0&0&0&0\\{\sqrt {5}}&0&2{\sqrt {2}}&0&0&0\\0&2{\sqrt {2}}&0&3&0&0\\0&0&3&0&2{\sqrt {2}}&0\\0&0&0&2{\sqrt {2}}&0&{\sqrt {5}}\\0&0&0&0&{\sqrt {5}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{y}&={\frac {1}{2}}{\begin{pmatrix}0&-i{\sqrt {5}}&0&0&0&0\\i{\sqrt {5}}&0&-2i{\sqrt {2}}&0&0&0\\0&2i{\sqrt {2}}&0&-3i&0&0\\0&0&3i&0&-2i{\sqrt {2}}&0\\0&0&0&2i{\sqrt {2}}&0&-i{\sqrt {5}}\\0&0&0&0&i{\sqrt {5}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{z}&={\frac {1}{2}}{\begin{pmatrix}5&0&0&0&0&0\\0&3&0&0&0&0\\0&0&1&0&0&0\\0&0&0&-1&0&0\\0&0&0&0&-3&0\\0&0&0&0&0&-5\end{pmatrix}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d16c65731050552d06fc06cd60728b1eaea676b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -29.332ex; margin-bottom: -0.172ex; width:59.044ex; height:60.176ex;" alt="{\displaystyle {\begin{aligned}{\boldsymbol {J}}_{x}&={\frac {1}{2}}{\begin{pmatrix}0&{\sqrt {5}}&0&0&0&0\\{\sqrt {5}}&0&2{\sqrt {2}}&0&0&0\\0&2{\sqrt {2}}&0&3&0&0\\0&0&3&0&2{\sqrt {2}}&0\\0&0&0&2{\sqrt {2}}&0&{\sqrt {5}}\\0&0&0&0&{\sqrt {5}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{y}&={\frac {1}{2}}{\begin{pmatrix}0&-i{\sqrt {5}}&0&0&0&0\\i{\sqrt {5}}&0&-2i{\sqrt {2}}&0&0&0\\0&2i{\sqrt {2}}&0&-3i&0&0\\0&0&3i&0&-2i{\sqrt {2}}&0\\0&0&0&2i{\sqrt {2}}&0&-i{\sqrt {5}}\\0&0&0&0&i{\sqrt {5}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{z}&={\frac {1}{2}}{\begin{pmatrix}5&0&0&0&0&0\\0&3&0&0&0&0\\0&0&1&0&0&0\\0&0&0&-1&0&0\\0&0&0&0&-3&0\\0&0&0&0&0&-5\end{pmatrix}}.\end{aligned}}}"></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Spin_(physics)#Higher_spins" title="Spin (physics)">Spin (physics) § Higher spins</a></div> <div class="mw-heading mw-heading3"><h3 id="Isomorphism_with_𝖘𝖚(2)">Isomorphism with 𝖘𝖚(2)</h3>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=12" title="Edit section: Isomorphism with 𝖘𝖚(2)">edit</a>]</div> The Lie algebras <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4f1d3d3bf3da64b92af1a1018ce00545808b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.179ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(3)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {su}}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369267d5eae15a7478ebb52c71fcec1449a10dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.244ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(2)}"> are isomorphic. One basis for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {su}}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369267d5eae15a7478ebb52c71fcec1449a10dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.244ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(2)}"> is given by<a href="#cite_note-13">[10]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {t}}_{1}={\frac {1}{2}}{\begin{bmatrix}0&-i\\-i&0\end{bmatrix}},\quad {\boldsymbol {t}}_{2}={\frac {1}{2}}{\begin{bmatrix}0&-1\\1&0\end{bmatrix}},\quad {\boldsymbol {t}}_{3}={\frac {1}{2}}{\begin{bmatrix}-i&0\\0&i\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>i</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {t}}_{1}={\frac {1}{2}}{\begin{bmatrix}0&-i\\-i&0\end{bmatrix}},\quad {\boldsymbol {t}}_{2}={\frac {1}{2}}{\begin{bmatrix}0&-1\\1&0\end{bmatrix}},\quad {\boldsymbol {t}}_{3}={\frac {1}{2}}{\begin{bmatrix}-i&0\\0&i\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0117b11bb7d14277fcd7cc885a33235109480262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.423ex; height:6.176ex;" alt="{\displaystyle {\boldsymbol {t}}_{1}={\frac {1}{2}}{\begin{bmatrix}0&-i\\-i&0\end{bmatrix}},\quad {\boldsymbol {t}}_{2}={\frac {1}{2}}{\begin{bmatrix}0&-1\\1&0\end{bmatrix}},\quad {\boldsymbol {t}}_{3}={\frac {1}{2}}{\begin{bmatrix}-i&0\\0&i\end{bmatrix}}.}"></dd></dl> These are related to the <a href="/wiki/Pauli_matrix" class="mw-redirect" title="Pauli matrix">Pauli matrices</a> by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {t}}_{i}\longleftrightarrow {\frac {1}{2i}}\sigma _{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">⟷</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {t}}_{i}\longleftrightarrow {\frac {1}{2i}}\sigma _{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae6399637414517fb7892246fa2628b2b8e869d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.949ex; height:5.176ex;" alt="{\displaystyle {\boldsymbol {t}}_{i}\longleftrightarrow {\frac {1}{2i}}\sigma _{i}.}"></dd></dl> The Pauli matrices abide by the physicists' convention for Lie algebras. In that convention, Lie algebra elements are multiplied by i, the exponential map (below) is defined with an extra factor of i in the exponent and the <a href="/wiki/Structure_constant" class="mw-redirect" title="Structure constant">structure constants</a> remain the same, but the definition of them acquires a factor of i. Likewise, commutation relations acquire a factor of i. The commutation relations for the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {t}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {t}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c0976a680fe420e26a0760edcbed33e92aed2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.765ex; height:2.343ex;" alt="{\displaystyle {\boldsymbol {t}}_{i}}"> are <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\boldsymbol {t}}_{i},{\boldsymbol {t}}_{j}]=\varepsilon _{ijk}{\boldsymbol {t}}_{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\boldsymbol {t}}_{i},{\boldsymbol {t}}_{j}]=\varepsilon _{ijk}{\boldsymbol {t}}_{k},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28856ed20f15fbc3ba113622cf650f1fecffe0e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.183ex; height:3.009ex;" alt="{\displaystyle [{\boldsymbol {t}}_{i},{\boldsymbol {t}}_{j}]=\varepsilon _{ijk}{\boldsymbol {t}}_{k},}"></dd></dl> where <a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">εijk</a> is the totally anti-symmetric symbol with ε123 = 1. The isomorphism between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4f1d3d3bf3da64b92af1a1018ce00545808b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.179ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(3)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {su}}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369267d5eae15a7478ebb52c71fcec1449a10dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.244ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(2)}"> can be set up in several ways. For later convenience, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4f1d3d3bf3da64b92af1a1018ce00545808b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.179ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(3)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {su}}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369267d5eae15a7478ebb52c71fcec1449a10dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.244ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(2)}"> are identified by mapping <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {L}}_{x}\longleftrightarrow {\boldsymbol {t}}_{1},\quad {\boldsymbol {L}}_{y}\longleftrightarrow {\boldsymbol {t}}_{2},\quad {\boldsymbol {L}}_{z}\longleftrightarrow {\boldsymbol {t}}_{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">⟷</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">⟷</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">⟷</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {L}}_{x}\longleftrightarrow {\boldsymbol {t}}_{1},\quad {\boldsymbol {L}}_{y}\longleftrightarrow {\boldsymbol {t}}_{2},\quad {\boldsymbol {L}}_{z}\longleftrightarrow {\boldsymbol {t}}_{3},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73bd9d4a47bd7665e4f0c58662cec8da2caa0a49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.74ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {L}}_{x}\longleftrightarrow {\boldsymbol {t}}_{1},\quad {\boldsymbol {L}}_{y}\longleftrightarrow {\boldsymbol {t}}_{2},\quad {\boldsymbol {L}}_{z}\longleftrightarrow {\boldsymbol {t}}_{3},}"></dd></dl> and extending by linearity. <div class="mw-heading mw-heading2"><h2 id="Exponential_map">Exponential map</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=13" title="Edit section: Exponential map">edit</a>]</div> The exponential map for SO(3), is, since SO(3) is a matrix Lie group, defined using the standard <a href="/wiki/Matrix_exponential" title="Matrix exponential">matrix exponential</a> series, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}\exp :{\mathfrak {so}}(3)\to \operatorname {SO} (3)\\A\mapsto e^{A}=\sum _{k=0}^{\infty }{\frac {1}{k!}}A^{k}=I+A+{\tfrac {1}{2}}A^{2}+\cdots .\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>exp</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>A</mi> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mi>I</mi> <mo>+</mo> <mi>A</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\exp :{\mathfrak {so}}(3)\to \operatorname {SO} (3)\\A\mapsto e^{A}=\sum _{k=0}^{\infty }{\frac {1}{k!}}A^{k}=I+A+{\tfrac {1}{2}}A^{2}+\cdots .\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa1a3092df3649d460df0ca5c400c6f1848e47cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:47.258ex; height:6.509ex;" alt="{\displaystyle {\begin{cases}\exp :{\mathfrak {so}}(3)\to \operatorname {SO} (3)\\A\mapsto e^{A}=\sum _{k=0}^{\infty }{\frac {1}{k!}}A^{k}=I+A+{\tfrac {1}{2}}A^{2}+\cdots .\end{cases}}}"></dd></dl> For any <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">skew-symmetric matrix</a> A ∈ 𝖘𝖔(3), eA is always in SO(3). The proof uses the elementary properties of the matrix exponential <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(e^{A}\right)^{\textsf {T}}e^{A}=e^{A^{\textsf {T}}}e^{A}=e^{A^{\textsf {T}}+A}=e^{-A+A}=e^{A-A}=e^{A}\left(e^{A}\right)^{\textsf {T}}=e^{0}=I.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mo>+</mo> <mi>A</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>A</mi> <mo>+</mo> <mi>A</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>−</mo> <mi>A</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mi>I</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(e^{A}\right)^{\textsf {T}}e^{A}=e^{A^{\textsf {T}}}e^{A}=e^{A^{\textsf {T}}+A}=e^{-A+A}=e^{A-A}=e^{A}\left(e^{A}\right)^{\textsf {T}}=e^{0}=I.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26552a11b20b05e67e348953bc03296380702863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:66.501ex; height:3.843ex;" alt="{\displaystyle \left(e^{A}\right)^{\textsf {T}}e^{A}=e^{A^{\textsf {T}}}e^{A}=e^{A^{\textsf {T}}+A}=e^{-A+A}=e^{A-A}=e^{A}\left(e^{A}\right)^{\textsf {T}}=e^{0}=I.}"></dd></dl> since the matrices A and AT commute, this can be easily proven with the skew-symmetric matrix condition. This is not enough to show that 𝖘𝖔(3) is the corresponding Lie algebra for SO(3), and shall be proven separately. The level of difficulty of proof depends on how a matrix group Lie algebra is defined. <a href="#CITEREFHall2003">Hall (2003)</a> defines the Lie algebra as the set of matrices <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{A\in \operatorname {M} (n,\mathbb {R} )\left|e^{tA}\in \operatorname {SO} (3)\forall t\right.\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mi>A</mi> <mo>∈</mo> <mi mathvariant="normal">M</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mrow> <mo>|</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>A</mi> </mrow> </msup> <mo>∈</mo> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mi mathvariant="normal">∀</mi> <mi>t</mi> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{A\in \operatorname {M} (n,\mathbb {R} )\left|e^{tA}\in \operatorname {SO} (3)\forall t\right.\right\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9409c19331edd1319b169798d86fc2a7a7487a5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:31.596ex; height:3.509ex;" alt="{\displaystyle \left\{A\in \operatorname {M} (n,\mathbb {R} )\left|e^{tA}\in \operatorname {SO} (3)\forall t\right.\right\},}"></dd></dl> in which case it is trivial. <a href="#CITEREFRossmann2002">Rossmann (2002)</a> uses for a definition derivatives of smooth curve segments in SO(3) through the identity taken at the identity, in which case it is harder.<a href="#cite_note-14">[11]</a> For a fixed A ≠ 0, etA, −∞ < t < ∞ is a <a href="/wiki/One-parameter_subgroup" class="mw-redirect" title="One-parameter subgroup">one-parameter subgroup</a> along a <a href="/wiki/Geodesic" title="Geodesic">geodesic</a> in SO(3). That this gives a one-parameter subgroup follows directly from properties of the exponential map.<a href="#cite_note-15">[12]</a> The exponential map provides a <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphism</a> between a neighborhood of the origin in the 𝖘𝖔(3) and a neighborhood of the identity in the SO(3).<a href="#cite_note-16">[13]</a> For a proof, see <a href="/wiki/Closed_subgroup_theorem" class="mw-redirect" title="Closed subgroup theorem">Closed subgroup theorem</a>. The exponential map is <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a>. This follows from the fact that every R ∈ SO(3), since every rotation leaves an axis fixed (<a href="/wiki/Euler%27s_rotation_theorem" title="Euler's rotation theorem">Euler's rotation theorem</a>), and is conjugate to a <a href="/wiki/Block_diagonal_matrix" class="mw-redirect" title="Block diagonal matrix">block diagonal matrix</a> of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D={\begin{pmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\end{pmatrix}}=e^{\theta L_{z}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D={\begin{pmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\end{pmatrix}}=e^{\theta L_{z}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/097f9f9753d513863b0ba88b5c601ab42b4b2397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:34.506ex; height:9.176ex;" alt="{\displaystyle D={\begin{pmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\end{pmatrix}}=e^{\theta L_{z}},}"></dd></dl> such that A = BDB−1, and that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Be^{\theta L_{z}}B^{-1}=e^{B\theta L_{z}B^{-1}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </msup> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>θ</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Be^{\theta L_{z}}B^{-1}=e^{B\theta L_{z}B^{-1}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2119ada2c6b41c467bb81692c4ecc132a8e329d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.96ex; height:3.343ex;" alt="{\displaystyle Be^{\theta L_{z}}B^{-1}=e^{B\theta L_{z}B^{-1}},}"></dd></dl> together with the fact that 𝖘𝖔(3) is closed under the <a href="/wiki/Adjoint_representation" title="Adjoint representation">adjoint action</a> of SO(3), meaning that BθLzB−1 ∈ 𝖘𝖔(3). Thus, e.g., it is easy to check the popular identity <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\pi L_{x}/2}e^{\theta L_{z}}e^{\pi L_{x}/2}=e^{\theta L_{y}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\pi L_{x}/2}e^{\theta L_{z}}e^{\pi L_{x}/2}=e^{\theta L_{y}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ad2422addeca40a7338218449fea4d115c1ea3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:24.949ex; height:2.843ex;" alt="{\displaystyle e^{-\pi L_{x}/2}e^{\theta L_{z}}e^{\pi L_{x}/2}=e^{\theta L_{y}}.}"></dd></dl> As shown above, every element A ∈ 𝖘𝖔(3) is associated with a vector ω = θ u, where u = (x,y,z) is a unit magnitude vector. Since u is in the null space of A, if one now rotates to a new basis, through some other orthogonal matrix O, with u as the z axis, the final column and row of the rotation matrix in the new basis will be zero. Thus, we know in advance from the formula for the exponential that exp(OAOT) must leave u fixed. It is mathematically impossible to supply a straightforward formula for such a basis as a function of u, because its existence would violate the <a href="/wiki/Hairy_ball_theorem" title="Hairy ball theorem">hairy ball theorem</a>; but direct exponentiation is possible, and <a href="/wiki/Axis%E2%80%93angle_representation#Exponential_map_from_𝖘𝖔(3)_to_SO(3)" title="Axis–angle representation">yields</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\exp({\tilde {\boldsymbol {\omega }}})&=\exp(\theta ({\boldsymbol {u\cdot L}}))=\exp \left(\theta {\begin{bmatrix}0&-z&y\\z&0&-x\\-y&x&0\end{bmatrix}}\right)\\[4pt]&={\boldsymbol {I}}+2cs({\boldsymbol {u\cdot L}})+2s^{2}({\boldsymbol {u\cdot L}})^{2}\\[4pt]&={\begin{bmatrix}2\left(x^{2}-1\right)s^{2}+1&2xys^{2}-2zcs&2xzs^{2}+2ycs\\2xys^{2}+2zcs&2\left(y^{2}-1\right)s^{2}+1&2yzs^{2}-2xcs\\2xzs^{2}-2ycs&2yzs^{2}+2xcs&2\left(z^{2}-1\right)s^{2}+1\end{bmatrix}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ω</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> <mo mathvariant="bold">⋅</mo> <mi mathvariant="bold-italic">L</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>z</mi> </mtd> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>y</mi> </mtd> <mtd> <mi>x</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">I</mi> </mrow> <mo>+</mo> <mn>2</mn> <mi>c</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> <mo mathvariant="bold">⋅</mo> <mi mathvariant="bold-italic">L</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> <mo mathvariant="bold">⋅</mo> <mi mathvariant="bold-italic">L</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> <mi>x</mi> <mi>y</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>z</mi> <mi>c</mi> <mi>s</mi> </mtd> <mtd> <mn>2</mn> <mi>x</mi> <mi>z</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>y</mi> <mi>c</mi> <mi>s</mi> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> <mi>y</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>z</mi> <mi>c</mi> <mi>s</mi> </mtd> <mtd> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> <mi>y</mi> <mi>z</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mi>c</mi> <mi>s</mi> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> <mi>z</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mi>c</mi> <mi>s</mi> </mtd> <mtd> <mn>2</mn> <mi>y</mi> <mi>z</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>c</mi> <mi>s</mi> </mtd> <mtd> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\exp({\tilde {\boldsymbol {\omega }}})&=\exp(\theta ({\boldsymbol {u\cdot L}}))=\exp \left(\theta {\begin{bmatrix}0&-z&y\\z&0&-x\\-y&x&0\end{bmatrix}}\right)\\[4pt]&={\boldsymbol {I}}+2cs({\boldsymbol {u\cdot L}})+2s^{2}({\boldsymbol {u\cdot L}})^{2}\\[4pt]&={\begin{bmatrix}2\left(x^{2}-1\right)s^{2}+1&2xys^{2}-2zcs&2xzs^{2}+2ycs\\2xys^{2}+2zcs&2\left(y^{2}-1\right)s^{2}+1&2yzs^{2}-2xcs\\2xzs^{2}-2ycs&2yzs^{2}+2xcs&2\left(z^{2}-1\right)s^{2}+1\end{bmatrix}},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/287344e780fc2e78c2663d1416868b30eefb3269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:69.418ex; height:25.509ex;" alt="{\displaystyle {\begin{aligned}\exp({\tilde {\boldsymbol {\omega }}})&=\exp(\theta ({\boldsymbol {u\cdot L}}))=\exp \left(\theta {\begin{bmatrix}0&-z&y\\z&0&-x\\-y&x&0\end{bmatrix}}\right)\\[4pt]&={\boldsymbol {I}}+2cs({\boldsymbol {u\cdot L}})+2s^{2}({\boldsymbol {u\cdot L}})^{2}\\[4pt]&={\begin{bmatrix}2\left(x^{2}-1\right)s^{2}+1&2xys^{2}-2zcs&2xzs^{2}+2ycs\\2xys^{2}+2zcs&2\left(y^{2}-1\right)s^{2}+1&2yzs^{2}-2xcs\\2xzs^{2}-2ycs&2yzs^{2}+2xcs&2\left(z^{2}-1\right)s^{2}+1\end{bmatrix}},\end{aligned}}}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle c=\cos {\frac {\theta }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle c=\cos {\frac {\theta }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c6829664d944e124b69068d2a8581af98ca5281" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.262ex; height:3.676ex;" alt="{\textstyle c=\cos {\frac {\theta }{2}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle s=\sin {\frac {\theta }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle s=\sin {\frac {\theta }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b874e9bbc837a55627945ed5669f9f1ab3113f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.09ex; height:3.676ex;" alt="{\textstyle s=\sin {\frac {\theta }{2}}}">. This is recognized as a matrix for a rotation around axis u by the angle θ: cf. <a href="/wiki/Rodrigues%27_rotation_formula" title="Rodrigues' rotation formula">Rodrigues' rotation formula</a>. <div class="mw-heading mw-heading2"><h2 id="Logarithm_map">Logarithm map</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=14" title="Edit section: Logarithm map">edit</a>]</div> Given R ∈ SO(3), let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\tfrac {1}{2}}\left(R-R^{\mathrm {T} }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mi>R</mi> <mo>−</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\tfrac {1}{2}}\left(R-R^{\mathrm {T} }\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fac798d86a511f66c8d113123b600d0cd57ec56d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.804ex; height:3.509ex;" alt="{\displaystyle A={\tfrac {1}{2}}\left(R-R^{\mathrm {T} }\right)}"> denote the antisymmetric part and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \|A\|={\sqrt {-{\frac {1}{2}}\operatorname {Tr} \left(A^{2}\right)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>A</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>Tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \|A\|={\sqrt {-{\frac {1}{2}}\operatorname {Tr} \left(A^{2}\right)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/772c27848ef001a5e3d7642524dd20031020b065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:21.187ex; height:4.676ex;" alt="{\textstyle \|A\|={\sqrt {-{\frac {1}{2}}\operatorname {Tr} \left(A^{2}\right)}}.}"> Then, the logarithm of R is given by<a href="#cite_note-Engø_2001-11">[9]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log R={\frac {\sin ^{-1}\|A\|}{\|A\|}}A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo fence="false" stretchy="false">‖</mo> <mi>A</mi> <mo fence="false" stretchy="false">‖</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>A</mi> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log R={\frac {\sin ^{-1}\|A\|}{\|A\|}}A.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8400301197be3c7315ee6dc2400cb39b6cffff6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.091ex; height:6.676ex;" alt="{\displaystyle \log R={\frac {\sin ^{-1}\|A\|}{\|A\|}}A.}"></dd></dl> This is manifest by inspection of the mixed symmetry form of Rodrigues' formula, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{X}=I+{\frac {\sin \theta }{\theta }}X+2{\frac {\sin ^{2}{\frac {\theta }{2}}}{\theta ^{2}}}X^{2},\quad \theta =\|X\|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <mo>=</mo> <mi>I</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mi>θ</mi> </mfrac> </mrow> <mi>X</mi> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>θ</mi> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>X</mi> <mo fence="false" stretchy="false">‖</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{X}=I+{\frac {\sin \theta }{\theta }}X+2{\frac {\sin ^{2}{\frac {\theta }{2}}}{\theta ^{2}}}X^{2},\quad \theta =\|X\|,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36169c5a8500d0e3bbeda6639670680a0b3cd036" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:43.318ex; height:6.843ex;" alt="{\displaystyle e^{X}=I+{\frac {\sin \theta }{\theta }}X+2{\frac {\sin ^{2}{\frac {\theta }{2}}}{\theta ^{2}}}X^{2},\quad \theta =\|X\|,}"></dd></dl> where the first and last term on the right-hand side are symmetric. <div class="mw-heading mw-heading2"><h2 id="Uniform_random_sampling">Uniform random sampling</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=15" title="Edit section: Uniform random sampling">edit</a>]</div> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SO(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SO(3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16c677fee782e584fd417726201ce27c567f1e11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.244ex; height:2.843ex;" alt="{\displaystyle SO(3)}"> is doubly covered by the group of unit quaternions, which is isomorphic to the 3-sphere. Since the <a href="/wiki/Haar_measure" title="Haar measure">Haar measure</a> on the unit quaternions is just the 3-area measure in 4 dimensions, the Haar measure on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SO(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SO(3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16c677fee782e584fd417726201ce27c567f1e11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.244ex; height:2.843ex;" alt="{\displaystyle SO(3)}"> is just the pushforward of the 3-area measure. Consequently, generating a uniformly random rotation in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> is equivalent to generating a uniformly random point on the 3-sphere. This can be accomplished by the following<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\sqrt {1-u_{1}}}\sin(2\pi u_{2}),{\sqrt {1-u_{1}}}\cos(2\pi u_{2}),{\sqrt {u_{1}}}\sin(2\pi u_{3}),{\sqrt {u_{1}}}\cos(2\pi u_{3}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </msqrt> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </msqrt> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </msqrt> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </msqrt> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\sqrt {1-u_{1}}}\sin(2\pi u_{2}),{\sqrt {1-u_{1}}}\cos(2\pi u_{2}),{\sqrt {u_{1}}}\sin(2\pi u_{3}),{\sqrt {u_{1}}}\cos(2\pi u_{3}))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e19e1ebfc8ab0113fa2904819021088392909c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:71.204ex; height:3.509ex;" alt="{\displaystyle ({\sqrt {1-u_{1}}}\sin(2\pi u_{2}),{\sqrt {1-u_{1}}}\cos(2\pi u_{2}),{\sqrt {u_{1}}}\sin(2\pi u_{3}),{\sqrt {u_{1}}}\cos(2\pi u_{3}))}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1},u_{2},u_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{1},u_{2},u_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5a52b84973e274c62eef0f3afce7c377f11d2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.22ex; height:2.009ex;" alt="{\displaystyle u_{1},u_{2},u_{3}}"> are uniformly random samples of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}">.<a href="#cite_note-17">[14]</a> <div class="mw-heading mw-heading2"><h2 id="Baker–Campbell–Hausdorff_formula">Baker–Campbell–Hausdorff formula</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=16" title="Edit section: Baker–Campbell–Hausdorff formula">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula" title="Baker–Campbell–Hausdorff formula">Baker–Campbell–Hausdorff formula</a></div> Suppose X and Y in the Lie algebra are given. Their exponentials, exp(X) and exp(Y), are rotation matrices, which can be multiplied. Since the exponential map is a surjection, for some Z in the Lie algebra, exp(Z) = exp(X) exp(Y), and one may tentatively write <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=C(X,Y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=C(X,Y),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2ba1bf19a610b8c4f4b7e7e82439b2d0219b76f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.789ex; height:2.843ex;" alt="{\displaystyle Z=C(X,Y),}"></dd></dl> for C some expression in X and Y. When exp(X) and exp(Y) commute, then Z = X + Y, mimicking the behavior of complex exponentiation. The general case is given by the more elaborate <a href="/wiki/BCH_formula" class="mw-redirect" title="BCH formula">BCH formula</a>, a series expansion of nested Lie brackets.<a href="#cite_note-18">[15]</a> For matrices, the Lie bracket is the same operation as the <a href="/wiki/Commutator" title="Commutator">commutator</a>, which monitors lack of commutativity in multiplication. This general expansion unfolds as follows,<a href="#cite_note-19">[nb 4]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=C(X,Y)=X+Y+{\frac {1}{2}}[X,Y]+{\tfrac {1}{12}}[X,[X,Y]]-{\frac {1}{12}}[Y,[X,Y]]+\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=C(X,Y)=X+Y+{\frac {1}{2}}[X,Y]+{\tfrac {1}{12}}[X,[X,Y]]-{\frac {1}{12}}[Y,[X,Y]]+\cdots .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deb50254cdee1dcfdbb89e3c9994d201051418bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:72.244ex; height:5.176ex;" alt="{\displaystyle Z=C(X,Y)=X+Y+{\frac {1}{2}}[X,Y]+{\tfrac {1}{12}}[X,[X,Y]]-{\frac {1}{12}}[Y,[X,Y]]+\cdots .}"></dd></dl> The infinite expansion in the BCH formula for SO(3) reduces to a compact form, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=\alpha X+\beta Y+\gamma [X,Y],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mi>α</mi> <mi>X</mi> <mo>+</mo> <mi>β</mi> <mi>Y</mi> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=\alpha X+\beta Y+\gamma [X,Y],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cdd2c4da063f9ec30359beaacd3767b962f3875" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.023ex; height:2.843ex;" alt="{\displaystyle Z=\alpha X+\beta Y+\gamma [X,Y],}"></dd></dl> for suitable trigonometric function coefficients (α, β, γ). <style data-mw-deduplicate="TemplateStyles:r1214851843">.mw-parser-output .hidden-begin{box-sizing:border-box;width:100%;padding:5px;border:none;font-size:95%}.mw-parser-output .hidden-title{font-weight:bold;line-height:1.6;text-align:left}.mw-parser-output .hidden-content{text-align:left}@media all and (max-width:500px){.mw-parser-output .hidden-begin{width:auto!important;clear:none!important;float:none!important}}</style><div class="hidden-begin mw-collapsible mw-collapsed" style=""><div class="hidden-title skin-nightmode-reset-color" style="color:green;background:lightgrey;">The trigonometric coefficients</div><div class="hidden-content mw-collapsible-content" style=""> The (α, β, γ) are given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\phi \cot \left({\frac {\phi }{2}}\right)\gamma ,\qquad \beta =\theta \cot \left({\frac {\theta }{2}}\right)\gamma ,\qquad \gamma ={\frac {\sin ^{-1}d}{d}}{\frac {c}{\theta \phi }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>ϕ</mi> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>γ</mi> <mo>,</mo> <mspace width="2em" /> <mi>β</mi> <mo>=</mo> <mi>θ</mi> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>γ</mi> <mo>,</mo> <mspace width="2em" /> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>d</mi> </mrow> <mi>d</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mi>θ</mi> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\phi \cot \left({\frac {\phi }{2}}\right)\gamma ,\qquad \beta =\theta \cot \left({\frac {\theta }{2}}\right)\gamma ,\qquad \gamma ={\frac {\sin ^{-1}d}{d}}{\frac {c}{\theta \phi }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9c7379f942b183ea07c848a566f943c1c3ea09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:59.358ex; height:6.343ex;" alt="{\displaystyle \alpha =\phi \cot \left({\frac {\phi }{2}}\right)\gamma ,\qquad \beta =\theta \cot \left({\frac {\theta }{2}}\right)\gamma ,\qquad \gamma ={\frac {\sin ^{-1}d}{d}}{\frac {c}{\theta \phi }},}"></dd></dl> where <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}c&={\frac {1}{2}}\sin \theta \sin \phi -2\sin ^{2}{\frac {\theta }{2}}\sin ^{2}{\frac {\phi }{2}}\cos(\angle (u,v)),\quad a=c\cot \left({\frac {\phi }{2}}\right),\quad b=c\cot \left({\frac {\theta }{2}}\right),\\d&={\sqrt {a^{2}+b^{2}+2ab\cos(\angle (u,v))+c^{2}\sin ^{2}(\angle (u,v))}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>−</mo> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∠</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mo>=</mo> <mi>c</mi> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>b</mi> <mo>=</mo> <mi>c</mi> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∠</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∠</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c&={\frac {1}{2}}\sin \theta \sin \phi -2\sin ^{2}{\frac {\theta }{2}}\sin ^{2}{\frac {\phi }{2}}\cos(\angle (u,v)),\quad a=c\cot \left({\frac {\phi }{2}}\right),\quad b=c\cot \left({\frac {\theta }{2}}\right),\\d&={\sqrt {a^{2}+b^{2}+2ab\cos(\angle (u,v))+c^{2}\sin ^{2}(\angle (u,v))}},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18a5e82ce8fd897568c813162242f9c0e8fd95f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:82.511ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}c&={\frac {1}{2}}\sin \theta \sin \phi -2\sin ^{2}{\frac {\theta }{2}}\sin ^{2}{\frac {\phi }{2}}\cos(\angle (u,v)),\quad a=c\cot \left({\frac {\phi }{2}}\right),\quad b=c\cot \left({\frac {\theta }{2}}\right),\\d&={\sqrt {a^{2}+b^{2}+2ab\cos(\angle (u,v))+c^{2}\sin ^{2}(\angle (u,v))}},\end{aligned}}}"></dd></dl> for <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\|X\|,\quad \phi =\|Y\|,\quad \angle (u,v)=\cos ^{-1}{\frac {\langle X,Y\rangle }{\|X\|\|Y\|}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>X</mi> <mo fence="false" stretchy="false">‖</mo> <mo>,</mo> <mspace width="1em" /> <mi>ϕ</mi> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>Y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∠</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>X</mi> <mo fence="false" stretchy="false">‖</mo> <mo fence="false" stretchy="false">‖</mo> <mi>Y</mi> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\|X\|,\quad \phi =\|Y\|,\quad \angle (u,v)=\cos ^{-1}{\frac {\langle X,Y\rangle }{\|X\|\|Y\|}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f481953ae0b2ffe0de3b0de678ba39ea050a38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.583ex; height:6.509ex;" alt="{\displaystyle \theta =\|X\|,\quad \phi =\|Y\|,\quad \angle (u,v)=\cos ^{-1}{\frac {\langle X,Y\rangle }{\|X\|\|Y\|}}.}"></dd></dl> The inner product is the <a href="/wiki/Hilbert%E2%80%93Schmidt_inner_product" class="mw-redirect" title="Hilbert–Schmidt inner product">Hilbert–Schmidt inner product</a> and the norm is the associated norm. Under the hat-isomorphism, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle u,v\rangle ={\frac {1}{2}}\operatorname {Tr} X^{\mathrm {T} }Y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>Tr</mi> <mo>⁡</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u,v\rangle ={\frac {1}{2}}\operatorname {Tr} X^{\mathrm {T} }Y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f64d759bacef13c360870791ad5c6db6d46dfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.597ex; height:5.176ex;" alt="{\displaystyle \langle u,v\rangle ={\frac {1}{2}}\operatorname {Tr} X^{\mathrm {T} }Y,}"></dd></dl> which explains the factors for θ and φ. This drops out in the expression for the angle. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Rotation_formalisms_in_three_dimensions#Rodrigues_parameters_and_Gibbs_representation" title="Rotation formalisms in three dimensions">Rotation formalisms in three dimensions § Rodrigues parameters and Gibbs representation</a></div> </div></div> It is worthwhile to write this composite rotation generator as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha X+\beta Y+\gamma [X,Y]{\underset {{\mathfrak {so}}(3)}{=}}X+Y+{\frac {1}{2}}[X,Y]+{\frac {1}{12}}[X,[X,Y]]-{\frac {1}{12}}[Y,[X,Y]]+\cdots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mi>X</mi> <mo>+</mo> <mi>β</mi> <mi>Y</mi> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>=</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </munder> </mrow> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha X+\beta Y+\gamma [X,Y]{\underset {{\mathfrak {so}}(3)}{=}}X+Y+{\frac {1}{2}}[X,Y]+{\frac {1}{12}}[X,[X,Y]]-{\frac {1}{12}}[Y,[X,Y]]+\cdots ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11e64023ced100d20590d64041bc325a08fba789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:79.915ex; height:5.509ex;" alt="{\displaystyle \alpha X+\beta Y+\gamma [X,Y]{\underset {{\mathfrak {so}}(3)}{=}}X+Y+{\frac {1}{2}}[X,Y]+{\frac {1}{12}}[X,[X,Y]]-{\frac {1}{12}}[Y,[X,Y]]+\cdots ,}"></dd></dl> to emphasize that this is a Lie algebra identity. The above identity holds for all <a href="/wiki/Faithful_representation" title="Faithful representation">faithful representations</a> of 𝖘𝖔(3). The <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> of a Lie algebra homomorphism is an <a href="/wiki/Ideal_(Lie_algebra)" class="mw-redirect" title="Ideal (Lie algebra)">ideal</a>, but 𝖘𝖔(3), being <a href="/wiki/Simple_(abstract_algebra)" title="Simple (abstract algebra)">simple</a>, has no nontrivial ideals and all nontrivial representations are hence faithful. It holds in particular in the doublet or spinor representation. The same explicit formula thus follows in a simpler way through Pauli matrices, cf. the <a href="/wiki/Pauli_matrices#Exponential_of_a_Pauli_vector" title="Pauli matrices">2×2 derivation for SU(2)</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214851843"><div class="hidden-begin mw-collapsible mw-collapsed" style=""><div class="hidden-title skin-nightmode-reset-color" style="color:green;background:lightgrey;">The SU(2) case</div><div class="hidden-content mw-collapsible-content" style=""> The <a href="/wiki/Pauli_matrices#Exponential_of_a_Pauli_vector" title="Pauli matrices">Pauli vector version</a> of the same BCH formula is the somewhat simpler group composition law of SU(2), <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{ia'\left({\hat {u}}\cdot {\vec {\sigma }}\right)}e^{ib'\left({\hat {v}}\cdot {\vec {\sigma }}\right)}=\exp \left({\frac {c'}{\sin c'}}\sin a'\sin b'\left(\left(i\cot b'{\hat {u}}+i\cot a'{\hat {v}}\right)\cdot {\vec {\sigma }}+{\frac {1}{2}}\left[i{\hat {u}}\cdot {\vec {\sigma }},i{\hat {v}}\cdot {\vec {\sigma }}\right]\right)\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msup> <mi>a</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msup> <mi>b</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>c</mi> <mo>′</mo> </msup> <mrow> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>cot</mi> <mo>⁡</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>i</mi> <mi>cot</mi> <mo>⁡</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{ia'\left({\hat {u}}\cdot {\vec {\sigma }}\right)}e^{ib'\left({\hat {v}}\cdot {\vec {\sigma }}\right)}=\exp \left({\frac {c'}{\sin c'}}\sin a'\sin b'\left(\left(i\cot b'{\hat {u}}+i\cot a'{\hat {v}}\right)\cdot {\vec {\sigma }}+{\frac {1}{2}}\left[i{\hat {u}}\cdot {\vec {\sigma }},i{\hat {v}}\cdot {\vec {\sigma }}\right]\right)\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8fd1e518377c5c96334d61622523e44a4cd258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:87.008ex; height:6.176ex;" alt="{\displaystyle e^{ia'\left({\hat {u}}\cdot {\vec {\sigma }}\right)}e^{ib'\left({\hat {v}}\cdot {\vec {\sigma }}\right)}=\exp \left({\frac {c'}{\sin c'}}\sin a'\sin b'\left(\left(i\cot b'{\hat {u}}+i\cot a'{\hat {v}}\right)\cdot {\vec {\sigma }}+{\frac {1}{2}}\left[i{\hat {u}}\cdot {\vec {\sigma }},i{\hat {v}}\cdot {\vec {\sigma }}\right]\right)\right),}"></dd></dl> where <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos c'=\cos a'\cos b'-{\hat {u}}\cdot {\hat {v}}\sin a'\sin b',}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mi>cos</mi> <mo>⁡</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos c'=\cos a'\cos b'-{\hat {u}}\cdot {\hat {v}}\sin a'\sin b',}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bbf9895667dbaca6b9fef2a614d66d02518493d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.848ex; height:2.843ex;" alt="{\displaystyle \cos c'=\cos a'\cos b'-{\hat {u}}\cdot {\hat {v}}\sin a'\sin b',}"></dd></dl> the <a href="/wiki/Spherical_law_of_cosines" title="Spherical law of cosines">spherical law of cosines</a>. (Note a', b', c' are angles, not the a, b, c above.) This is manifestly of the same format as above, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=\alpha 'X+\beta 'Y+\gamma '[X,Y],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <msup> <mi>α</mi> <mo>′</mo> </msup> <mi>X</mi> <mo>+</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mi>Y</mi> <mo>+</mo> <msup> <mi>γ</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=\alpha 'X+\beta 'Y+\gamma '[X,Y],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c19c2a544b1f7101840f86cc505df9d42b697df8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.099ex; height:3.009ex;" alt="{\displaystyle Z=\alpha 'X+\beta 'Y+\gamma '[X,Y],}"></dd></dl> with <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=ia'{\hat {u}}\cdot \mathbf {\sigma } ,\quad Y=ib'{\hat {v}}\cdot \mathbf {\sigma } \in {\mathfrak {su}}(2),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>i</mi> <msup> <mi>a</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>Y</mi> <mo>=</mo> <mi>i</mi> <msup> <mi>b</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=ia'{\hat {u}}\cdot \mathbf {\sigma } ,\quad Y=ib'{\hat {v}}\cdot \mathbf {\sigma } \in {\mathfrak {su}}(2),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/539200d0258ea4ea87c1db9604742ee0ad7d2760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.774ex; height:3.009ex;" alt="{\displaystyle X=ia'{\hat {u}}\cdot \mathbf {\sigma } ,\quad Y=ib'{\hat {v}}\cdot \mathbf {\sigma } \in {\mathfrak {su}}(2),}"></dd></dl> so that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\alpha '&={\frac {c'}{\sin c'}}{\frac {\sin a'}{a'}}\cos b'\\\beta '&={\frac {c'}{\sin c'}}{\frac {\sin b'}{b'}}\cos a'\\\gamma '&={\frac {1}{2}}{\frac {c'}{\sin c'}}{\frac {\sin a'}{a'}}{\frac {\sin b'}{b'}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>α</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>c</mi> <mo>′</mo> </msup> <mrow> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> </mrow> <msup> <mi>a</mi> <mo>′</mo> </msup> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>β</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>c</mi> <mo>′</mo> </msup> <mrow> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> </mrow> <msup> <mi>b</mi> <mo>′</mo> </msup> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>γ</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>c</mi> <mo>′</mo> </msup> <mrow> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> </mrow> <msup> <mi>a</mi> <mo>′</mo> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> </mrow> <msup> <mi>b</mi> <mo>′</mo> </msup> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\alpha '&={\frac {c'}{\sin c'}}{\frac {\sin a'}{a'}}\cos b'\\\beta '&={\frac {c'}{\sin c'}}{\frac {\sin b'}{b'}}\cos a'\\\gamma '&={\frac {1}{2}}{\frac {c'}{\sin c'}}{\frac {\sin a'}{a'}}{\frac {\sin b'}{b'}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7db9d6cc4f9d35ab4b099b09e8018818c55fb6a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; margin-top: -0.203ex; width:26.193ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}\alpha '&={\frac {c'}{\sin c'}}{\frac {\sin a'}{a'}}\cos b'\\\beta '&={\frac {c'}{\sin c'}}{\frac {\sin b'}{b'}}\cos a'\\\gamma '&={\frac {1}{2}}{\frac {c'}{\sin c'}}{\frac {\sin a'}{a'}}{\frac {\sin b'}{b'}}.\end{aligned}}}"></dd></dl> For uniform normalization of the generators in the Lie algebra involved, express the Pauli matrices in terms of t-matrices, σ → 2i t, so that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a'\mapsto -{\frac {\theta }{2}},\quad b'\mapsto -{\frac {\phi }{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo stretchy="false">↦</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo stretchy="false">↦</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a'\mapsto -{\frac {\theta }{2}},\quad b'\mapsto -{\frac {\phi }{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6354f9ce67ed7574e56ffc4c65abd39ab6aa47b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.665ex; height:5.343ex;" alt="{\displaystyle a'\mapsto -{\frac {\theta }{2}},\quad b'\mapsto -{\frac {\phi }{2}}.}"></dd></dl> To verify then these are the same coefficients as above, compute the ratios of the coefficients, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\alpha '}{\gamma '}}&=\theta \cot {\frac {\theta }{2}}&={\frac {\alpha }{\gamma }}\\{\frac {\beta '}{\gamma '}}&=\phi \cot {\frac {\phi }{2}}&={\frac {\beta }{\gamma }}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>α</mi> <mo>′</mo> </msup> <msup> <mi>γ</mi> <mo>′</mo> </msup> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>θ</mi> <mi>cot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mi>γ</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>β</mi> <mo>′</mo> </msup> <msup> <mi>γ</mi> <mo>′</mo> </msup> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ϕ</mi> <mi>cot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mi>γ</mi> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\alpha '}{\gamma '}}&=\theta \cot {\frac {\theta }{2}}&={\frac {\alpha }{\gamma }}\\{\frac {\beta '}{\gamma '}}&=\phi \cot {\frac {\phi }{2}}&={\frac {\beta }{\gamma }}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611bf931fa0ae2fd5dde0222404e55771122b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.289ex; margin-bottom: -0.216ex; width:24.253ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {\alpha '}{\gamma '}}&=\theta \cot {\frac {\theta }{2}}&={\frac {\alpha }{\gamma }}\\{\frac {\beta '}{\gamma '}}&=\phi \cot {\frac {\phi }{2}}&={\frac {\beta }{\gamma }}.\end{aligned}}}"></dd></dl> Finally, γ = γ' given the identity d = sin 2c'. </div></div> For the general n × n case, one might use Ref.<a href="#cite_note-20">[16]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214851843"><div class="hidden-begin mw-collapsible mw-collapsed" style=""><div class="hidden-title skin-nightmode-reset-color" style="color:green;background:lightgrey;">The quaternion case</div><div class="hidden-content mw-collapsible-content" style=""> The <a href="/wiki/Quaternion" title="Quaternion">quaternion</a> formulation of the composition of two rotations RB and RA also yields directly the <a href="/wiki/Axis_of_rotation" class="mw-redirect" title="Axis of rotation">rotation axis</a> and angle of the composite rotation RC = RBRA. Let the quaternion associated with a spatial rotation R is constructed from its <a href="/wiki/Axis_of_rotation" class="mw-redirect" title="Axis of rotation">rotation axis</a> S and the rotation angle φ this axis. The associated quaternion is given by, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\cos {\frac {\phi }{2}}+\sin {\frac {\phi }{2}}\mathbf {S} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\cos {\frac {\phi }{2}}+\sin {\frac {\phi }{2}}\mathbf {S} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75ddd7ad477f57f3ed531218aef24eb163110d88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.754ex; height:5.343ex;" alt="{\displaystyle S=\cos {\frac {\phi }{2}}+\sin {\frac {\phi }{2}}\mathbf {S} .}"></dd></dl> Then the composition of the rotation RR with RA is the rotation RC = RBRA with rotation axis and angle defined by the product of the quaternions <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\mathbf {A} \quad {\text{ and }}\quad B=\cos {\frac {\beta }{2}}+\sin {\frac {\beta }{2}}\mathbf {B} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="1em" /> <mi>B</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\mathbf {A} \quad {\text{ and }}\quad B=\cos {\frac {\beta }{2}}+\sin {\frac {\beta }{2}}\mathbf {B} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb16979fc48ade8f81aa314262101a7e41fe03f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.972ex; height:5.343ex;" alt="{\displaystyle A=\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\mathbf {A} \quad {\text{ and }}\quad B=\cos {\frac {\beta }{2}}+\sin {\frac {\beta }{2}}\mathbf {B} ,}"></dd></dl> that is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=\cos {\frac {\gamma }{2}}+\sin {\frac {\gamma }{2}}\mathbf {C} =\left(\cos {\frac {\beta }{2}}+\sin {\frac {\beta }{2}}\mathbf {B} \right)\left(\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\mathbf {A} \right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=\cos {\frac {\gamma }{2}}+\sin {\frac {\gamma }{2}}\mathbf {C} =\left(\cos {\frac {\beta }{2}}+\sin {\frac {\beta }{2}}\mathbf {B} \right)\left(\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\mathbf {A} \right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9db2dd23c88ff87547e521b0b236c6f585c29f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:63.357ex; height:6.176ex;" alt="{\displaystyle C=\cos {\frac {\gamma }{2}}+\sin {\frac {\gamma }{2}}\mathbf {C} =\left(\cos {\frac {\beta }{2}}+\sin {\frac {\beta }{2}}\mathbf {B} \right)\left(\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\mathbf {A} \right).}"></dd></dl> Expand this product to obtain <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\frac {\gamma }{2}}+\sin {\frac {\gamma }{2}}\mathbf {C} =\left(\cos {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}-\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} \right)+\left(\sin {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}\mathbf {B} +\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\mathbf {A} +\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \times \mathbf {A} \right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\gamma }{2}}+\sin {\frac {\gamma }{2}}\mathbf {C} =\left(\cos {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}-\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} \right)+\left(\sin {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}\mathbf {B} +\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\mathbf {A} +\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \times \mathbf {A} \right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1da63d7a0e5eb4d57db489162bd6618a9408d028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:112.171ex; height:6.176ex;" alt="{\displaystyle \cos {\frac {\gamma }{2}}+\sin {\frac {\gamma }{2}}\mathbf {C} =\left(\cos {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}-\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} \right)+\left(\sin {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}\mathbf {B} +\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\mathbf {A} +\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \times \mathbf {A} \right).}"></dd></dl> Divide both sides of this equation by the identity, which is the <a href="/wiki/Spherical_law_of_cosines" title="Spherical law of cosines">law of cosines on a sphere</a>, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\frac {\gamma }{2}}=\cos {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}-\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\gamma }{2}}=\cos {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}-\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/071c736128aedb24fbedffadf91aedc9d1e526a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.022ex; height:5.343ex;" alt="{\displaystyle \cos {\frac {\gamma }{2}}=\cos {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}-\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} ,}"></dd></dl> and compute <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan {\frac {\gamma }{2}}\mathbf {C} ={\frac {\tan {\frac {\beta }{2}}\mathbf {B} +\tan {\frac {\alpha }{2}}\mathbf {A} +\tan {\frac {\beta }{2}}\tan {\frac {\alpha }{2}}\mathbf {B} \times \mathbf {A} }{1-\tan {\frac {\beta }{2}}\tan {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan {\frac {\gamma }{2}}\mathbf {C} ={\frac {\tan {\frac {\beta }{2}}\mathbf {B} +\tan {\frac {\alpha }{2}}\mathbf {A} +\tan {\frac {\beta }{2}}\tan {\frac {\alpha }{2}}\mathbf {B} \times \mathbf {A} }{1-\tan {\frac {\beta }{2}}\tan {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6beebd8e0b921524bd82b4675d58dc984250fdc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:51.426ex; height:8.843ex;" alt="{\displaystyle \tan {\frac {\gamma }{2}}\mathbf {C} ={\frac {\tan {\frac {\beta }{2}}\mathbf {B} +\tan {\frac {\alpha }{2}}\mathbf {A} +\tan {\frac {\beta }{2}}\tan {\frac {\alpha }{2}}\mathbf {B} \times \mathbf {A} }{1-\tan {\frac {\beta }{2}}\tan {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} }}.}"></dd></dl> This is Rodrigues' formula for the axis of a composite rotation defined in terms of the axes of the two rotations. He derived this formula in 1840 (see page 408).<a href="#cite_note-21">[17]</a> The three rotation axes A, B, and C form a spherical triangle and the dihedral angles between the planes formed by the sides of this triangle are defined by the rotation angles. </div></div> <div class="mw-heading mw-heading2"><h2 id="Infinitesimal_rotations">Infinitesimal rotations</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=17" title="Edit section: Infinitesimal rotations">edit</a>]</div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Infinitesimal_rotation_matrix" title="Infinitesimal rotation matrix">Infinitesimal rotation matrix</a>.[<a class="external text" href="https://en.wikipedia.org/w/index.php?title=Infinitesimal_rotation_matrix&action=edit">edit</a>]</div><div class="excerpt"> An <a href="/wiki/Infinitesimal_rotation_matrix" title="Infinitesimal rotation matrix">infinitesimal rotation matrix</a> or differential rotation matrix is a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> representing an <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitely</a> small <a href="/wiki/Rotation" title="Rotation">rotation</a>. While a <a href="/wiki/Rotation_matrix" title="Rotation matrix">rotation matrix</a> is an <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{\mathsf {T}}=R^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{\mathsf {T}}=R^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85439c87cf2430c0c01e9eb25f6d6cee89ee3238" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.311ex; height:2.676ex;" alt="{\displaystyle R^{\mathsf {T}}=R^{-1}}"> representing an element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SO(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SO(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8243929248730d44fbb8ee9d267361713332ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.477ex; height:2.843ex;" alt="{\displaystyle SO(n)}"> (the <a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">special orthogonal group</a>), the <a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">differential</a> of a rotation is a <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">skew-symmetric matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\mathsf {T}}=-A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\mathsf {T}}=-A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f020a5cc94f0fd27929041f589affa0e575a083" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.744ex; height:2.843ex;" alt="{\displaystyle A^{\mathsf {T}}=-A}"> in the <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57c18bd074c7bdc79ea650563edde2e2bc080321" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.412ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(n)}"> (the <a href="/wiki/Special_orthogonal_Lie_algebra" class="mw-redirect" title="Special orthogonal Lie algebra">special orthogonal Lie algebra</a>), which is not itself a rotation matrix. An infinitesimal rotation matrix has the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I+d\theta \,A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>+</mo> <mi>d</mi> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I+d\theta \,A,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3da72e5ca97ba81447afc60fae1a089f4c1a20f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.096ex; height:2.509ex;" alt="{\displaystyle I+d\theta \,A,}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is the identity matrix, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75ae6ca1248d081ef3fcfdd3e17ba0e3f6c02ee9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.306ex; height:2.176ex;" alt="{\displaystyle d\theta }"> is vanishingly small, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathfrak {so}}(n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in {\mathfrak {so}}(n).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/392ece43ea9ed9515d3cf7e3e4e2f36f1b8474db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.602ex; height:2.843ex;" alt="{\displaystyle A\in {\mathfrak {so}}(n).}"> For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=L_{x},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=L_{x},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/622657009d214614f9d7329bab84c67762534d8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.244ex; height:2.509ex;" alt="{\displaystyle A=L_{x},}"> representing an infinitesimal three-dimensional rotation about the x-axis, a basis element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(3),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(3),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0b216599a18df448fe258d3e160a969c64c7dc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.826ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(3),}"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dL_{x}={\begin{bmatrix}1&0&0\\0&1&-d\theta \\0&d\theta &1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mi>d</mi> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>d</mi> <mi>θ</mi> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dL_{x}={\begin{bmatrix}1&0&0\\0&1&-d\theta \\0&d\theta &1\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d81596610fbd23af0d094a16c6d0c4d33e2cab2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:23.797ex; height:9.176ex;" alt="{\displaystyle dL_{x}={\begin{bmatrix}1&0&0\\0&1&-d\theta \\0&d\theta &1\end{bmatrix}}.}"></dd></dl> The computation rules for infinitesimal rotation matrices are as usual except that infinitesimals of second order are routinely dropped. With these rules, these matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals.<a href="#cite_note-22">[18]</a> It turns out that the order in which infinitesimal rotations are applied is irrelevant.</div></div> <div class="mw-heading mw-heading2"><h2 id="Realizations_of_rotations">Realizations of rotations</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=18" title="Edit section: Realizations of rotations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Rotation_formalisms_in_three_dimensions" title="Rotation formalisms in three dimensions">Rotation formalisms in three dimensions</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Charts_on_SO(3)" title="Charts on SO(3)">Charts on SO(3)</a></div> We have seen that there are a variety of ways to represent rotations: <ul><li>as orthogonal matrices with determinant 1,</li> <li>by axis and rotation angle</li> <li>in <a href="/wiki/Quaternion" title="Quaternion">quaternion</a> algebra with <a href="/wiki/Versor" title="Versor">versors</a> and the map <a href="/wiki/3-sphere" title="3-sphere">3-sphere</a> S3 → SO(3) (see <a href="/wiki/Quaternions_and_spatial_rotation" title="Quaternions and spatial rotation">quaternions and spatial rotations</a>)</li> <li>in <a href="/wiki/Geometric_algebra" title="Geometric algebra">geometric algebra</a> as a <a href="/wiki/Rotor_(mathematics)" title="Rotor (mathematics)">rotor</a></li> <li>as a sequence of three rotations about three fixed axes; see <a href="/wiki/Euler_angle" class="mw-redirect" title="Euler angle">Euler angles</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Spherical_harmonics">Spherical harmonics</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=19" title="Edit section: Spherical harmonics">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Spherical_harmonics" title="Spherical harmonics">Spherical harmonics</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Representation_of_a_Lie_group#An_example:_The_rotation_group_SO.283.29" title="Representation of a Lie group">Representations of SO(3)</a></div> The group SO(3) of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}\left(\mathbf {S} ^{2}\right)=\operatorname {span} \left\{Y_{m}^{\ell },\ell \in \mathbb {N} ^{+},-\ell \leq m\leq \ell \right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mi>span</mi> <mo>⁡</mo> <mrow> <mo>{</mo> <mrow> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msubsup> <mo>,</mo> <mi>ℓ</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>,</mo> <mo>−</mo> <mi>ℓ</mi> <mo>≤</mo> <mi>m</mi> <mo>≤</mo> <mi>ℓ</mi> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}\left(\mathbf {S} ^{2}\right)=\operatorname {span} \left\{Y_{m}^{\ell },\ell \in \mathbb {N} ^{+},-\ell \leq m\leq \ell \right\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3bc85e3aa871572201b562763217a9b989a84f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.891ex; height:3.343ex;" alt="{\displaystyle L^{2}\left(\mathbf {S} ^{2}\right)=\operatorname {span} \left\{Y_{m}^{\ell },\ell \in \mathbb {N} ^{+},-\ell \leq m\leq \ell \right\},}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{m}^{\ell }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{m}^{\ell }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eec221a8066161cb86ec6711256ef71c3cf5791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.026ex; height:2.843ex;" alt="{\displaystyle Y_{m}^{\ell }}"> are <a href="/wiki/Spherical_harmonics" title="Spherical harmonics">spherical harmonics</a>. Its elements are square integrable complex-valued functions<a href="#cite_note-23">[nb 5]</a> on the sphere. The inner product on this space is given by <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f,g\rangle =\int _{\mathbf {S} ^{2}}{\overline {f}}g\,d\Omega =\int _{0}^{2\pi }\int _{0}^{\pi }{\overline {f}}g\sin \theta \,d\theta \,d\phi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo accent="false">¯</mo> </mover> </mrow> <mi>g</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo accent="false">¯</mo> </mover> </mrow> <mi>g</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ϕ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle =\int _{\mathbf {S} ^{2}}{\overline {f}}g\,d\Omega =\int _{0}^{2\pi }\int _{0}^{\pi }{\overline {f}}g\sin \theta \,d\theta \,d\phi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2088f1508e232caad82c50f0cb9c3dac8e27a3c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.632ex; height:6.176ex;" alt="{\displaystyle \langle f,g\rangle =\int _{\mathbf {S} ^{2}}{\overline {f}}g\,d\Omega =\int _{0}^{2\pi }\int _{0}^{\pi }{\overline {f}}g\sin \theta \,d\theta \,d\phi .}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(H1)</td></tr></tbody></table> If f is an arbitrary square integrable function defined on the unit sphere S2, then it can be expressed as<a href="#cite_note-Gelfand_M_S-24">[19]</a> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f\rangle =\sum _{\ell =1}^{\infty }\sum _{m=-\ell }^{m=\ell }\left|Y_{m}^{\ell }\right\rangle \left\langle Y_{m}^{\ell }|f\right\rangle ,\qquad f(\theta ,\phi )=\sum _{\ell =1}^{\infty }\sum _{m=-\ell }^{m=\ell }f_{\ell m}Y_{m}^{\ell }(\theta ,\phi ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi>ℓ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mi>ℓ</mi> </mrow> </munderover> <mrow> <mo>|</mo> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msubsup> <mo>⟩</mo> </mrow> <mrow> <mo>⟨</mo> <mrow> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> </mrow> <mo>⟩</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi>ℓ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mi>ℓ</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <mi>m</mi> </mrow> </msub> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f\rangle =\sum _{\ell =1}^{\infty }\sum _{m=-\ell }^{m=\ell }\left|Y_{m}^{\ell }\right\rangle \left\langle Y_{m}^{\ell }|f\right\rangle ,\qquad f(\theta ,\phi )=\sum _{\ell =1}^{\infty }\sum _{m=-\ell }^{m=\ell }f_{\ell m}Y_{m}^{\ell }(\theta ,\phi ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e84afefce7669a38b9d21715f3277a0ced631d5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64.115ex; height:7.509ex;" alt="{\displaystyle |f\rangle =\sum _{\ell =1}^{\infty }\sum _{m=-\ell }^{m=\ell }\left|Y_{m}^{\ell }\right\rangle \left\langle Y_{m}^{\ell }|f\right\rangle ,\qquad f(\theta ,\phi )=\sum _{\ell =1}^{\infty }\sum _{m=-\ell }^{m=\ell }f_{\ell m}Y_{m}^{\ell }(\theta ,\phi ),}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(H2)</td></tr></tbody></table> where the expansion coefficients are given by <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\ell m}=\left\langle Y_{m}^{\ell },f\right\rangle =\int _{\mathbf {S} ^{2}}{\overline {Y_{m}^{\ell }}}f\,d\Omega =\int _{0}^{2\pi }\int _{0}^{\pi }{\overline {Y_{m}^{\ell }}}(\theta ,\phi )f(\theta ,\phi )\sin \theta \,d\theta \,d\phi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>⟨</mo> <mrow> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msubsup> <mo>,</mo> <mi>f</mi> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msubsup> <mo accent="false">¯</mo> </mover> </mrow> <mi>f</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msubsup> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ϕ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\ell m}=\left\langle Y_{m}^{\ell },f\right\rangle =\int _{\mathbf {S} ^{2}}{\overline {Y_{m}^{\ell }}}f\,d\Omega =\int _{0}^{2\pi }\int _{0}^{\pi }{\overline {Y_{m}^{\ell }}}(\theta ,\phi )f(\theta ,\phi )\sin \theta \,d\theta \,d\phi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce2a69e69fff86e2f243bfb7514a82d5f64c350" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:66.801ex; height:6.176ex;" alt="{\displaystyle f_{\ell m}=\left\langle Y_{m}^{\ell },f\right\rangle =\int _{\mathbf {S} ^{2}}{\overline {Y_{m}^{\ell }}}f\,d\Omega =\int _{0}^{2\pi }\int _{0}^{\pi }{\overline {Y_{m}^{\ell }}}(\theta ,\phi )f(\theta ,\phi )\sin \theta \,d\theta \,d\phi .}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(H3)</td></tr></tbody></table> The Lorentz group action restricts to that of SO(3) and is expressed as <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Pi (R)f)(\theta (x),\phi (x))=\sum _{\ell =1}^{\infty }\sum _{m=-\ell }^{m=\ell }\sum _{m'=-\ell }^{m'=\ell }D_{mm'}^{(\ell )}(R)f_{\ell m'}Y_{m}^{\ell }\left(\theta \left(R^{-1}x\right),\phi \left(R^{-1}x\right)\right),\qquad R\in \operatorname {SO} (3),\quad x\in \mathbf {S} ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi>ℓ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mi>ℓ</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>m</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <mi>ℓ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>m</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>ℓ</mi> </mrow> </munderover> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <msup> <mi>m</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>ℓ</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <msup> <mi>m</mi> <mo>′</mo> </msup> </mrow> </msub> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mi>ϕ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>R</mi> <mo>∈</mo> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Pi (R)f)(\theta (x),\phi (x))=\sum _{\ell =1}^{\infty }\sum _{m=-\ell }^{m=\ell }\sum _{m'=-\ell }^{m'=\ell }D_{mm'}^{(\ell )}(R)f_{\ell m'}Y_{m}^{\ell }\left(\theta \left(R^{-1}x\right),\phi \left(R^{-1}x\right)\right),\qquad R\in \operatorname {SO} (3),\quad x\in \mathbf {S} ^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc3368a8ffafb5960be67166351209e979b6569d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:103.48ex; height:7.843ex;" alt="{\displaystyle (\Pi (R)f)(\theta (x),\phi (x))=\sum _{\ell =1}^{\infty }\sum _{m=-\ell }^{m=\ell }\sum _{m'=-\ell }^{m'=\ell }D_{mm'}^{(\ell )}(R)f_{\ell m'}Y_{m}^{\ell }\left(\theta \left(R^{-1}x\right),\phi \left(R^{-1}x\right)\right),\qquad R\in \operatorname {SO} (3),\quad x\in \mathbf {S} ^{2}.}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(H4)</td></tr></tbody></table> This action is unitary, meaning that <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \Pi (R)f,\Pi (R)g\rangle =\langle f,g\rangle \qquad \forall f,g\in \mathbf {S} ^{2},\quad \forall R\in \operatorname {SO} (3).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo>,</mo> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mi>R</mi> <mo>∈</mo> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \Pi (R)f,\Pi (R)g\rangle =\langle f,g\rangle \qquad \forall f,g\in \mathbf {S} ^{2},\quad \forall R\in \operatorname {SO} (3).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bcbe64a890bd54a4b6bedb81b97cec938975ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.926ex; height:3.176ex;" alt="{\displaystyle \langle \Pi (R)f,\Pi (R)g\rangle =\langle f,g\rangle \qquad \forall f,g\in \mathbf {S} ^{2},\quad \forall R\in \operatorname {SO} (3).}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(H5)</td></tr></tbody></table> The D(ℓ) can be obtained from the D(m, n) of above using <a href="/wiki/Clebsch%E2%80%93Gordan_coefficients" title="Clebsch–Gordan coefficients">Clebsch–Gordan decomposition</a>, but they are more easily directly expressed as an exponential of an odd-dimensional su(2)-representation (the 3-dimensional one is exactly 𝖘𝖔(3)).<a href="#cite_note-25">[20]</a><a href="#cite_note-26">[21]</a> In this case the space L2(S2) decomposes neatly into an infinite direct sum of irreducible odd finite-dimensional representations V2i + 1, i = 0, 1, ... according to<a href="#cite_note-27">[22]</a> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}\left(\mathbf {S} ^{2}\right)=\sum _{i=0}^{\infty }V_{2i+1}\equiv \bigoplus _{i=0}^{\infty }\operatorname {span} \left\{Y_{m}^{2i+1}\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>≡</mo> <munderover> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>span</mi> <mo>⁡</mo> <mrow> <mo>{</mo> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}\left(\mathbf {S} ^{2}\right)=\sum _{i=0}^{\infty }V_{2i+1}\equiv \bigoplus _{i=0}^{\infty }\operatorname {span} \left\{Y_{m}^{2i+1}\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ae5af5894e798d0b87223ab1c3db3b16bb292ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.251ex; height:6.843ex;" alt="{\displaystyle L^{2}\left(\mathbf {S} ^{2}\right)=\sum _{i=0}^{\infty }V_{2i+1}\equiv \bigoplus _{i=0}^{\infty }\operatorname {span} \left\{Y_{m}^{2i+1}\right\}.}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(H6)</td></tr></tbody></table> This is characteristic of infinite-dimensional unitary representations of SO(3). If Π is an infinite-dimensional unitary representation on a <a href="/wiki/Separable_space" title="Separable space">separable</a><a href="#cite_note-28">[nb 6]</a> Hilbert space, then it decomposes as a direct sum of finite-dimensional unitary representations.<a href="#cite_note-Gelfand_M_S-24">[19]</a> Such a representation is thus never irreducible. All irreducible finite-dimensional representations (Π, V) can be made unitary by an appropriate choice of inner product,<a href="#cite_note-Gelfand_M_S-24">[19]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f,g\rangle _{U}\equiv \int _{\operatorname {SO} (3)}\langle \Pi (R)f,\Pi (R)g\rangle \,dg={\frac {1}{8\pi ^{2}}}\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{2\pi }\langle \Pi (R)f,\Pi (R)g\rangle \sin \theta \,d\phi \,d\theta \,d\psi ,\quad f,g\in V,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>≡</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo>,</mo> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>8</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo>,</mo> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ϕ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ψ</mi> <mo>,</mo> <mspace width="1em" /> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle _{U}\equiv \int _{\operatorname {SO} (3)}\langle \Pi (R)f,\Pi (R)g\rangle \,dg={\frac {1}{8\pi ^{2}}}\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{2\pi }\langle \Pi (R)f,\Pi (R)g\rangle \sin \theta \,d\phi \,d\theta \,d\psi ,\quad f,g\in V,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45dbd7a26dd69a84707bfdbe4bf50e407b117bbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:96.892ex; height:6.509ex;" alt="{\displaystyle \langle f,g\rangle _{U}\equiv \int _{\operatorname {SO} (3)}\langle \Pi (R)f,\Pi (R)g\rangle \,dg={\frac {1}{8\pi ^{2}}}\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{2\pi }\langle \Pi (R)f,\Pi (R)g\rangle \sin \theta \,d\phi \,d\theta \,d\psi ,\quad f,g\in V,}"></dd></dl> where the integral is the unique invariant integral over SO(3) normalized to 1, here expressed using the <a href="/wiki/Euler_angles" title="Euler angles">Euler angles</a> parametrization. The inner product inside the integral is any inner product on V. <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=20" title="Edit section: Generalizations">edit</a>]</div> The rotation group generalizes quite naturally to n-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> with its standard Euclidean structure. The group of all proper and improper rotations in n dimensions is called the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> O(n), and the subgroup of proper rotations is called the <a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">special orthogonal group</a> SO(n), which is a <a href="/wiki/Lie_group" title="Lie group">Lie group</a> of dimension n(n − 1)/2. In <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>, one works in a 4-dimensional vector space, known as <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a> rather than 3-dimensional Euclidean space. Unlike Euclidean space, Minkowski space has an inner product with an indefinite <a href="/wiki/Metric_signature" title="Metric signature">signature</a>. However, one can still define generalized rotations which preserve this inner product. Such generalized rotations are known as <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a> and the group of all such transformations is called the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a>. The rotation group SO(3) can be described as a subgroup of <a href="/wiki/SE(3)" class="mw-redirect" title="SE(3)">E+(3)</a>, the <a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean group</a> of <a href="/wiki/Euclidean_group#Direct_and_indirect_isometries" title="Euclidean group">direct isometries</a> of Euclidean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b00b2b4fd27c2cbffa02df568472f77b194a6db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}.}"> This larger group is the group of all motions of a <a href="/wiki/Rigid_body" title="Rigid body">rigid body</a>: each of these is a combination of a rotation about an arbitrary axis and a translation, or put differently, a combination of an element of SO(3) and an arbitrary translation. In general, the rotation group of an object is the <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> within the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For <a href="/wiki/Chirality_(mathematics)" title="Chirality (mathematics)">chiral</a> objects it is the same as the full symmetry group. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=21" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal group</a></li> <li><a href="/wiki/Angular_momentum" title="Angular momentum">Angular momentum</a></li> <li><a href="/wiki/Coordinate_rotation" class="mw-redirect" title="Coordinate rotation">Coordinate rotations</a></li> <li><a href="/wiki/Charts_on_SO(3)" title="Charts on SO(3)">Charts on SO(3)</a></li> <li><a href="/wiki/Representation_of_a_Lie_group#An_example:_The_rotation_group_SO(3)" title="Representation of a Lie group">Representations of SO(3)</a></li> <li><a href="/wiki/Euler_angles" title="Euler angles">Euler angles</a></li> <li><a href="/wiki/Rodrigues%27_rotation_formula" title="Rodrigues' rotation formula">Rodrigues' rotation formula</a></li> <li><a href="/wiki/Infinitesimal_rotation" class="mw-redirect" title="Infinitesimal rotation">Infinitesimal rotation</a></li> <li><a href="/wiki/Pin_group" title="Pin group">Pin group</a></li> <li><a href="/wiki/Quaternions_and_spatial_rotation" title="Quaternions and spatial rotation">Quaternions and spatial rotations</a></li> <li><a href="/wiki/Rigid_body" title="Rigid body">Rigid body</a></li> <li><a href="/wiki/Spherical_harmonics" title="Spherical harmonics">Spherical harmonics</a></li> <li><a href="/wiki/Plane_of_rotation" title="Plane of rotation">Plane of rotation</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a></li> <li><a href="/wiki/Pauli_matrix" class="mw-redirect" title="Pauli matrix">Pauli matrix</a></li> <li><a href="/wiki/Plate_trick" title="Plate trick">Plate trick</a></li> <li><a href="/wiki/Three-dimensional_rotation_operator" class="mw-redirect" title="Three-dimensional rotation operator">Three-dimensional rotation operator</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=22" title="Edit section: Footnotes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-6">'<a href="#cite_ref-6">^</a> This is effected by first applying a rotation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\theta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aedf5ba36e3a9362424d65717feae5cc7cbada2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.112ex; height:2.009ex;" alt="{\displaystyle g_{\theta }}"> through φ about the z-axis to take the x-axis to the line L, the intersection between the planes xy and x'y, the latter being the rotated xy-plane. Then rotate with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\theta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aedf5ba36e3a9362424d65717feae5cc7cbada2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.112ex; height:2.009ex;" alt="{\displaystyle g_{\theta }}"> through θ about L to obtain the new z-axis from the old one, and finally rotate by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\psi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\psi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc8ca165904d0990611ca4aefb77482a62a7e821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.411ex; height:2.343ex;" alt="{\displaystyle g_{\psi }}"> through an angle ψ about the new z-axis, where ψ is the angle between L and the new x-axis. In the equation, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\theta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aedf5ba36e3a9362424d65717feae5cc7cbada2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.112ex; height:2.009ex;" alt="{\displaystyle g_{\theta }}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\psi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\psi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc8ca165904d0990611ca4aefb77482a62a7e821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.411ex; height:2.343ex;" alt="{\displaystyle g_{\psi }}"> are expressed in a temporary rotated basis at each step, which is seen from their simple form. To transform these back to the original basis, observe that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} _{\theta }=g_{\phi }g_{\theta }g_{\phi }^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{\theta }=g_{\phi }g_{\theta }g_{\phi }^{-1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fc73f074699b35ce50916b3cfb95e6fc3ac2510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.97ex; height:3.676ex;" alt="{\displaystyle \mathbf {g} _{\theta }=g_{\phi }g_{\theta }g_{\phi }^{-1}.}"> Here boldface means that the rotation is expressed in the original basis. Likewise, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} _{\psi }=g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }g_{\psi }\left[g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }\right]^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <msup> <mrow> <mo>[</mo> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{\psi }=g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }g_{\psi }\left[g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }\right]^{-1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f0c1b4f9dae05dcdb68b7dbac5631fc70e8cffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.734ex; height:5.176ex;" alt="{\displaystyle \mathbf {g} _{\psi }=g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }g_{\psi }\left[g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }\right]^{-1}.}"></dd></dl> Thus <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} _{\psi }\mathbf {g} _{\theta }\mathbf {g} _{\phi }=g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }g_{\psi }\left[g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }\right]^{-1}*g_{\phi }g_{\theta }g_{\phi }^{-1}*g_{\phi }=g_{\phi }g_{\theta }g_{\psi }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <msup> <mrow> <mo>[</mo> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>∗</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>∗</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{\psi }\mathbf {g} _{\theta }\mathbf {g} _{\phi }=g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }g_{\psi }\left[g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }\right]^{-1}*g_{\phi }g_{\theta }g_{\phi }^{-1}*g_{\phi }=g_{\phi }g_{\theta }g_{\psi }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f40e8079a8f380e4f598c51335f31f825fbfa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.16ex; height:5.176ex;" alt="{\displaystyle \mathbf {g} _{\psi }\mathbf {g} _{\theta }\mathbf {g} _{\phi }=g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }g_{\psi }\left[g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }\right]^{-1}*g_{\phi }g_{\theta }g_{\phi }^{-1}*g_{\phi }=g_{\phi }g_{\theta }g_{\psi }.}"></dd></dl> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> For an alternative derivation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4f1d3d3bf3da64b92af1a1018ce00545808b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.179ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(3)}">, see <a href="/wiki/Classical_group" title="Classical group">Classical group</a>. </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Specifically, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {U}}{\boldsymbol {J}}_{\alpha }{\boldsymbol {U}}^{\dagger }=i{\boldsymbol {L}}_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">U</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <mi>i</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {U}}{\boldsymbol {J}}_{\alpha }{\boldsymbol {U}}^{\dagger }=i{\boldsymbol {L}}_{\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/763718a9caccd90d88c6b227a2b187ea3c0ca7e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.785ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {U}}{\boldsymbol {J}}_{\alpha }{\boldsymbol {U}}^{\dagger }=i{\boldsymbol {L}}_{\alpha }}"> for <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {U}}=\left({\begin{array}{ccc}-{\frac {i}{\sqrt {2}}}&0&{\frac {i}{\sqrt {2}}}\\{\frac {1}{\sqrt {2}}}&0&{\frac {1}{\sqrt {2}}}\\0&i&0\\\end{array}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">U</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {U}}=\left({\begin{array}{ccc}-{\frac {i}{\sqrt {2}}}&0&{\frac {i}{\sqrt {2}}}\\{\frac {1}{\sqrt {2}}}&0&{\frac {1}{\sqrt {2}}}\\0&i&0\\\end{array}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5240ab232bfd7fbe2d8058ef9506c2e5411029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:24.659ex; height:12.176ex;" alt="{\displaystyle {\boldsymbol {U}}=\left({\begin{array}{ccc}-{\frac {i}{\sqrt {2}}}&0&{\frac {i}{\sqrt {2}}}\\{\frac {1}{\sqrt {2}}}&0&{\frac {1}{\sqrt {2}}}\\0&i&0\\\end{array}}\right).}"></dd></dl> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> For a full proof, see <a href="/wiki/Derivative_of_the_exponential_map" title="Derivative of the exponential map">Derivative of the exponential map</a>. Issues of convergence of this series to the correct element of the Lie algebra are here swept under the carpet. Convergence is guaranteed when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|X\|+\|Y\|<\log 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>X</mi> <mo fence="false" stretchy="false">‖</mo> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>Y</mi> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|X\|+\|Y\|<\log 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fc45dd2dad2bbfe24b25c14b20f4e2bc4c159ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.863ex; height:2.843ex;" alt="{\displaystyle \|X\|+\|Y\|<\log 2}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|Z\|<\log 2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>Z</mi> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <mi>log</mi> <mo>⁡</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|Z\|<\log 2.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd02484062e1cb0ff1fdd3bf40ad0de15a9cef1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.272ex; height:2.843ex;" alt="{\displaystyle \|Z\|<\log 2.}"> The series may still converge even if these conditions are not fulfilled. A solution always exists since exp is onto in the cases under consideration. </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> The elements of L2(S2) are actually equivalence classes of functions. two functions are declared equivalent if they differ merely on a set of <a href="/wiki/Measure_zero" class="mw-redirect" title="Measure zero">measure zero</a>. The integral is the Lebesgue integral in order to obtain a complete inner product space. </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> A Hilbert space is separable if and only if it has a countable basis. All separable Hilbert spaces are isomorphic. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=23" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> Jacobson (2009), p. 34, Ex. 14. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> n × n real matrices are identical to linear transformations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> expressed in its <a href="/wiki/Standard_basis" title="Standard basis">standard basis</a>. </li> <li id="cite_note-coxeter-3"><a href="#cite_ref-coxeter_3-0">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFCoxeter1973" class="citation book cs1">Coxeter, H. S. M. (1973). Regular polytopes (Third ed.). New York. p. 53. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-61480-8" title="Special:BookSources/0-486-61480-8"><bdi>0-486-61480-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Regular+polytopes&rft.place=New+York&rft.pages=53&rft.edition=Third&rft.date=1973&rft.isbn=0-486-61480-8&rft.aulast=Coxeter&rft.aufirst=H.+S.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3D+rotation+group" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>) </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Proposition 1.17 </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFRossmann2002">Rossmann 2002</a> p. 95. </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> These expressions were, in fact, seminal in the development of quantum mechanics in the 1930s, cf. Ch III, § 16, B.L. van der Waerden, 1932/1932 </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Proposition 3.24 </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="#CITEREFRossmann2002">Rossmann 2002</a> </li> <li id="cite_note-Engø_2001-11">^ <a href="#cite_ref-Engø_2001_11-0">a</a> <a href="#cite_ref-Engø_2001_11-1">b</a> <a href="#CITEREFEngø2001">Engø 2001</a> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Example 3.27 </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> See <a href="#CITEREFRossmann2002">Rossmann 2002</a>, theorem 3, section 2.2. </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <a href="#CITEREFRossmann2002">Rossmann 2002</a> Section 1.1. </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <a href="#CITEREFHall2003">Hall 2003</a> Theorem 2.27. </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShoemake1992" class="citation cs2">Shoemake, Ken (1992-01-01), Kirk, DAVID (ed.), <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/B9780080507552500361">"III.6 - Uniform Random Rotations"</a>, Graphics Gems III (IBM Version), San Francisco: Morgan Kaufmann, pp. 124–132, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-409673-8" title="Special:BookSources/978-0-12-409673-8"><bdi>978-0-12-409673-8</bdi></a>, retrieved 2022-07-29</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Graphics+Gems+III+%28IBM+Version%29&rft.atitle=III.6+-+Uniform+Random+Rotations&rft.pages=124-132&rft.date=1992-01-01&rft.isbn=978-0-12-409673-8&rft.aulast=Shoemake&rft.aufirst=Ken&rft_id=https%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2FB9780080507552500361&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3D+rotation+group" class="Z3988"> </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <a href="#CITEREFHall2003">Hall 2003</a>, Ch. 3; <a href="#CITEREFVaradarajan1984">Varadarajan 1984</a>, §2.15 </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <a href="#CITEREFCurtrightFairlieZachos2014">Curtright, Fairlie & Zachos 2014</a> Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group. </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> Rodrigues, O. (1840), Des lois géométriques qui régissent les déplacements d'un système solide dans l'espace, et la variation des coordonnées provenant de ses déplacements con- sidérés indépendamment des causes qui peuvent les produire, Journal de Mathématiques Pures et Appliquées de Liouville 5, 380–440. </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> (<a href="#CITEREFGoldsteinPooleSafko2002">Goldstein, Poole & Safko 2002</a>, §4.8) </li> <li id="cite_note-Gelfand_M_S-24">^ <a href="#cite_ref-Gelfand_M_S_24-0">a</a> <a href="#cite_ref-Gelfand_M_S_24-1">b</a> <a href="#cite_ref-Gelfand_M_S_24-2">c</a> <a href="#CITEREFGelfandMinlosShapiro1963">Gelfand, Minlos & Shapiro 1963</a> </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> In Quantum Mechanics – non-relativistic theory by <a href="/wiki/Course_of_Theoretical_Physics" title="Course of Theoretical Physics">Landau and Lifshitz</a> the lowest order D are calculated analytically. </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <a href="#CITEREFCurtrightFairlieZachos2014">Curtright, Fairlie & Zachos 2014</a> A formula for D(ℓ) valid for all ℓ is given. </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <a href="#CITEREFHall2003">Hall 2003</a> Section 4.3.5. </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2>[<a href="/w/index.php?title=3D_rotation_group&action=edit&section=24" title="Edit section: Bibliography">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoas2006" class="citation cs2"><a href="/wiki/Mary_L._Boas" title="Mary L. Boas">Boas, Mary L.</a> (2006), <a href="/wiki/Mathematical_Methods_in_the_Physical_Sciences" title="Mathematical Methods in the Physical Sciences">Mathematical Methods in the Physical Sciences</a> (3rd ed.), John Wiley & sons, pp. 120, 127, 129, 155ff and 535, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0471198260" title="Special:BookSources/978-0471198260"><bdi>978-0471198260</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods+in+the+Physical+Sciences&rft.pages=120%2C+127%2C+129%2C+155ff+and+535&rft.edition=3rd&rft.pub=John+Wiley+%26+sons&rft.date=2006&rft.isbn=978-0471198260&rft.aulast=Boas&rft.aufirst=Mary+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3D+rotation+group" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCurtrightFairlieZachos2014" class="citation cs2"><a href="/wiki/David_Fairlie" title="David Fairlie">Curtright, T. L.</a>; <a href="/wiki/Thomas_Curtright" title="Thomas Curtright">Fairlie, D. B.</a>; <a href="/wiki/Cosmas_Zachos" title="Cosmas Zachos">Zachos, C. K.</a> (2014), "A compact formula for rotations as spin matrix polynomials", SIGMA, 10: 084, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1402.3541">1402.3541</a>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014SIGMA..10..084C">2014SIGMA..10..084C</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3842%2FSIGMA.2014.084">10.3842/SIGMA.2014.084</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18776942">18776942</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIGMA&rft.atitle=A+compact+formula+for+rotations+as+spin+matrix+polynomials&rft.volume=10&rft.pages=084&rft.date=2014&rft_id=info%3Aarxiv%2F1402.3541&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18776942%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.3842%2FSIGMA.2014.084&rft_id=info%3Abibcode%2F2014SIGMA..10..084C&rft.aulast=Curtright&rft.aufirst=T.+L.&rft.au=Fairlie%2C+D.+B.&rft.au=Zachos%2C+C.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3D+rotation+group" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEngø2001" class="citation cs2">Engø, Kenth (2001), "On the BCH-formula in 𝖘𝖔(3)", BIT Numerical Mathematics, 41 (3): 629–632, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1021979515229">10.1023/A:1021979515229</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0006-3835">0006-3835</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:126053191">126053191</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=BIT+Numerical+Mathematics&rft.atitle=On+the+BCH-formula+in+%F0%9D%96%98%F0%9D%96%94%283%29&rft.volume=41&rft.issue=3&rft.pages=629-632&rft.date=2001&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A126053191%23id-name%3DS2CID&rft.issn=0006-3835&rft_id=info%3Adoi%2F10.1023%2FA%3A1021979515229&rft.aulast=Eng%C3%B8&rft.aufirst=Kenth&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3D+rotation+group" class="Z3988"> <a rel="nofollow" class="external autonumber" href="http://www.ii.uib.no/publikasjoner/texrap/pdf/2000-201.pdf">[1]</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGelfandMinlosShapiro1963" class="citation cs2"><a href="/wiki/Israel_Gelfand" title="Israel Gelfand">Gelfand, I.M.</a>; <a href="/wiki/Robert_Adol%27fovich_Minlos" class="mw-redirect" title="Robert Adol'fovich Minlos">Minlos, R.A.</a>; Shapiro, Z.Ya. (1963), Representations of the Rotation and Lorentz Groups and their Applications, New York: Pergamon Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Representations+of+the+Rotation+and+Lorentz+Groups+and+their+Applications&rft.place=New+York&rft.pub=Pergamon+Press&rft.date=1963&rft.aulast=Gelfand&rft.aufirst=I.M.&rft.au=Minlos%2C+R.A.&rft.au=Shapiro%2C+Z.Ya.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3D+rotation+group" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldsteinPooleSafko2002" class="citation cs2"><a href="/wiki/Herbert_Goldstein" title="Herbert Goldstein">Goldstein, Herbert</a>; <a href="/w/index.php?title=Charles_P._Poole&action=edit&redlink=1" class="new" title="Charles P. Poole (page does not exist)">Poole, Charles P.</a>; Safko, John L. (2002), Classical Mechanics (third ed.), <a href="/wiki/Addison_Wesley" class="mw-redirect" title="Addison Wesley">Addison Wesley</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-65702-9" title="Special:BookSources/978-0-201-65702-9"><bdi>978-0-201-65702-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Mechanics&rft.edition=third&rft.pub=Addison+Wesley&rft.date=2002&rft.isbn=978-0-201-65702-9&rft.aulast=Goldstein&rft.aufirst=Herbert&rft.au=Poole%2C+Charles+P.&rft.au=Safko%2C+John+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3D+rotation+group" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2015" class="citation cs2">Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3319134666" title="Special:BookSources/978-3319134666"><bdi>978-3319134666</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+Groups%2C+Lie+Algebras%2C+and+Representations%3A+An+Elementary+Introduction&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer&rft.date=2015&rft.isbn=978-3319134666&rft.aulast=Hall&rft.aufirst=Brian+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3D+rotation+group" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2003" class="citation book cs1">Hall, Brian C. (2003). Lie groups, Lie algebras, and representations : an elementary introduction. Graduate Texts in Mathematics. Vol. 222. New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-40122-9" title="Special:BookSources/0-387-40122-9"><bdi>0-387-40122-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+groups%2C+Lie+algebras%2C+and+representations+%3A+an+elementary+introduction&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=2003&rft.isbn=0-387-40122-9&rft.aulast=Hall&rft.aufirst=Brian+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3D+rotation+group" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacobson2009" class="citation cs2"><a href="/wiki/Nathan_Jacobson" title="Nathan Jacobson">Jacobson, Nathan</a> (2009), Basic algebra, vol. 1 (2nd ed.), Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-47189-1" title="Special:BookSources/978-0-486-47189-1"><bdi>978-0-486-47189-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+algebra&rft.edition=2nd&rft.pub=Dover+Publications&rft.date=2009&rft.isbn=978-0-486-47189-1&rft.aulast=Jacobson&rft.aufirst=Nathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3D+rotation+group" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoshi2007" class="citation cs2">Joshi, A. 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L.</a> (1952), Group Theory and Quantum Mechanics, Springer Publishing, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3642658624" title="Special:BookSources/978-3642658624"><bdi>978-3642658624</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Group+Theory+and+Quantum+Mechanics&rft.pub=Springer+Publishing&rft.date=1952&rft.isbn=978-3642658624&rft.aulast=van+der+Waerden&rft.aufirst=B.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3D+rotation+group" class="Z3988"> (translation of the original 1932 edition, Die Gruppentheoretische Methode in Der Quantenmechanik).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVaradarajan1984" class="citation book cs1">Varadarajan, V. S. (1984). Lie groups, Lie algebras, and their representations. New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90969-1" title="Special:BookSources/978-0-387-90969-1"><bdi>978-0-387-90969-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+groups%2C+Lie+algebras%2C+and+their+representations&rft.place=New+York&rft.pub=Springer-Verlag&rft.date=1984&rft.isbn=978-0-387-90969-1&rft.aulast=Varadarajan&rft.aufirst=V.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3D+rotation+group" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVeltman't_Hooftde_Wit2007" class="citation web cs1"><a href="/wiki/Martinus_Veltman" class="mw-redirect" title="Martinus Veltman">Veltman, M.</a>; <a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft, G.</a>; <a href="/wiki/Bernard_de_Wit" title="Bernard de Wit">de Wit, B.</a> (2007). <a rel="nofollow" class="external text" href="http://www.staff.science.uu.nl/~hooft101/lectures/lieg07.pdf">"Lie Groups in Physics (online lecture)"</a> (PDF). Retrieved 2016-10-24.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Lie+Groups+in+Physics+%28online+lecture%29&rft.date=2007&rft.aulast=Veltman&rft.aufirst=M.&rft.au=%27t+Hooft%2C+G.&rft.au=de+Wit%2C+B.&rft_id=http%3A%2F%2Fwww.staff.science.uu.nl%2F~hooft101%2Flectures%2Flieg07.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3D+rotation+group" class="Z3988">.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=3D_rotation_group&oldid=1254216031">https://en.wikipedia.org/w/index.php?title=3D_rotation_group&oldid=1254216031</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Lie_groups" title="Category:Lie groups">Lie groups</a></li><li><a href="/wiki/Category:Rotational_symmetry" title="Category:Rotational symmetry">Rotational symmetry</a></li><li><a href="/wiki/Category:Rotation_in_three_dimensions" title="Category:Rotation in three dimensions">Rotation in three dimensions</a></li><li><a href="/wiki/Category:Euclidean_solid_geometry" title="Category:Euclidean solid geometry">Euclidean solid geometry</a></li><li><a href="/wiki/Category:3-manifolds" title="Category:3-manifolds">3-manifolds</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">CS1 maint: location missing publisher</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Articles_with_excerpts" title="Category:Articles with excerpts">Articles with excerpts</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 29 October 2024, at 23:22 (UTC).</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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3D rotation group - Wikipedia