Scaling (geometry)

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href="https://eu.wikipedia.org/wiki/Eskalatze_(matematika)" title="Eskalatze (matematika) – Basque" lang="eu" hreflang="eu" data-title="Eskalatze (matematika)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%AC%D8%A7%D9%86%D8%B3" title="تجانس – Persian" lang="fa" hreflang="fa" data-title="تجانس" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Agrandissement_et_r%C3%A9duction" title="Agrandissement et réduction – French" lang="fr" hreflang="fr" data-title="Agrandissement et réduction" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Scalare_(geometrie)" title="Scalare (geometrie) – Romanian" lang="ro" hreflang="ro" data-title="Scalare (geometrie)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%85%E0%AE%B3%E0%AE%B5%E0%AF%81%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D_(%E0%AE%B5%E0%AE%9F%E0%AE%BF%E0%AE%B5%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D)" title="அளவுமாற்றம் (வடிவவியல்) – Tamil" lang="ta" hreflang="ta" data-title="அளவுமாற்றம் (வடிவவியல்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%B8%AE%E6%94%BE" title="縮放 – Cantonese" 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srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Scaling_(geometry)" title="Special:EditPage/Scaling (geometry)">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Scaling%22+geometry">"Scaling" geometry</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Scaling%22+geometry+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Scaling%22+geometry&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Scaling%22+geometry+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Scaling%22+geometry">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Scaling%22+geometry&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">April 2008</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sierpinski_triangle_evolution.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Sierpinski_triangle_evolution.svg/220px-Sierpinski_triangle_evolution.svg.png" decoding="async" width="220" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Sierpinski_triangle_evolution.svg/330px-Sierpinski_triangle_evolution.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Sierpinski_triangle_evolution.svg/440px-Sierpinski_triangle_evolution.svg.png 2x" data-file-width="680" data-file-height="111" /></a><figcaption>Each iteration of the <a href="/wiki/Sierpinski_triangle" class="mw-redirect" title="Sierpinski triangle">Sierpinski triangle</a> contains triangles related to the next iteration by a scale factor of 1/2</figcaption></figure> <p>In <a href="/wiki/Affine_geometry" title="Affine geometry">affine geometry</a>, <b>uniform scaling</b> (or <b>isotropic scaling</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>) is a <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> that enlarges (increases) or shrinks (diminishes) objects by a <i><a href="/wiki/Scale_factor" class="mw-redirect" title="Scale factor">scale factor</a></i> that is the same in all directions (<a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropically</a>). The result of uniform scaling is <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a> (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a <a href="/wiki/Photograph" title="Photograph">photograph</a>, or when creating a <a href="/wiki/Scale_model" title="Scale model">scale model</a> of a building, car, airplane, etc. </p><p>More general is <b>scaling</b> with a separate scale factor for each axis direction. <b>Non-uniform scaling</b> (<b><a href="/wiki/Anisotropic" class="mw-redirect" title="Anisotropic">anisotropic</a> scaling</b>) is obtained when at least one of the scaling factors is different from the others; a special case is <b>directional scaling</b> or <b>stretching</b> (in one direction). Non-uniform scaling changes the <a href="/wiki/Shape" title="Shape">shape</a> of the object; e.g. a square may change into a rectangle, or into a parallelogram if the sides of the square are not parallel to the scaling axes (the angles between lines parallel to the axes are preserved, but not all angles). It occurs, for example, when a faraway billboard is viewed from an <a href="/wiki/Oblique_angle" class="mw-redirect" title="Oblique angle">oblique angle</a>, or when the shadow of a flat object falls on a surface that is not parallel to it. </p><p>When the scale factor is larger than 1, (uniform or non-uniform) scaling is sometimes also called <b>dilation</b> or <b>enlargement</b>. When the scale factor is a positive number smaller than 1, scaling is sometimes also called <b>contraction</b> or <b>reduction</b>. </p><p>In the most general sense, a scaling includes the case in which the directions of scaling are not perpendicular. It also includes the case in which one or more scale factors are equal to zero (<a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">projection</a>), and the case of one or more negative scale factors (a directional scaling by -1 is equivalent to a <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a>). </p><p>Scaling is a <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a>, and a special case of <a href="/wiki/Homothetic_transformation" class="mw-redirect" title="Homothetic transformation">homothetic transformation</a> (scaling about a point). In most cases, the homothetic transformations are non-linear transformations. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Uniform_scaling">Uniform scaling</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scaling_(geometry)&action=edit&section=1" title="Edit section: Uniform scaling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>scale factor</b> is usually a decimal which scales, or multiplies, some quantity. In the equation <i>y</i> = <i>Cx</i>, <i>C</i> is the scale factor for <i>x</i>. <i>C</i> is also the <a href="/wiki/Coefficient" title="Coefficient">coefficient</a> of <i>x</i>, and may be called the <a href="/wiki/Constant_of_proportionality" class="mw-redirect" title="Constant of proportionality">constant of proportionality</a> of <i>y</i> to <i>x</i>. For example, doubling distances corresponds to a scale factor of two for distance, while cutting a cake in half results in pieces with a scale factor for volume of one half. The basic equation for it is image over preimage. </p><p>In the field of measurements, the scale factor of an instrument is sometimes referred to as sensitivity. The ratio of any two corresponding lengths in two similar geometric figures is also called a scale. </p> <div class="mw-heading mw-heading2"><h2 id="Matrix_representation">Matrix representation<span class="anchor" id="Matrix"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scaling_(geometry)&action=edit&section=2" title="Edit section: Matrix representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A scaling can be represented by a scaling <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>. To scale an object by a <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vector</a> <i>v</i> = (<i>v<sub>x</sub>, v<sub>y</sub>, v<sub>z</sub></i>), each point <i>p</i> = (<i>p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub></i>) would need to be multiplied with this scaling matrix: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4561ec79e4ad2fb898cf7b2f4ec0873a01fb31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:21.303ex; height:9.509ex;" alt="{\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}.}"></span></dd></dl> <p>As shown below, the multiplication will give the expected result: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b981072c2d00c1ce373cd483c00fd6d927f668" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:40.26ex; height:9.509ex;" alt="{\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\end{bmatrix}}.}"></span></dd></dl> <p>Such a scaling changes the <a href="/wiki/Diameter" title="Diameter">diameter</a> of an object by a factor between the scale factors, the <a href="/wiki/Area" title="Area">area</a> by a factor between the smallest and the largest product of two scale factors, and the <a href="/wiki/Volume" title="Volume">volume</a> by the product of all three. </p><p>The scaling is uniform <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the scaling factors are equal (<i>v<sub>x</sub> = v<sub>y</sub> = v<sub>z</sub></i>). If all except one of the scale factors are equal to 1, we have directional scaling. </p><p>In the case where <i>v<sub>x</sub> = v<sub>y</sub> = v<sub>z</sub> = k</i>, scaling increases the area of any surface by a factor of <i>k</i><sup>2</sup> and the volume of any solid object by a factor of <i>k</i><sup>3</sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Scaling_in_arbitrary_dimensions">Scaling in arbitrary dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scaling_(geometry)&action=edit&section=3" title="Edit section: Scaling in arbitrary dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-dimensional space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}"></span>, uniform scaling by a factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is accomplished by <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, that is, multiplying each coordinate of each point by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>. As a special case of linear transformation, it can be achieved also by multiplying each point (viewed as a column vector) with a <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a> whose entries on the diagonal are all equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, namely <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle vI}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle vI}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfeb357337ab60f8563b4823b52b4f0759d7e2f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.299ex; height:2.176ex;" alt="{\displaystyle vI}"></span> . </p><p>Non-uniform scaling is accomplished by multiplication with any <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrix</a>. The <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of the matrix are the scale factors, and the corresponding <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a> are the axes along which each scale factor applies. A special case is a diagonal matrix, with arbitrary numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1},v_{2},\ldots v_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1},v_{2},\ldots v_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a83425826075fe66bf9f8426ffeb4aaa5102366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.888ex; height:2.009ex;" alt="{\displaystyle v_{1},v_{2},\ldots v_{n}}"></span> along the diagonal: the axes of scaling are then the coordinate axes, and the transformation scales along each axis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> by the factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dffe5726650f6daac54829972a94f38eb8ec127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}"></span>. </p><p>In uniform scaling with a non-zero scale factor, all non-zero vectors retain their direction (as seen from the origin), or all have the direction reversed, depending on the sign of the scaling factor. In non-uniform scaling only the vectors that belong to an <a href="/wiki/Eigenspace" class="mw-redirect" title="Eigenspace">eigenspace</a> will retain their direction. A vector that is the sum of two or more non-zero vectors belonging to different eigenspaces will be tilted towards the eigenspace with largest eigenvalue. </p> <div class="mw-heading mw-heading2"><h2 id="Using_homogeneous_coordinates">Using homogeneous coordinates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scaling_(geometry)&action=edit&section=4" title="Edit section: Using homogeneous coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a>, often used in <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>, points are represented using <a href="/wiki/Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a>. To scale an object by a <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vector</a> <i>v</i> = (<i>v<sub>x</sub>, v<sub>y</sub>, v<sub>z</sub></i>), each homogeneous coordinate vector <i>p</i> = (<i>p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub></i>, 1) would need to be multiplied with this <a href="/wiki/Projective_transformation" class="mw-redirect" title="Projective transformation">projective transformation</a> matrix: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </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>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba17744c20c70ba66748f68725966e236de30eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:24.789ex; height:12.843ex;" alt="{\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}.}"></span></dd></dl> <p>As shown below, the multiplication will give the expected result: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\\1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </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>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\\1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/539be147e9c632c58c8a87e4ecf6c2f3d4f857db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:43.745ex; height:12.843ex;" alt="{\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\\1\end{bmatrix}}.}"></span></dd></dl> <p>Since the last component of a homogeneous coordinate can be viewed as the denominator of the other three components, a uniform scaling by a common factor <i>s</i> (uniform scaling) can be accomplished by using this scaling matrix: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{v}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</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> <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> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{v}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2ac4cca4d86727353d98875c9208b7e434bca8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:22.165ex; height:13.176ex;" alt="{\displaystyle S_{v}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}.}"></span></dd></dl> <p>For each vector <i>p</i> = (<i>p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub></i>, 1) we would have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{v}p={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\{\frac {1}{s}}\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mi>p</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> <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> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{v}p={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\{\frac {1}{s}}\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c88fc85c0623bcfb8533ea577f3b210174d3921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:38.821ex; height:13.509ex;" alt="{\displaystyle S_{v}p={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\{\frac {1}{s}}\end{bmatrix}},}"></span></dd></dl> <p>which would be equivalent to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}sp_{x}\\sp_{y}\\sp_{z}\\1\end{bmatrix}}.}"> <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>s</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}sp_{x}\\sp_{y}\\sp_{z}\\1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5117bb588482f8e8ceb7a94ba373faa08bb615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:7.931ex; height:12.843ex;" alt="{\displaystyle {\begin{bmatrix}sp_{x}\\sp_{y}\\sp_{z}\\1\end{bmatrix}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Function_dilation_and_contraction">Function dilation and contraction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scaling_(geometry)&action=edit&section=5" title="Edit section: Function dilation and contraction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b3d8f37f5458c22b61eaf26e5af0523acb63e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.074ex; height:2.843ex;" alt="{\displaystyle P(x,y)}"></span>, the dilation associates it with the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'(x',y')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P'(x',y')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ef542f849d3fb2e73ff45a8706615db44d47797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.209ex; height:3.009ex;" alt="{\displaystyle P'(x',y')}"></span> through the equations </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}x'=mx\\y'=ny\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>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>m</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>n</mi> <mi>y</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}x'=mx\\y'=ny\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2d6e3f3b53239e5b1fe063fbf06cef61d66688" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.978ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}x'=mx\\y'=ny\end{cases}}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n\in \mathbb {R} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m,n\in \mathbb {R} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c237b7356bbd82218545e71dea5f732c9fadc763" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.499ex; height:2.843ex;" alt="{\displaystyle m,n\in \mathbb {R} ^{+}}"></span>.</dd></dl> <p>Therefore, given a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"></span>, the equation of the dilated function is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=nf\left({\frac {x}{m}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>n</mi> <mi>f</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>m</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=nf\left({\frac {x}{m}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c01749716b8057353ed62deae0771511d2d933b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14ex; height:4.843ex;" alt="{\displaystyle y=nf\left({\frac {x}{m}}\right).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Particular_cases">Particular cases</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scaling_(geometry)&action=edit&section=6" title="Edit section: Particular cases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}"></span>, the transformation is horizontal; when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f27527902d05e4c32bcbe28d425d7790f8ae191" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m>1}"></span>, it is a dilation, when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93622197d9405d5f48e765b73f7dee1375d235c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m<1}"></span>, it is a contraction. </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6100c5ebd48c6fd848709f2be624465203eb173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=1}"></span>, the transformation is vertical; when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}"></span> it is a dilation, when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dffd3f7a2c3b20dc8e615dfd85f820287a8fc06b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n<1}"></span>, it is a contraction. </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=1/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=1/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e790a49b1854fd6fea537626e48924d3be147bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.858ex; height:2.843ex;" alt="{\displaystyle m=1/n}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1/m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1/m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d878944b5631971a2f551fc66798b7aea9502ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.858ex; height:2.843ex;" alt="{\displaystyle n=1/m}"></span>, the transformation is a <a href="/wiki/Squeeze_mapping" title="Squeeze mapping">squeeze mapping</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scaling_(geometry)&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output 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class="citation web cs1">Durand; Cutler. <a rel="nofollow" class="external text" href="http://groups.csail.mit.edu/graphics/classes/6.837/F03/lectures/04_transformations.ppt">"Transformations"</a> <span class="cs1-format">(PowerPoint)</span>. 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Retrieved <span class="nowrap">12 September</span> 2008</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Transformations&rft.pub=Massachusetts+Institute+of+Technology&rft.au=Durand&rft.au=Cutler&rft_id=http%3A%2F%2Fgroups.csail.mit.edu%2Fgraphics%2Fclasses%2F6.837%2FF03%2Flectures%2F04_transformations.ppt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScaling+%28geometry%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scaling_(geometry)&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/3D_projection" title="3D projection">3D projection</a></li> <li><a href="/wiki/3D_rendering" title="3D rendering">3D rendering</a></li> <li>(<a href="/wiki/Image-based_modeling_and_rendering" title="Image-based modeling and rendering">Image-based</a></li> <li><a href="/wiki/Spectral_rendering" title="Spectral rendering">Spectral</a></li> <li><a href="/wiki/Unbiased_rendering" title="Unbiased rendering">Unbiased</a>)</li> <li><a href="/wiki/Aliasing" title="Aliasing">Aliasing</a></li> <li><a href="/wiki/Anisotropic_filtering" title="Anisotropic filtering">Anisotropic filtering</a></li> <li><a href="/wiki/Cel_shading" title="Cel shading">Cel shading</a></li> <li><a href="/wiki/Fluid_animation" title="Fluid animation">Fluid animation</a></li> <li><a href="/wiki/Computer_graphics_lighting" title="Computer graphics lighting">Lighting</a> <ul><li><a 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transformation">Affine transformation</a></li> <li><a href="/wiki/Back-face_culling" title="Back-face culling">Back-face culling</a></li> <li><a href="/wiki/Clipping_(computer_graphics)" title="Clipping (computer graphics)">Clipping</a></li> <li><a href="/wiki/Collision_detection" title="Collision detection">Collision detection</a></li> <li><a href="/wiki/Planar_projection" title="Planar projection">Planar projection</a></li> <li><a href="/wiki/Reflection_(computer_graphics)" title="Reflection (computer graphics)">Reflection</a></li> <li><a href="/wiki/Rendering_(computer_graphics)" title="Rendering (computer graphics)">Rendering</a> <ul><li><a href="/wiki/Beam_tracing" title="Beam tracing">Beam tracing</a></li> <li><a href="/wiki/Cone_tracing" title="Cone tracing">Cone tracing</a></li> <li><a href="/wiki/Checkerboard_rendering" title="Checkerboard rendering">Checkerboard rendering</a></li> <li><a href="/wiki/Ray_tracing_(graphics)" title="Ray tracing (graphics)">Ray tracing</a></li> <li><a href="/wiki/Path_tracing" title="Path tracing">Path tracing</a></li> <li><a href="/wiki/Ray_casting" title="Ray casting">Ray casting</a></li> <li><a href="/wiki/Scanline_rendering" title="Scanline rendering">Scanline rendering</a></li></ul></li> <li><a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">Rotation</a></li> <li><a class="mw-selflink selflink">Scaling</a></li> <li><a href="/wiki/Shadow_mapping" title="Shadow mapping">Shadow mapping</a></li> <li><a href="/wiki/Shadow_volume" title="Shadow volume">Shadow volume</a></li> <li><a href="/wiki/Shear_matrix" class="mw-redirect" title="Shear matrix">Shear matrix</a></li> <li><a href="/wiki/Shader" title="Shader">Shader</a></li> <li><a href="/wiki/Texel_(graphics)" title="Texel (graphics)">Texel</a></li> <li><a href="/wiki/Translation_(geometry)" title="Translation (geometry)">Translation</a></li> <li><a href="/wiki/Volume_rendering" title="Volume rendering">Volume rendering</a></li> <li><a href="/wiki/Voxel" title="Voxel">Voxel</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Graphics_software" title="Graphics software">Graphics software</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_3D_computer_graphics_software" title="List of 3D computer graphics software">3D computer graphics software</a> <ul><li><a href="/wiki/List_of_3D_animation_software" title="List of 3D animation software">animation</a></li> <li><a href="/wiki/List_of_3D_modeling_software" title="List of 3D modeling software">modeling</a></li> <li><a href="/wiki/List_of_3D_rendering_software" title="List of 3D rendering software">rendering</a></li></ul></li> <li><a href="/wiki/Raster_graphics_editor" title="Raster graphics editor">Raster graphics editors</a></li> <li><a href="/wiki/Comparison_of_vector_graphics_editors" title="Comparison of vector graphics editors">Vector graphics editors</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algorithms</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_computer_graphics_algorithms" class="mw-redirect" title="List of computer graphics algorithms">List of computer graphics algorithms</a></li></ul> </div></td></tr></tbody></table></div>    </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=Scaling_(geometry)&oldid=1253327046">https://en.wikipedia.org/w/index.php?title=Scaling_(geometry)&oldid=1253327046</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" 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Scaling (geometry) - Wikipedia