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<title>Zernike-Polynom</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Zernike-Polynom</h1>

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Die Zernike-Polynome sind nach <a href="Frits_Zernike" title="Frits Zernike">Frits Zernike</a> benannte <a href="Orthogonale_Polynome" title="Orthogonale Polynome">orthogonale Polynome</a> auf dem <a href="Einheitskreis" title="Einheitskreis">Einheitskreis</a>. Sie sind <a href="Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> bezüglich des <a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukts</a>

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<annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle =\iint _{x^{2}+y^{2}\leq 1}f(x,y)g(x,y)\mathrm {d} x\mathrm {d} y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/983eb2dde656607141864202905767deb38285fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.399ex; height:6.009ex;" alt="{\displaystyle \langle f,g\rangle =\iint _{x^{2}+y^{2}\leq 1}f(x,y)g(x,y)\mathrm {d} x\mathrm {d} y}" loading="lazy">

in <a href="Kartesische_Koordinaten" class="mw-redirect" title="Kartesische Koordinaten">kartesischen</a> beziehungsweise

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<annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle =\int _{\phi =0}^{2\pi }\int _{\rho =0}^{1}f(\rho ,\phi )g(\rho ,\phi )\rho \mathrm {d} \rho \mathrm {d} \phi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a71a476e8f0cdd06eadc107ef70602eff1fb18da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.821ex; height:6.509ex;" alt="{\displaystyle \langle f,g\rangle =\int _{\phi =0}^{2\pi }\int _{\rho =0}^{1}f(\rho ,\phi )g(\rho ,\phi )\rho \mathrm {d} \rho \mathrm {d} \phi }" loading="lazy">

in <a href="Polarkoordinaten" title="Polarkoordinaten">Polarkoordinaten</a>.

Die Zernike Polynome spielen insbesondere in der <a href="Wellenoptik" title="Wellenoptik">Wellenoptik</a> eine wichtige Rolle. Es gibt bezüglich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> <a href="Gerade_und_ungerade_Funktionen" title="Gerade und ungerade Funktionen">gerade und ungerade</a> Zernike-Polynome.

Die geraden Zernike-Polynome sind definiert durch:

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<annotation encoding="application/x-tex">{\displaystyle Z_{n}^{+m}(\rho ,\phi )=R_{n}^{m}(\rho )\,\cos m\phi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b75d0bc7da432549d86d32969462ffa6c270f5fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.34ex; height:3.009ex;" alt="{\displaystyle Z_{n}^{+m}(\rho ,\phi )=R_{n}^{m}(\rho )\,\cos m\phi }" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{n}^{+m}(\rho ,-\phi )=Z_{n}^{+m}(\rho ,\phi )}">

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<annotation encoding="application/x-tex">{\displaystyle Z_{n}^{+m}(\rho ,-\phi )=Z_{n}^{+m}(\rho ,\phi )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a01dfe634aaa7b80105b6a4bbc72f7c789e3091c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.092ex; height:3.009ex;" alt="{\displaystyle Z_{n}^{+m}(\rho ,-\phi )=Z_{n}^{+m}(\rho ,\phi )}" loading="lazy">,</dd></dl>

und die ungeraden durch

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<annotation encoding="application/x-tex">{\displaystyle Z_{n}^{-m}(\rho ,\phi )=R_{n}^{m}(\rho )\,\sin m\phi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b81c4920ff426480c15138768ee8d1281aa9e916.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.084ex; height:3.009ex;" alt="{\displaystyle Z_{n}^{-m}(\rho ,\phi )=R_{n}^{m}(\rho )\,\sin m\phi }" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{n}^{-m}(\rho ,-\phi )=-Z_{n}^{-m}(\rho ,\phi )}">

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<annotation encoding="application/x-tex">{\displaystyle Z_{n}^{-m}(\rho ,-\phi )=-Z_{n}^{-m}(\rho ,\phi )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d7193fbc7fe6bd8bf1c7ae30c3ca8638f0894c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.9ex; height:3.009ex;" alt="{\displaystyle Z_{n}^{-m}(\rho ,-\phi )=-Z_{n}^{-m}(\rho ,\phi )}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy"> nichtnegative ganze Zahlen sind. Zusätzlich wird gefordert, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\leq n}">

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<annotation encoding="application/x-tex">{\displaystyle m\leq n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0017737947454c2911336b2d038c91f5a7a70bc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.534ex; height:2.176ex;" alt="{\displaystyle m\leq n}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-m}">

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<annotation encoding="application/x-tex">{\displaystyle n-m}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a9677b812ea9ee4d4538767f9aef960c69aca59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.276ex; height:2.176ex;" alt="{\displaystyle n-m}" loading="lazy"> <a href="Parit%C3%A4t_(Mathematik)" title="Parität (Mathematik)">gerade</a> ist. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> ist der <a href="Azimut" title="Azimut">azimutale</a> <a href="Winkel" title="Winkel">Winkel</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> ist der normierte radiale Abstand.

Die Radialpolynome <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{n}^{m}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{n}^{m}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74f45574773b5834e452a9eadc309fc49e24f1ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.439ex; height:2.509ex;" alt="{\displaystyle R_{n}^{m}}" loading="lazy"> sind definiert gemäß

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<annotation encoding="application/x-tex">{\displaystyle R_{n}^{m}(\rho )=\!\sum _{k=0}^{(n-m)/2}\!\!\!{\frac {(-1)^{k}\,(n-k)!}{k!\,((n+m)/2-k)!\,((n-m)/2-k)!}}\;\rho ^{n-2\,k}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6becf2a8338dd9a21acd7fa249a2654d379a2b25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:58.731ex; height:7.843ex;" alt="{\displaystyle R_{n}^{m}(\rho )=\!\sum _{k=0}^{(n-m)/2}\!\!\!{\frac {(-1)^{k}\,(n-k)!}{k!\,((n+m)/2-k)!\,((n-m)/2-k)!}}\;\rho ^{n-2\,k}}" loading="lazy">,</dd></dl>

wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-m}">

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<annotation encoding="application/x-tex">{\displaystyle n-m}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a9677b812ea9ee4d4538767f9aef960c69aca59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.276ex; height:2.176ex;" alt="{\displaystyle n-m}" loading="lazy"> gerade ist und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{n}^{m}(\rho )=0}">

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<annotation encoding="application/x-tex">{\displaystyle R_{n}^{m}(\rho )=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e2b491fbd5185f6614f0941386485fac63b60a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.711ex; height:2.843ex;" alt="{\displaystyle R_{n}^{m}(\rho )=0}" loading="lazy">, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-m}">

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<annotation encoding="application/x-tex">{\displaystyle n-m}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a9677b812ea9ee4d4538767f9aef960c69aca59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.276ex; height:2.176ex;" alt="{\displaystyle n-m}" loading="lazy"> ungerade ist.

In dieser Form sind sie zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{n}^{m}(1)=1}">

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<annotation encoding="application/x-tex">{\displaystyle R_{n}^{m}(1)=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a0eed401fe2ed686a59ec8d4ab07788e80af33f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.672ex; height:2.843ex;" alt="{\displaystyle R_{n}^{m}(1)=1}" loading="lazy"> normiert.

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

Zernike-Polynome sind ein Produkt eines radiusabhängigen Teils <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{n}^{m}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{n}^{m}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74f45574773b5834e452a9eadc309fc49e24f1ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.439ex; height:2.509ex;" alt="{\displaystyle R_{n}^{m}}" loading="lazy"> und eines winkelabhängigen Teils <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{m}}">

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<annotation encoding="application/x-tex">{\displaystyle G^{m}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be782949ced5d6399c52285439fa2510e8a7ba19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.502ex; height:2.343ex;" alt="{\displaystyle G^{m}}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle Z_{n}^{\pm m}(\rho ,\phi )=R_{n}^{m}(\rho )\cdot G^{m}(\phi )\!.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb28e01782ab8f44ebde74bed51abe17bd111ac8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.277ex; height:3.009ex;" alt="{\displaystyle Z_{n}^{\pm m}(\rho ,\phi )=R_{n}^{m}(\rho )\cdot G^{m}(\phi )\!.}" loading="lazy"></dd></dl>

Zernike-Polynome werden üblicherweise in <a href="Polarkoordinaten" title="Polarkoordinaten">Polarkoordinaten</a> angegeben. Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\rho \cos \phi }">

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<annotation encoding="application/x-tex">{\displaystyle x=\rho \cos \phi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c9f8373160cbdf92967ce9359c0413de3da45ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.901ex; height:2.676ex;" alt="{\displaystyle x=\rho \cos \phi }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\rho \sin \phi }">

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<annotation encoding="application/x-tex">{\displaystyle y=\rho \sin \phi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ad94299f418a6490320bd9279d1f58770a2bb17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.471ex; height:2.676ex;" alt="{\displaystyle y=\rho \sin \phi }" loading="lazy"> umgewandelt auf <a href="Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesische Koordinaten</a> sind die Zernike-Polynome <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{n}^{\pm m}}">

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<annotation encoding="application/x-tex">{\displaystyle Z_{n}^{\pm m}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a1eee15cb47ee4bd4e1ac4b1d24c483c68625f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.662ex; height:2.843ex;" alt="{\displaystyle Z_{n}^{\pm m}}" loading="lazy"> <a href="Univariat" title="Univariat">bivariate</a> Polynome in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">.<a href="#cite_note-1">[1]</a>

Eine Rotation des Koordinatensystems um den Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi /m}">

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<annotation encoding="application/x-tex">{\displaystyle 2\pi /m}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b65ed23a2ee5630fc5b0ddf642e2bc4b0278ade9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.697ex; height:2.843ex;" alt="{\displaystyle 2\pi /m}" loading="lazy"> ändert den Wert des Polynoms nicht:

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<annotation encoding="application/x-tex">{\displaystyle G^{m}(\phi +2\pi /m)=G^{m}(\phi )\!.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a82e3da4aec4385821cd0bc70b55d44d24fd3b00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.289ex; height:2.843ex;" alt="{\displaystyle G^{m}(\phi +2\pi /m)=G^{m}(\phi )\!.}" loading="lazy"></dd></dl>

Der radiusabhängige Teil <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{n}^{m}(\rho )}">

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<annotation encoding="application/x-tex">{\displaystyle R_{n}^{m}(\rho )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1a2c064d93f16e9368035d31d00ed33f91049e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.45ex; height:2.843ex;" alt="{\displaystyle R_{n}^{m}(\rho )}" loading="lazy">ist ein Polynom über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> vom Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy">, welches nur Potenzen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ^{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07764bf0f4aa3157d6441dea49a9b3b46ca1db2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.291ex; height:3.176ex;" alt="{\displaystyle \rho ^{k}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\leq k\leq n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\leq k\leq n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad9110789af2413623ac536cc3fc5d7f483c2594.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.843ex; height:2.343ex;" alt="{\displaystyle m\leq k\leq n}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k-m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k-m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0bd4b821c71d96cc24539d20dc27c24c48b3b98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.092ex; height:2.343ex;" alt="{\displaystyle k-m}" loading="lazy"> gerade enthält.

Dadurch sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ^{k}\cos(m\phi )}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ^{k}\cos(m\phi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0f5ef51ed71983f7688065a839371044218b00b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.024ex; height:3.176ex;" alt="{\displaystyle \rho ^{k}\cos(m\phi )}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ^{k}\sin(m\phi )}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ^{k}\sin(m\phi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bdcd445e95e59fe21e5d9eb09b6c245552660129.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.769ex; height:3.176ex;" alt="{\displaystyle \rho ^{k}\sin(m\phi )}" loading="lazy"> und somit auch die Zernike-Polynome in kartesischen Koordinaten als Polynome in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> darstellbar, vgl. <a href="Formelsammlung_Trigonometrie#Winkelfunktionen_für_weitere_Vielfache" title="Formelsammlung Trigonometrie">Winkelfunktionen für weitere Vielfache</a>.

<caption>Zernike-Polynome in Polarkoordinaten und in kartesischen Koordinaten<a href="#cite_note-2">[2]</a>

</caption>

<th>Polynom

</th>

<th>Polarkoordinaten

</th>

<th>Kartesische Koordinaten

</th>

<th>Wellenoptische Interpretation

</th></tr>

<tr>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{0}^{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5baa8a00b21e107517c792cea4d126460b8db373.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.763ex; height:3.176ex;" alt="{\displaystyle Z_{0}^{0}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">

</td>

<td><a href="Mittelwert" title="Mittelwert">Mittelwert</a> (Piston)

</td></tr>

<tr>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{1}^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b16740f09bc2a0bee5a6ac65dfdb332f02b04d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.041ex; height:3.343ex;" alt="{\displaystyle Z_{1}^{-1}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho \sin \phi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/332b9d0e62cf6ec21ba6d52536ccba5dba20c678.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.217ex; height:2.676ex;" alt="{\displaystyle \rho \sin \phi }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">

</td>

<td><a href="Querachse" title="Querachse">Verschwenkung in der horizontalen Achse</a>

</td></tr>

<tr>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{1}^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2c8ae1cfba1fed2c28c8254f58bdece0a9c7197.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.763ex; height:3.176ex;" alt="{\displaystyle Z_{1}^{1}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho \cos \phi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79d9ed92a822be73add7d7b0e75ec046b1d6632d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.473ex; height:2.676ex;" alt="{\displaystyle \rho \cos \phi }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy">

</td>

<td><a href="Gierachse" title="Gierachse">Verschwenkung in der vertikalen Achse</a>

</td></tr>

<tr>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{2}^{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a0a8b7db6b94ab9de5f92f985ac6a2f5af72012.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.763ex; height:3.176ex;" alt="{\displaystyle Z_{2}^{0}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1+2\rho ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/083fdfd912a59866503eaecb053bf63e7b392a9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.23ex; height:3.176ex;" alt="{\displaystyle -1+2\rho ^{2}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1+2(x^{2}+y^{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6baeafa776d78ab97db343f746593ea8b21be017.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.222ex; height:3.176ex;" alt="{\displaystyle -1+2(x^{2}+y^{2})}" loading="lazy">

</td>

<td><a href="Entfernungseinstellung" title="Entfernungseinstellung">Defokussierung</a>

</td></tr>

<tr>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{2}^{-2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77c45f3cc6cac1a2aac80f2c09680048298f0d30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.041ex; height:3.343ex;" alt="{\displaystyle Z_{2}^{-2}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ^{2}\sin(2\phi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3dbbfd4068189aec46121c9090535156e437a682.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.856ex; height:3.176ex;" alt="{\displaystyle \rho ^{2}\sin(2\phi )}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2xy}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0cf28a82088fc2021c794fee1de241f90622d5ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.648ex; height:2.509ex;" alt="{\displaystyle 2xy}" loading="lazy">

</td>

<td><a href="Astigmatismus_(Optik)" class="mw-redirect" title="Astigmatismus (Optik)">Astigmatismus</a> schräg (45°) zu den Hauptachsen

</td></tr>

<tr>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{2}^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f7e4d2471234452cfc1f6ed8a78c25cc2ecce58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.763ex; height:3.176ex;" alt="{\displaystyle Z_{2}^{2}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ^{2}\cos(2\phi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06d9ac9748f6ddd333bafe758888b1beab7a97c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.112ex; height:3.176ex;" alt="{\displaystyle \rho ^{2}\cos(2\phi )}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}-y^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7f07ef330c7bbc2bd1c835de4093c922c1ce1d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.439ex; height:3.009ex;" alt="{\displaystyle x^{2}-y^{2}}" loading="lazy">

</td>

<td>Astigmatismus in den Hauptachsen

</td></tr>

<tr>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{3}^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6c703cb0a959724ac0d1d88dcb5cf9f7c0d1f41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.041ex; height:3.343ex;" alt="{\displaystyle Z_{3}^{-1}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (3\rho ^{3}-2\rho )\sin \phi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2806d2e2d9b5e65eff24a3bf6d18253eba0e2aa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.448ex; height:3.176ex;" alt="{\displaystyle (3\rho ^{3}-2\rho )\sin \phi }" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3y(x^{2}+y^{2})-2y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98d6753047a9e1628932f1ec717481d6600a301c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.725ex; height:3.176ex;" alt="{\displaystyle 3y(x^{2}+y^{2})-2y}" loading="lazy">

</td>

<td><a href="Koma_(Optik)" class="mw-redirect" title="Koma (Optik)">Koma</a>, horizontale Achse

</td></tr>

<tr>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{3}^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75c59959f6d864a931728a8cd62cd226725a2be1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.763ex; height:3.176ex;" alt="{\displaystyle Z_{3}^{1}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (3\rho ^{3}-2\rho )\cos \phi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbfb7c71255fd9836141aef30f5368d27b2948c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.703ex; height:3.176ex;" alt="{\displaystyle (3\rho ^{3}-2\rho )\cos \phi }" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3x(x^{2}+y^{2})-2x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/446b794ed9189b9d57208bc1dc24b8415581b563.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.073ex; height:3.176ex;" alt="{\displaystyle 3x(x^{2}+y^{2})-2x}" loading="lazy">

</td>

<td>Koma, vertikale Achse

</td></tr>

<tr>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{4}^{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/630a6431d475fb2bad58e28c499f3fedfd1d3e95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.763ex; height:3.176ex;" alt="{\displaystyle Z_{4}^{0}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 6\rho ^{4}-6\rho ^{2}+1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00ab809a659da630a79712232fd9007be86aecc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.681ex; height:3.176ex;" alt="{\displaystyle 6\rho ^{4}-6\rho ^{2}+1}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 6(x^{2}+y^{2})^{2}-6(x^{2}+y^{2})+1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0607b58a07499fd449ab3a754ebf1f63e2915d4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.719ex; height:3.176ex;" alt="{\displaystyle 6(x^{2}+y^{2})^{2}-6(x^{2}+y^{2})+1}" loading="lazy">

</td>

<td><a href="Sph%C3%A4rische_Aberration" class="mw-redirect" title="Sphärische Aberration">Sphärische Aberration</a>

</td></tr></tbody></table>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{n}^{m}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74f45574773b5834e452a9eadc309fc49e24f1ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.439ex; height:2.509ex;" alt="{\displaystyle R_{n}^{m}}" loading="lazy"> ist eine bezüglich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> gerade (ungerade) Funktion, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> gerade (ungerade) ist.

Der radiusabhängige Teil <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{n}^{m}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{n}^{m}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74f45574773b5834e452a9eadc309fc49e24f1ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.439ex; height:2.509ex;" alt="{\displaystyle R_{n}^{m}}" loading="lazy"> stellt einen Spezialfall der <a href="Jacobi-Polynom" title="Jacobi-Polynom">Jacobi-Polynome</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}^{(\alpha ,\beta )}(z)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{n}^{(\alpha ,\beta )}(z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0de03e1b31dda46293bb72db292a1956553bcc44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.682ex; height:3.509ex;" alt="{\displaystyle P_{n}^{(\alpha ,\beta )}(z)}" loading="lazy"> dar.

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{n}^{m}(\rho )=(-1)^{(n-m)/2}\rho ^{m}P_{(n-m)/2}^{(m,0)}(1-2\rho ^{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cd5503ea9f1e5b78f91a209e3e37f3008942826.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:41.655ex; height:4.176ex;" alt="{\displaystyle R_{n}^{m}(\rho )=(-1)^{(n-m)/2}\rho ^{m}P_{(n-m)/2}^{(m,0)}(1-2\rho ^{2})}" loading="lazy"></dd></dl>

Die Reihe der radiusabhängigen Polynome beginnt mit

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{0}^{0}(\rho )=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f39c5c254d0bf69e797ddd68d600dfec05371ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.09ex; height:3.176ex;" alt="{\displaystyle R_{0}^{0}(\rho )=1}" loading="lazy"></dd></dl>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{1}^{1}(\rho )=\rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef20e2f33df0046e8cef25e7a3119cf33a4c832a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.13ex; height:3.176ex;" alt="{\displaystyle R_{1}^{1}(\rho )=\rho }" loading="lazy"></dd></dl>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{2}^{0}(\rho )=2\rho ^{2}-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90e434d1820fd5d5ce508c79e843af0efe4cb2d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.349ex; height:3.343ex;" alt="{\displaystyle R_{2}^{0}(\rho )=2\rho ^{2}-1}" loading="lazy"></dd></dl>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{2}^{2}(\rho )=\rho ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19f558e299e925811a8a50a7afbed3d4dde05313.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.184ex; height:3.343ex;" alt="{\displaystyle R_{2}^{2}(\rho )=\rho ^{2}}" loading="lazy"></dd></dl>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{3}^{1}(\rho )=3\rho ^{3}-2\rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d52edf5fb2bd12eea7d95876def41e9840a1d072.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.551ex; height:3.343ex;" alt="{\displaystyle R_{3}^{1}(\rho )=3\rho ^{3}-2\rho }" loading="lazy"></dd></dl>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{3}^{3}(\rho )=\rho ^{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4a6d5b5987b4ef3e0dde406ea4f61970bd091b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.184ex; height:3.343ex;" alt="{\displaystyle R_{3}^{3}(\rho )=\rho ^{3}}" loading="lazy"></dd></dl>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{4}^{0}(\rho )=6\rho ^{4}-6\rho ^{2}+1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c491dd6476e6dfe66ba2f657dadca2c51e712a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.608ex; height:3.343ex;" alt="{\displaystyle R_{4}^{0}(\rho )=6\rho ^{4}-6\rho ^{2}+1}" loading="lazy"></dd></dl>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{4}^{2}(\rho )=4\rho ^{4}-3\rho ^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8dd4cab2cff9980241f03e03574e2b962b23f1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.606ex; height:3.343ex;" alt="{\displaystyle R_{4}^{2}(\rho )=4\rho ^{4}-3\rho ^{2}}" loading="lazy"></dd></dl>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{4}^{4}(\rho )=\rho ^{4}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/069c549749339383059ce43b1f195d0d2242a35b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.184ex; height:3.343ex;" alt="{\displaystyle R_{4}^{4}(\rho )=\rho ^{4}}" loading="lazy"></dd></dl>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{5}^{1}(\rho )=10\rho ^{5}-12\rho ^{3}+3\rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88bb6bc788f4e6c04918a4460f91b9b29d47bd38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.135ex; height:3.343ex;" alt="{\displaystyle R_{5}^{1}(\rho )=10\rho ^{5}-12\rho ^{3}+3\rho }" loading="lazy"></dd></dl>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{5}^{3}(\rho )=5\rho ^{5}-4\rho ^{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/375965c021312ec8a4f937689de0f06df3812888.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.606ex; height:3.343ex;" alt="{\displaystyle R_{5}^{3}(\rho )=5\rho ^{5}-4\rho ^{3}}" loading="lazy"></dd></dl>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{5}^{5}(\rho )=\rho ^{5}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6294149f6b14eb27d38e44012a0274c3302e3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.184ex; height:3.343ex;" alt="{\displaystyle R_{5}^{5}(\rho )=\rho ^{5}}" loading="lazy"></dd></dl>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{6}^{0}(\rho )=20\rho ^{6}-30\rho ^{4}+12\rho ^{2}-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44f18d0b527b09b3046fcecbd9466ecb5f8225e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.355ex; height:3.343ex;" alt="{\displaystyle R_{6}^{0}(\rho )=20\rho ^{6}-30\rho ^{4}+12\rho ^{2}-1}" loading="lazy"></dd></dl>

Allgemein ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{n}^{n}(\rho )=\rho ^{n}.}">

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{n}^{n}(\rho )=\rho ^{n}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea0d0218b3fe6e7ffb8912031e367eff1fbe6f07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.159ex; height:2.843ex;" alt="{\displaystyle R_{n}^{n}(\rho )=\rho ^{n}.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Anwendungen">Anwendungen</h2></div>

In der <a href="Optik" title="Optik">Optik</a> werden Zernike-Polynome benutzt, um <a href="Abbildungsfehler" title="Abbildungsfehler">Abbildungsfehler</a> optischer Systeme quantitativ zu erfassen. Dies findet zum Beispiel in der <a href="Adaptive_Optik" title="Adaptive Optik">adaptiven Optik</a> Anwendung.

Seit einigen Jahren ist die Verwendung der Zernike-Polynome auch in der <a href="Optometrie" title="Optometrie">Optometrie</a> und <a href="Augenheilkunde" title="Augenheilkunde">Augenheilkunde</a> üblich. Hier führen Abweichungen der <a href="Hornhaut" title="Hornhaut">Cornea</a> beziehungsweise der <a href="Linse_(Optik)" title="Linse (Optik)">Linse</a> von der idealen Form zu Abbildungsfehlern.

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noresize noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"></div><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Zernike_polynomials?uselang=de">Commons: Zernike-Polynom</a> – Sammlung von Bildern</div>

<ul><li><a href="Frits_Zernike" title="Frits Zernike">Frits Zernike</a>: „Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode.“ Physica 1, 689–704, 1934.</li>

<li>Born and Wolf: Principles of Optics. Oxford: Pergamon, 1970.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ZernikePolynomial.html">Zernike Polynomial.</a> In: <a href="MathWorld" title="MathWorld">MathWorld</a> (englisch).</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> <a rel="nofollow" class="external text" href="https://e-l.unifi.it/pluginfile.php/1055875/mod_resource/content/1/Appunti_2020_Lezione%2014_4_Zernikepolynomialsaguidefinal.pdf">Lakshminarayanan & Fleck, Review, 2011: Zernike polynomials: A guide</a>.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> <a rel="nofollow" class="external text" href="https://wp.optics.arizona.edu/jcwyant/wp-content/uploads/sites/13/2016/08/Zernike_Polynomials_For_The_Web.pdf">Wyant - Zernike Polynomials For The Web</a>.

</li>

</ol></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2026-01-05" href="https://de.wikipedia.org/wiki/?title=Zernike-Polynom&oldid=263053739">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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