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.mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Lens_(optics)" class="mw-redirect" title="Lens (optics)">Lens (optics)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Geometric_lens.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Geometric_lens.gif/220px-Geometric_lens.gif" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/b/bf/Geometric_lens.gif 1.5x" data-file-width="254" data-file-height="184" /></a><figcaption>A lens contained between two circular arcs of radius <span class="texhtml mvar" style="font-style:italic;">R</span>, and centers at <span class="texhtml"><i>O</i><sub>1</sub></span> and <span class="texhtml"><i>O</i><sub>2</sub></span></figcaption></figure> <p>In 2-dimensional <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>lens</b> is a <a href="/wiki/Convex_set" title="Convex set">convex</a> region bounded by two <a href="/wiki/Circular_arc" title="Circular arc">circular arcs</a> joined to each other at their endpoints. In order for this shape to be convex, both arcs must bow outwards (convex-convex). This shape can be formed as the <a href="/wiki/Intersection" title="Intersection">intersection</a> of two <a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">circular disks</a>. It can also be formed as the union of two <a href="/wiki/Circular_segment" title="Circular segment">circular segments</a> (regions between the <a href="/wiki/Chord_(geometry)" title="Chord (geometry)">chord</a> of a circle and the circle itself), joined along a common chord. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Types">Types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lens_(geometry)&action=edit&section=1" title="Edit section: Types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Geometric_lens_examples.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Geometric_lens_examples.png/220px-Geometric_lens_examples.png" decoding="async" width="220" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Geometric_lens_examples.png/330px-Geometric_lens_examples.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Geometric_lens_examples.png/440px-Geometric_lens_examples.png 2x" data-file-width="1105" data-file-height="469" /></a><figcaption>Example of two asymmetric lenses (left and right) and one symmetric lens (in the middle)</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Vesica_piscis_circles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Vesica_piscis_circles.svg/220px-Vesica_piscis_circles.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Vesica_piscis_circles.svg/330px-Vesica_piscis_circles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Vesica_piscis_circles.svg/440px-Vesica_piscis_circles.svg.png 2x" data-file-width="1000" data-file-height="666" /></a><figcaption>The <a href="/wiki/Vesica_piscis" title="Vesica piscis">Vesica piscis</a> is the intersection of two <a href="/wiki/Disk_(geometry)" class="mw-redirect" title="Disk (geometry)">disks</a> with the same radius, R, and with the distance between centers also equal to R.</figcaption></figure> <p>If the two arcs of a lens have equal radius, it is called a <b>symmetric lens</b>, otherwise is an <b>asymmetric lens</b>. </p><p>The <a href="/wiki/Vesica_piscis" title="Vesica piscis">vesica piscis</a> is one form of a symmetric lens, formed by arcs of two circles whose centers each lie on the opposite arc. The arcs meet at angles of 120° at their endpoints. </p> <div class="mw-heading mw-heading2"><h2 id="Area">Area</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lens_(geometry)&action=edit&section=2" title="Edit section: Area"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Symmetric</dt></dl> <p>The <a href="/wiki/Area" title="Area">area</a> of a symmetric lens can be expressed in terms of the radius <i>R</i> and arc lengths <i>θ</i> in radians: </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 A=R^{2}\left(\theta -\sin \theta \right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=R^{2}\left(\theta -\sin \theta \right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ddfd86ef30955965a0e8bae886121c07d570840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.154ex; height:3.176ex;" alt="{\displaystyle A=R^{2}\left(\theta -\sin \theta \right).}" /></span></dd></dl> <dl><dt>Asymmetric</dt></dl> <p>The area of an asymmetric lens formed from circles of radii <i>R</i> and <i>r</i> with distance <i>d</i> between their centers is<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> </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 A=r^{2}\cos ^{-1}\left({\frac {d^{2}+r^{2}-R^{2}}{2dr}}\right)+R^{2}\cos ^{-1}\left({\frac {d^{2}+R^{2}-r^{2}}{2dR}}\right)-2\Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>d</mi> <mi>R</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>2</mn> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=r^{2}\cos ^{-1}\left({\frac {d^{2}+r^{2}-R^{2}}{2dr}}\right)+R^{2}\cos ^{-1}\left({\frac {d^{2}+R^{2}-r^{2}}{2dR}}\right)-2\Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3382e342174ef505f0c61b19ab2ecfd7ff90a60c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.467ex; height:6.343ex;" alt="{\displaystyle A=r^{2}\cos ^{-1}\left({\frac {d^{2}+r^{2}-R^{2}}{2dr}}\right)+R^{2}\cos ^{-1}\left({\frac {d^{2}+R^{2}-r^{2}}{2dR}}\right)-2\Delta }" /></span></dd></dl> <p>where </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 \Delta ={\frac {1}{4}}{\sqrt {(-d+r+R)(d-r+R)(d+r-R)(d+r+R)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mo>−</mo> <mi>d</mi> <mo>+</mo> <mi>r</mi> <mo>+</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>−</mo> <mi>r</mi> <mo>+</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>+</mo> <mi>r</mi> <mo>−</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>+</mo> <mi>r</mi> <mo>+</mo> <mi>R</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ={\frac {1}{4}}{\sqrt {(-d+r+R)(d-r+R)(d+r-R)(d+r+R)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9cb5a10bc38785e944b1ae09389ed2e6ac414fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:57.239ex; height:5.176ex;" alt="{\displaystyle \Delta ={\frac {1}{4}}{\sqrt {(-d+r+R)(d-r+R)(d+r-R)(d+r+R)}}}" /></span></dd></dl> <p>is the <a href="/wiki/Triangle#Using_Heron's_formula" title="Triangle">area of a triangle</a> with sides <i>d</i>, <i>r</i>, and <i>R</i>. </p><p>The two circles overlap 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 d<r+R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo><</mo> <mi>r</mi> <mo>+</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d<r+R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcdbb4e1f168eced9d9167e5db1ad1c3cb854b9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.967ex; height:2.343ex;" alt="{\displaystyle d<r+R}" /></span>. For sufficiently large <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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" /></span>, the coordinate <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span> of the lens centre lies between the coordinates of the two circle centers: </p><p><span typeof="mw:File"><a href="/wiki/File:Two_overlapping_circles_with_large_distance.svg" class="mw-file-description" title="A lens contained between two circular arcs of radii R and r at distance of d"><img alt="A lens contained between two circular arcs of radii R and r at distance of d" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Two_overlapping_circles_with_large_distance.svg/300px-Two_overlapping_circles_with_large_distance.svg.png" decoding="async" width="300" height="241" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Two_overlapping_circles_with_large_distance.svg/450px-Two_overlapping_circles_with_large_distance.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Two_overlapping_circles_with_large_distance.svg/600px-Two_overlapping_circles_with_large_distance.svg.png 2x" data-file-width="858" data-file-height="690" /></a></span> </p><p>For small <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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" /></span> the coordinate <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span> of the lens centre lies outside the line that connects the circle centres: </p><p><span typeof="mw:File"><a href="/wiki/File:Two_overlapping_circles_with_small_distance.svg" class="mw-file-description" title="A lens contained between two circular arcs of radii R and r at distance of d"><img alt="A lens contained between two circular arcs of radii R and r at distance of d" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Two_overlapping_circles_with_small_distance.svg/300px-Two_overlapping_circles_with_small_distance.svg.png" decoding="async" width="300" height="272" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Two_overlapping_circles_with_small_distance.svg/450px-Two_overlapping_circles_with_small_distance.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Two_overlapping_circles_with_small_distance.svg/600px-Two_overlapping_circles_with_small_distance.svg.png 2x" data-file-width="965" data-file-height="876" /></a></span> </p><p>By eliminating <i>y</i> from the circle equations <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 x^{2}+y^{2}=r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37dd4f282df84a83620f71dc52345122e0e3a514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.64ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=r^{2}}" /></span> and <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 (x-d)^{2}+y^{2}=R^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>d</mi> <msup> <mo stretchy="false">)</mo> <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>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-d)^{2}+y^{2}=R^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c71eccc17a84cd132662514eb97492a2d3a922f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.221ex; height:3.176ex;" alt="{\displaystyle (x-d)^{2}+y^{2}=R^{2}}" /></span> the <a href="/wiki/Abscissa_and_ordinate" title="Abscissa and ordinate">abscissa</a> of the intersecting rims 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 x=(d^{2}+r^{2}-R^{2})/(2d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=(d^{2}+r^{2}-R^{2})/(2d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a0b40ed5600a1cc34420b0779ebf95e033b7922" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.462ex; height:3.176ex;" alt="{\displaystyle x=(d^{2}+r^{2}-R^{2})/(2d)}" /></span>.</dd></dl> <p>The sign of <i>x</i>, i.e., <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 d^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef0309e83b9f8917fb33be7c0c04fd6d871a4135" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.272ex; height:2.676ex;" alt="{\displaystyle d^{2}}" /></span> being larger or smaller than <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 R^{2}-r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{2}-r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec5b0edc54ee6d76ce15c5b5232ce5eca1104485" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.762ex; height:2.843ex;" alt="{\displaystyle R^{2}-r^{2}}" /></span>, distinguishes the two cases shown in the images. </p><p>The <a href="/wiki/Abscissa_and_ordinate" title="Abscissa and ordinate">ordinate</a> of the intersection 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={\sqrt {r^{2}-x^{2}}}={\frac {\sqrt {[(R-d)^{2}-r^{2}][r^{2}-(R+d)^{2}]}}{2d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</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> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>−</mo> <mi>d</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>d</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </msqrt> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\sqrt {r^{2}-x^{2}}}={\frac {\sqrt {[(R-d)^{2}-r^{2}][r^{2}-(R+d)^{2}]}}{2d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d68a8f7ed99cb57d2bb945073eb2d9646145cb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:50.005ex; height:6.343ex;" alt="{\displaystyle y={\sqrt {r^{2}-x^{2}}}={\frac {\sqrt {[(R-d)^{2}-r^{2}][r^{2}-(R+d)^{2}]}}{2d}}}" /></span>.</dd></dl> <p>Negative values under the square root indicate that the rims of the two circles do not touch because the circles are too far apart or one circle lies entirely within the other. </p><p>The value under the square root is a biquadratic polynomial of <i>d</i>. The four roots of this polynomial are associated with <i>y=0</i> and with the four values of <i>d</i> where the two circles have only one point in common. </p><p>The angles in the blue triangle of sides <i>d</i>, <i>r</i> and <i>R</i> are </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 \sin a_{r}=y/r;\quad \sin a_{R}=y/R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> <mo>;</mo> <mspace width="1em"></mspace> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>=</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin a_{r}=y/r;\quad \sin a_{R}=y/R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c0ba2a79457d642e8aebdfc9b79dae3c50a7d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.4ex; height:2.843ex;" alt="{\displaystyle \sin a_{r}=y/r;\quad \sin a_{R}=y/R}" /></span></dd></dl> <p>where <i>y</i> is the ordinate of the intersection. The branch of the arcsin 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 a_{r}>\pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>></mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{r}>\pi /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8caec0294557e04ef1dcf42c468424f66b07c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.959ex; height:2.843ex;" alt="{\displaystyle a_{r}>\pi /2}" /></span> is to be taken 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 d^{2}<R^{2}-r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{2}<R^{2}-r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8bb0be1d259019412a869e10c935b173c7b07a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.132ex; height:2.843ex;" alt="{\displaystyle d^{2}<R^{2}-r^{2}}" /></span>. </p><p>The <a href="/wiki/Triangle#Computing_the_area_of_a_triangle" title="Triangle">area</a> of the triangle is <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 \Delta ={\frac {1}{2}}yd}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>y</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ={\frac {1}{2}}yd}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b95a22b5dbee40a1efd45b745827e99a2dfcdf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.404ex; height:5.176ex;" alt="{\displaystyle \Delta ={\frac {1}{2}}yd}" /></span>. </p><p>The area of the asymmetric lens is <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 A=a_{r}r^{2}+a_{R}R^{2}-yd}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>y</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=a_{r}r^{2}+a_{R}R^{2}-yd}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebb83986531c71811272348e6237d88588e68c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.728ex; height:3.009ex;" alt="{\displaystyle A=a_{r}r^{2}+a_{R}R^{2}-yd}" /></span>, where the two angles are measured in radians. [This is an application of the <a href="/wiki/Inclusion-exclusion_principle" class="mw-redirect" title="Inclusion-exclusion principle">Inclusion-exclusion principle</a>: the two circular sectors centered at (0,0) and (d,0) with central angles <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 2a_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf0e0cf35e50941a9e56416b749dfd6d5b3422d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.366ex; height:2.509ex;" alt="{\displaystyle 2a_{r}}" /></span> and <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 2a_{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a_{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/460f1bd6a43490102d35ab04d29d3929490e0652" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.872ex; height:2.509ex;" alt="{\displaystyle 2a_{R}}" /></span> have areas <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 2a_{r}r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a_{r}r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d42fd0e91ec299f00c6416eb344121c33a15c121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.469ex; height:3.009ex;" alt="{\displaystyle 2a_{r}r^{2}}" /></span> and <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 2a_{R}R^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a_{R}R^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e162c7e4689dfb8fec5b12a85a9f3efca3977e62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.69ex; height:3.009ex;" alt="{\displaystyle 2a_{R}R^{2}}" /></span>. Their union covers the triangle, the flipped triangle with corner at (x,-y), and twice the lens area.] </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lens_(geometry)&action=edit&section=3" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A lens with a different shape forms the answer to <a href="/wiki/Mrs._Miniver%27s_problem" title="Mrs. Miniver's problem">Mrs. Miniver's problem</a>, on finding a lens with half the area of the union of the two circles. </p><p>Lenses are used to define <a href="/wiki/Beta_skeleton" title="Beta skeleton">beta skeletons</a>, geometric graphs defined on a set of points by connecting pairs of points by an edge whenever a lens determined by the two points is empty. </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=Lens_(geometry)&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Circle%E2%80%93circle_intersection" class="mw-redirect" title="Circle–circle intersection">Circle–circle intersection</a></li> <li><a href="/wiki/Lune_(geometry)" title="Lune (geometry)">Lune</a>, a related non-convex shape formed by two circular arcs, one bowing outwards and the other inwards</li> <li><a href="/wiki/Lemon_(geometry)" title="Lemon (geometry)">Lemon</a>, created by a lens rotated around an axis through its tips.<sup id="cite_ref-mathworld_2-0" class="reference"><a href="#cite_note-mathworld-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Lemon_(geometry).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Lemon_%28geometry%29.png/220px-Lemon_%28geometry%29.png" decoding="async" width="220" height="106" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Lemon_%28geometry%29.png/330px-Lemon_%28geometry%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Lemon_%28geometry%29.png/440px-Lemon_%28geometry%29.png 2x" data-file-width="476" data-file-height="229" /></a><figcaption>A <a href="/wiki/Lemon_(geometry)" title="Lemon (geometry)">lemon</a>.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lens_(geometry)&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Lens"><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 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navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Convex_analysis_and_variational_analysis" title="Template:Convex analysis and variational analysis"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Convex_analysis_and_variational_analysis" title="Template talk:Convex analysis and variational analysis"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Convex_analysis_and_variational_analysis" title="Special:EditPage/Template:Convex analysis and variational analysis"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Convex_analysis_and_variational_analysis174" style="font-size:114%;margin:0 4em"><a href="/wiki/Convex_analysis" title="Convex analysis">Convex analysis</a> and <a href="/wiki/Variational_analysis" title="Variational analysis">variational analysis</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Convex_combination" title="Convex combination">Convex combination</a></li> <li><a href="/wiki/Convex_function" title="Convex function">Convex function</a></li> <li><a href="/wiki/Convex_set" title="Convex set">Convex set</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_convexity_topics" title="List of convexity topics">Topics (list)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Choquet_theory" title="Choquet theory">Choquet theory</a></li> <li><a href="/wiki/Convex_geometry" title="Convex geometry">Convex geometry</a></li> <li><a href="/wiki/Convex_metric_space" title="Convex metric space">Convex metric space</a></li> <li><a href="/wiki/Convex_optimization" title="Convex optimization">Convex optimization</a></li> <li><a href="/wiki/Duality_(optimization)" title="Duality (optimization)">Duality</a></li> <li><a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li> <li><a href="/wiki/Legendre_transformation" title="Legendre transformation">Legendre transformation</a></li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex topological vector space</a></li> <li><a href="/wiki/Simplex" title="Simplex">Simplex</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Convex_conjugate" title="Convex conjugate">Convex conjugate</a></li> <li><a href="/wiki/Concave_function" title="Concave function">Concave</a></li> <li>(<a href="/wiki/Closed_convex_function" title="Closed convex function">Closed</a></li> <li><a href="/wiki/K-convex_function" title="K-convex function">K-</a></li> <li><a href="/wiki/Logarithmically_convex_function" title="Logarithmically convex function">Logarithmically</a></li> <li><a href="/wiki/Proper_convex_function" title="Proper convex function">Proper</a></li> <li><a href="/wiki/Pseudoconvex_function" title="Pseudoconvex function">Pseudo-</a></li> <li><a href="/wiki/Quasiconvex_function" title="Quasiconvex function">Quasi-</a>) <a href="/wiki/Convex_function" title="Convex function">Convex function</a></li> <li><a href="/wiki/Invex_function" title="Invex function">Invex function</a></li> <li><a href="/wiki/Legendre_transformation" title="Legendre transformation">Legendre transformation</a></li> <li><a href="/wiki/Semi-continuity" title="Semi-continuity">Semi-continuity</a></li> <li><a href="/wiki/Subderivative" title="Subderivative">Subderivative</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main <a href="/wiki/Category:Theorems_involving_convexity" title="Category:Theorems involving convexity">results (list)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carath%C3%A9odory%27s_theorem_(convex_hull)" title="Carathéodory's theorem (convex hull)">Carathéodory's theorem</a></li> <li><a href="/wiki/Ekeland%27s_variational_principle" title="Ekeland's variational principle">Ekeland's variational principle</a></li> <li><a href="/wiki/Fenchel%E2%80%93Moreau_theorem" title="Fenchel–Moreau theorem">Fenchel–Moreau theorem</a></li> <li><a href="/wiki/Fenchel-Young_inequality" class="mw-redirect" title="Fenchel-Young inequality">Fenchel-Young inequality</a></li> <li><a href="/wiki/Jensen%27s_inequality" title="Jensen's inequality">Jensen's inequality</a></li> <li><a href="/wiki/Hermite%E2%80%93Hadamard_inequality" title="Hermite–Hadamard inequality">Hermite–Hadamard inequality</a></li> <li><a href="/wiki/Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman theorem</a></li> <li><a href="/wiki/Mazur%27s_lemma" title="Mazur's lemma">Mazur's lemma</a></li> <li><a href="/wiki/Shapley%E2%80%93Folkman_lemma" title="Shapley–Folkman lemma">Shapley–Folkman lemma</a></li> <li><a href="/wiki/Ursescu_theorem#Robinson–Ursescu_theorem" title="Ursescu theorem">Robinson–Ursescu</a></li> <li><a href="/wiki/Ursescu_theorem#Simons'_theorem" title="Ursescu theorem">Simons</a></li> <li><a href="/wiki/Ursescu_theorem" title="Ursescu theorem">Ursescu</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Convex_hull" title="Convex hull">Convex hull</a></li> <li>(<a href="/wiki/Orthogonally_convex_set" class="mw-redirect" title="Orthogonally convex set">Orthogonally</a>, <a href="/wiki/Pseudoconvexity" title="Pseudoconvexity">Pseudo-</a>) <a href="/wiki/Convex_set" title="Convex set">Convex set</a></li> <li><a href="/wiki/Effective_domain" title="Effective domain">Effective domain</a></li> <li><a href="/wiki/Epigraph_(mathematics)" title="Epigraph (mathematics)">Epigraph</a></li> <li><a href="/wiki/Hypograph_(mathematics)" title="Hypograph (mathematics)">Hypograph</a></li> <li><a href="/wiki/John_ellipsoid" title="John ellipsoid">John ellipsoid</a></li> <li><a class="mw-selflink selflink">Lens</a></li> <li><a href="/wiki/Radial_set" title="Radial set">Radial set</a>/<a href="/wiki/Algebraic_interior" title="Algebraic interior">Algebraic interior</a></li> <li><a href="/wiki/Zonotope" class="mw-redirect" title="Zonotope">Zonotope</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Series</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">Convex series related</a> (<a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">(cs, lcs)-closed</a>, <a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">(cs, bcs)-complete</a>, <a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">(lower) ideally convex</a>, <a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">(H<i>x</i>)</a>, and <a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">(Hw<i>x</i>)</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Duality_(optimization)" title="Duality (optimization)">Duality</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_system" title="Dual system">Dual system</a></li> <li><a href="/wiki/Duality_gap" title="Duality gap">Duality gap</a></li> <li><a href="/wiki/Strong_duality" title="Strong duality">Strong duality</a></li> <li><a href="/wiki/Weak_duality" title="Weak duality">Weak duality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications and related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Convexity_in_economics" title="Convexity in economics">Convexity in economics</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" 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=Lens_(geometry)&oldid=1239970216">https://en.wikipedia.org/w/index.php?title=Lens_(geometry)&oldid=1239970216</a>"</div></div> <div 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