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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Torische Linse</span></h1>
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<p>Eine <b>torische Linse</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> ist eine Linse, die in zwei senkrecht zueinander stehenden Richtungen zwei unterschiedliche <a href="Brennweite" title="Brennweite">Brechwerte</a> hat. Mindestens eine der Linsenoberflächen hat dabei die Form eines „Käppchens“ von einem <a href="Torus" title="Torus">Torus</a> (s. Abb. 1), die andere ist meist <a href="Linse_(Optik)#Sphärische_Linsen" title="Linse (Optik)">sphärisch</a>. Torische Linsen werden für <a href="Brille" title="Brille">Brillen</a>, als <a href="Kontaktlinse" title="Kontaktlinse">Kontaktlinsen</a> und <a href="Intraokularlinse" title="Intraokularlinse">Intraokularlinsen</a> zur Korrektur des <a href="Astigmatismus_(Medizin)" title="Astigmatismus (Medizin)">Astigmatismus</a> verwendet. Weiterhin kann man damit <a href="Laserstrahl" class="mw-redirect" title="Laserstrahl">Laserstrahlen</a> zu elliptischen <a href="Fokus" title="Fokus">Foki</a> bündeln.
</p>
<div class="mw-heading mw-heading2"><h2 id="Torus">Torus</h2></div>
<p>Ein <a href="Torus" title="Torus">Torus</a> entsteht, wenn ein Kreis mit Radius <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> um eine Achse rotiert (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span>-Achse in Abb. 2), die in derselben Ebene liegt wie der Kreis. Der Mittelpunkt dieses Kreises folgt dabei einer kreisförmigen Bahn mit Radius <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> um die Rotationsachse. Wenn <span class="mwe-math-element mwe-math-element-inline"><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>r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo>></mo>
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R>r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/185691d6d238376261dda1cf90392a62e5df832f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle R>r}" loading="lazy"></span>, erhält man einen <i>Ringtorus</i>. Wenn <span class="mwe-math-element mwe-math-element-inline"><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=r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo>=</mo>
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R=r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ed97a934a158af5ecdf7587f66ec24b3234f52f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle R=r}" loading="lazy"></span>, ist die Öffnung in den Mittelpunkt des Rotationskreises zusammengeschrumpft; man spricht hier von einem <i>Horntorus</i>. Wenn <span class="mwe-math-element mwe-math-element-inline"><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<r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo><</mo>
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R<r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b07ed06b793f014ad6a6f8379116925aa30fe11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle R<r}" loading="lazy"></span>, spricht man von einem <i>Spindeltorus</i>; hier bleiben von der Öffnung nur zwei Vertiefungen übrig, deren Tiefe verschwindet, wenn <span class="mwe-math-element mwe-math-element-inline"><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\to 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R\to 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e01e56c6e9396e6f92efb3b3791ee01a4c1483a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.541ex; height:2.176ex;" alt="{\displaystyle R\to 0}" loading="lazy"></span> geht. Wenn <span class="mwe-math-element mwe-math-element-inline"><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=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7e38203eea12f5613721e2fc5e185b8eef9e800.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle R=0}" loading="lazy"></span>, ist der Torus in eine <a href="Sph%C3%A4re_(Mathematik)" title="Sphäre (Mathematik)">Sphäre</a> mit dem Radius <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> entartet. (Siehe Abb. 3.)
</p>
<div class="mw-heading mw-heading2"><h2 id="Beschreibung">Beschreibung</h2></div>
<p>Der größte Krümmungsradius der torischen Linsenoberfläche ist <span class="mwe-math-element mwe-math-element-inline"><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+r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo>+</mo>
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R+r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b9043bad9cd90562e777ea16dceecea8935d002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.653ex; height:2.343ex;" alt="{\displaystyle R+r}" loading="lazy"></span> (siehe Abb. 2); der entsprechende kleinste Brechwert ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=(n-1)/(R+r)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mo stretchy="false">(</mo>
<mi>R</mi>
<mo>+</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S=(n-1)/(R+r)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8cb9a8c50912eec45030e5682f188ce7da227df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.429ex; height:2.843ex;" alt="{\displaystyle S=(n-1)/(R+r)}" loading="lazy"></span>, wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_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"></span> der Brechungsindex des Glases ist. Dem kleinsten Krümmungsradius, <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span>, entspricht der größte Brechwert, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=(n-1)/r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s=(n-1)/r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/247359c813b866d6881a5a6854fe194cde459640.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.607ex; height:2.843ex;" alt="{\displaystyle s=(n-1)/r}" loading="lazy"></span>. Da <span class="mwe-math-element mwe-math-element-inline"><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+r>r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo>+</mo>
<mi>r</mi>
<mo>></mo>
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R+r>r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13e09186735666d84b4ec83d812db382ea9f0e66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.8ex; height:2.343ex;" alt="{\displaystyle R+r>r}" loading="lazy"></span> ist, ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S<s}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo><</mo>
<mi>s</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S<s}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c4da0fe05237408837250874ba019d475111b53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.688ex; height:2.176ex;" alt="{\displaystyle S<s}" loading="lazy"></span>. Die Differenz, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s-S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s-S}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/812068ade60c55305aca3066e53bd15830fdf55e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.43ex; height:2.343ex;" alt="{\displaystyle s-S}" loading="lazy"></span>, wird in der <a href="Augenheilkunde" title="Augenheilkunde">Augenheilkunde</a> und <a href="Augenoptik" class="mw-redirect" title="Augenoptik">Augenoptik</a> die <i>Zylinderkorrektur</i><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> genannt. Das Glas verhält sich etwa wie eine Kombination einer <a href="Linse_(Optik)" title="Linse (Optik)">sphärischen Linse</a> mit dem Brechwert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span> und einer <a href="Zylinderlinse" title="Zylinderlinse">Zylinderlinse</a> mit dem Brechwert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s-S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s-S}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/812068ade60c55305aca3066e53bd15830fdf55e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.43ex; height:2.343ex;" alt="{\displaystyle s-S}" loading="lazy"></span>.
</p><p>Die maximale und die minimale Krümmung sind beide kreisförmig – die Linsenoberfläche ist also keineswegs Teil eines <a href="Rotationsellipsoid" title="Rotationsellipsoid">Rotationsellipsoids</a> wie manchmal behauptet wird.
</p>
<div class="mw-heading mw-heading3"><h3 id="Wirkung">Wirkung</h3></div>
<p>Lichtstrahlen, die in der (<span class="mwe-math-element mwe-math-element-inline"><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,y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x,y}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ea0abffd33a692ded22accc104515a032851dff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}" loading="lazy"></span>)-Ebene des Torus (siehe Abb. 2) einfallen, werden entsprechend dem größten Krümmungsradius, <span class="mwe-math-element mwe-math-element-inline"><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+r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo>+</mo>
<mi>r</mi>
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<annotation encoding="application/x-tex">{\displaystyle R+r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b9043bad9cd90562e777ea16dceecea8935d002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.653ex; height:2.343ex;" alt="{\displaystyle R+r}" loading="lazy"></span>, also dem kleinsten Brechwert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=(n-1)/(R+r)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mo stretchy="false">(</mo>
<mi>R</mi>
<mo>+</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S=(n-1)/(R+r)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8cb9a8c50912eec45030e5682f188ce7da227df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.429ex; height:2.843ex;" alt="{\displaystyle S=(n-1)/(R+r)}" loading="lazy"></span>, gebrochen.
</p><p>Lichtstrahlen in einer Ebene durch die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span>-Achse des Torus (siehe Abb. 2) werden entsprechend dem kleinsten Krümmungsradius, <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span>, also dem größten Brechwert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=(n-1)/r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s=(n-1)/r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/247359c813b866d6881a5a6854fe194cde459640.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.607ex; height:2.843ex;" alt="{\displaystyle s=(n-1)/r}" loading="lazy"></span>, gebrochen.
</p><p>Es gibt somit zwei verschiedene Brechwerte in senkrecht zueinander stehenden Richtungen. In den Zwischenrichtungen verläuft der Brechwert fließend vom kleinsten zum größten Wert oder umgekehrt. Dies kompensiert die astigmatischen Abweichungen des Auges.
</p><p>Werden parallele Lichtstrahlen (Laser, Sonne) mit solchen Sammellinsen gebündelt, entstehen je nach Entfernung elliptische Foki.
</p>
<div class="mw-heading mw-heading2"><h2 id="Atorische_Linse">Atorische Linse</h2></div>
<p>Computergesteuerte Entwurf-, Schleif- und Polierverfahren ermöglichen es heute, in einem größeren Blickfeld gute Korrekturen zu erreichen, indem man bestimmte Abweichungen von der Torusform einbringt. In diesem Fall spricht man von einer <i>atorischen</i> Linse.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Von der Mathematik her würde man eher den Begriff <i>toroidale Linse</i> erwarten. Im Bereich der Augenheilkunde und Augenoptik ist jedoch der Begriff <i>torische Linse</i> üblich. Vermutlich basiert das darauf, dass man im Englischen unter <i><a href="https://en.wikipedia.org/wiki/Torus" class="extiw external" title="en:Torus">torus</a></i> nur den Rotationskörper eines <i>Kreises</i> versteht und den Begriff <i><a href="https://en.wikipedia.org/wiki/Torus" class="extiw external" title="en:Torus">toroid</a></i> nur für Rotationskörper anderer flacher Figuren verwendet.</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Diese korrigiert den Astigmatismus des Auges. Der Term <i>Zylinder</i> basiert hier auf eine mathematische Approximation, die nur bei kleinen Korrekturwerten gültig ist.</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">D. Meister: <i>Principles of Atoric Lens Design.</i> In <i>Lens Talk</i>, Vol. 27, No. 3, 1998 (<a rel="nofollow" class="external text" href="https://www.laramyk.com/wp-content/uploads/2010/05/Principles_of_Atoric_Lens_Design.pdf">PDF</a>)</span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">D. Volk: <i><style data-mw-deduplicate="TemplateStyles:r261891140">
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</style><span class="dewiki-iconexternal"><a class="external text" href="https://redirecter.toolforge.org/?url=http%3A%2F%2Fwww.oculist.net%2Fdownaton502%2Fprof%2Febook%2Fduanes%2Fpages%2Fv1%2Fv1c051b.html">Originals</a></span> vom 12. März 2012 im </i><a href="Internet_Archive" title="Internet Archive">Internet Archive</a><i>) <small class="archiv-bot"><span class="wp_boppel noviewer" aria-hidden="true" role="presentation"><span typeof="mw:File"><span title="i"></span></span></span> <b>Info:</b> Der Archivlink wurde automatisch eingesetzt und noch nicht geprüft. Bitte prüfe Original- und Archivlink gemäß Anleitung und entferne dann diesen Hinweis.</small><span style="display:none"><a rel="nofollow" class="external text" href="http://IABotmemento.invalid/http://www.oculist.net/downaton502/prof/ebook/duanes/pages/v1/v1c051b.html">@1</a></span><span style="display:none"><a rel="nofollow" class="external text" href="http://www.oculist.net/downaton502/prof/ebook/duanes/pages/v1/v1c051b.html">@2</a></span><span style="display:none">Vorlage:Webachiv/IABot/www.oculist.net</span></i>. (chapter 50 in Duane's Ophthalmology (Lippinkott, Wilkins & Williams / Wolters-Kluwer Health, Chicago, USA))</span>
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