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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Evolute</span></h1>
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<p>Die <b>Evolute</b> einer ebenen <a href="Kurve_(Mathematik)" title="Kurve (Mathematik)">Kurve</a> ist
</p>
<ul><li>die Bahn, auf der sich der <a href="Mittelpunkt" title="Mittelpunkt">Mittelpunkt</a> des <a href="Kr%C3%BCmmungskreis" title="Krümmungskreis">Krümmungskreises</a> der Kurve bewegt, wenn dieser die gesamte Kurve durchläuft.</li></ul>
<p>Oder auch:
</p>
<ul><li>die <a href="Einh%C3%BCllende" title="Einhüllende">Hüllkurve</a> (Enveloppe) der <a href="Normalenvektor" title="Normalenvektor">Normalen</a> der gegebenen Kurve.</li></ul>
<p>Evoluten stehen in engem Zusammenhang mit den <a href="Evolvente" title="Evolvente">Evolventen</a> einer gegebenen Kurve, denn es gilt: Eine Kurve ist die Evolute jeder ihrer Evolventen.
</p>
<div class="mw-heading mw-heading2"><h2 id="Evolute_einer_parametrisierten_Kurve">Evolute einer parametrisierten Kurve</h2></div>
<p>Beschreibt <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 {\vec {x}}={\vec {c}}(t),\;t\in [t_{1},t_{2}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
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</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
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<mover>
<mi>c</mi>
<mo stretchy="false">→<!-- → --></mo>
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<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
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<mi>t</mi>
<mo>∈<!-- ∈ --></mo>
<mo stretchy="false">[</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
<mo>,</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}={\vec {c}}(t),\;t\in [t_{1},t_{2}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c39b96578c926c9cfc44b8bab1621e67aac866f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.775ex; height:2.843ex;" alt="{\displaystyle {\vec {x}}={\vec {c}}(t),\;t\in [t_{1},t_{2}]}" loading="lazy"></span> eine <a href="Regul%C3%A4re_Kurve" class="mw-redirect" title="Reguläre Kurve">reguläre Kurve</a> in der euklidischen Ebene, deren Krümmung nirgends 0 ist, und sind <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 \rho (t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle \rho (t)}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a66f7a7ba345de9189b12ee79bdf43ba9499c25e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.851ex; height:2.843ex;" alt="{\displaystyle \rho (t)}" loading="lazy"></span> der <a href="Kr%C3%BCmmungskreis" title="Krümmungskreis">Krümmungskreisradius</a> und <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 {\vec {n}}(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
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<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
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<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}(t)}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5ea5159a54467be0b292241d2c4851b501ae9cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.044ex; height:2.843ex;" alt="{\displaystyle {\vec {n}}(t)}" loading="lazy"></span> die zum Krümmungsmittelpunkt weisende Einheitsnormale, so ist
</p>
<ul><li><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 {\vec {E}}(t)={\vec {c}}(t)+\rho (t){\vec {n}}(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
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<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">(</mo>
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<mo>+</mo>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
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<mo stretchy="false">(</mo>
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<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}(t)={\vec {c}}(t)+\rho (t){\vec {n}}(t)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b284eea679e32d87de49edbdc23a648181a46b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.13ex; height:3.343ex;" alt="{\displaystyle {\vec {E}}(t)={\vec {c}}(t)+\rho (t){\vec {n}}(t)}" loading="lazy"></span></li></ul>
<p>die <i>Evolute</i> der gegebenen Kurve.
</p><p>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 {\vec {c}}(t)=(x(t),y(t))^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">(</mo>
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<mi>T</mi>
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {c}}(t)=(x(t),y(t))^{T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5db6f482411029a42befda72e267bd9153870873.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.986ex; height:3.176ex;" alt="{\displaystyle {\vec {c}}(t)=(x(t),y(t))^{T}}" loading="lazy"></span> und <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 {\vec {E}}=(X,Y)^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}=(X,Y)^{T}}</annotation>
</semantics>
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</p>
<ul><li><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 \displaystyle X(t)=x(t)-{\frac {y'(t)\cdot {\Big (}x'(t)^{2}+y'(t)^{2}{\Big )}}{x'(t)\cdot y''(t)-x''(t)\cdot y'(t)}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b90aabe68e247c058382624852a50562fe0f6725.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.801ex; height:8.343ex;" alt="{\displaystyle \displaystyle X(t)=x(t)-{\frac {y'(t)\cdot {\Big (}x'(t)^{2}+y'(t)^{2}{\Big )}}{x'(t)\cdot y''(t)-x''(t)\cdot y'(t)}}}" loading="lazy"></span> und</li></ul>
<dl><dd><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 \displaystyle Y(t)=y(t)+{\frac {x'(t)\cdot {\Big (}x'(t)^{2}+y'(t)^{2}{\Big )}}{x'(t)\cdot y''(t)-x''(t)\cdot y'(t)}}}">
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<div class="mw-heading mw-heading2"><h2 id="Eigenschaften_der_Evolute">Eigenschaften der Evolute</h2></div>
<p>Um Eigenschaften einer regulären Kurve herzuleiten, ist es vorteilhaft, die <a href="Bogenl%C3%A4nge" class="mw-redirect" title="Bogenlänge">Bogenlänge</a> <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">
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<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> der gegebenen Kurve als <a href="Kurve_(Mathematik)#Differenzierbare_Kurven,_Krümmung" title="Kurve (Mathematik)">Parameter</a> zu verwenden. Denn dann gilt (s. <a href="Frenetsche_Formeln" title="Frenetsche Formeln">Frenetsche Formeln</a>) <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 \;|{\vec {c}}'|=1\;}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thickmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>c</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>′</mo>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mspace width="thickmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \;|{\vec {c}}'|=1\;}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19b29e07c43d295933af94c886057aad684d731c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.753ex; height:3.176ex;" alt="{\displaystyle \;|{\vec {c}}'|=1\;}" loading="lazy"></span> und <span style="white-space:nowrap"><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 \;{\vec {n}}'=-{\vec {c}}'/\rho \;}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thickmathspace"></mspace>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>′</mo>
</msup>
<mo>=</mo>
<mo>−<!-- − --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>c</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>′</mo>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>ρ<!-- ρ --></mi>
<mspace width="thickmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \;{\vec {n}}'=-{\vec {c}}'/\rho \;}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f75f78415af40da9b52abb8003b4597899b95849.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.549ex; height:3.176ex;" alt="{\displaystyle \;{\vec {n}}'=-{\vec {c}}'/\rho \;}" loading="lazy"></span>.</span> Daraus folgt für den Tangentenvektor der Evolute <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 \;{\vec {E}}={\vec {c}}+\rho {\vec {n}}\;}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thickmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>c</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>+</mo>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mspace width="thickmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \;{\vec {E}}={\vec {c}}+\rho {\vec {n}}\;}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d988bacc8081542319daee5834e8bac9633df64f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.825ex; height:3.343ex;" alt="{\displaystyle \;{\vec {E}}={\vec {c}}+\rho {\vec {n}}\;}" loading="lazy"></span>:
</p>
<dl><dd><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 {\vec {E}}'={\vec {c}}'+\rho '{\vec {n}}+\rho {\vec {n}}'=\rho '{\vec {n}}\;.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>′</mo>
</msup>
<mo>=</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>c</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>′</mo>
</msup>
<mo>+</mo>
<msup>
<mi>ρ<!-- ρ --></mi>
<mo>′</mo>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>+</mo>
<mi>ρ<!-- ρ --></mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>′</mo>
</msup>
<mo>=</mo>
<msup>
<mi>ρ<!-- ρ --></mi>
<mo>′</mo>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mspace width="thickmathspace"></mspace>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}'={\vec {c}}'+\rho '{\vec {n}}+\rho {\vec {n}}'=\rho '{\vec {n}}\;.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66d55f72cfc96533d06ba29a8f9930afb73e96de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.382ex; height:3.676ex;" alt="{\displaystyle {\vec {E}}'={\vec {c}}'+\rho '{\vec {n}}+\rho {\vec {n}}'=\rho '{\vec {n}}\;.}" loading="lazy"></span></dd></dl>
<p>Aus dieser Gleichung ergeben sich die folgenden Eigenschaften einer Evolute:
</p>
<ul><li>Die Evolute ist in Punkten mit <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 \rho '=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>ρ<!-- ρ --></mi>
<mo>′</mo>
</msup>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho '=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54b56dc936b40cd308dd7d06c581d0ec9cc3bef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.148ex; height:3.009ex;" alt="{\displaystyle \rho '=0}" loading="lazy"></span> <i>nicht regulär,</i> d. h., sie hat in Punkten maximaler oder minimaler Krümmung <i>Spitzen</i> (s. Parabel, Ellipse, Nephroide).</li>
<li>Die Normalen der gegebenen Kurve sind Tangenten der Evolute, d. h.: Die Evolute ist die <i>Einhüllende der Normalen</i> der gegebenen Kurve.</li>
<li>In Abschnitten der gegebenen Kurve, in denen <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 \rho '>0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>ρ<!-- ρ --></mi>
<mo>′</mo>
</msup>
<mo>></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho '>0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5891e7a1e41cff3ca7d211cf9d6015baa4a50a80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.148ex; height:3.009ex;" alt="{\displaystyle \rho '>0}" loading="lazy"></span> bzw. <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 \rho '<0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>ρ<!-- ρ --></mi>
<mo>′</mo>
</msup>
<mo><</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho '<0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3002b06b449489dd14e77c29ee16ed18b42acec4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.148ex; height:3.009ex;" alt="{\displaystyle \rho '<0}" loading="lazy"></span> gilt, ist sie eine Evolvente ihrer Evolute. (Im Bild ist die blaue Parabel eine Evolvente der roten Neilschen Parabel.)</li></ul>
<p><i>Beweis</i> der letzten Eigenschaft:<br>
In dem betrachteten Abschnitt sei <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 \rho '>0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>ρ<!-- ρ --></mi>
<mo>′</mo>
</msup>
<mo>></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho '>0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5891e7a1e41cff3ca7d211cf9d6015baa4a50a80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.148ex; height:3.009ex;" alt="{\displaystyle \rho '>0}" loading="lazy"></span>. Eine <a href="Evolvente" title="Evolvente">Evolvente</a> der Evolute lässt sich folgendermaßen beschreiben:
</p>
<dl><dd><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 {\vec {C}}_{0}={\vec {E}}-{\frac {{\vec {E}}'}{|{\vec {E}}'|}}\;{\Big (}\int _{0}^{s}|{\vec {E}}'(w)|\;\mathrm {d} w+l_{0}\;{\Big )}\;,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>C</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>′</mo>
</msup>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>′</mo>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<mspace width="thickmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo maxsize="1.623em" minsize="1.623em">(</mo>
</mrow>
</mrow>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
</mrow>
</msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mspace width="thickmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>w</mi>
<mo>+</mo>
<msub>
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mspace width="thickmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo maxsize="1.623em" minsize="1.623em">)</mo>
</mrow>
</mrow>
<mspace width="thickmathspace"></mspace>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {C}}_{0}={\vec {E}}-{\frac {{\vec {E}}'}{|{\vec {E}}'|}}\;{\Big (}\int _{0}^{s}|{\vec {E}}'(w)|\;\mathrm {d} w+l_{0}\;{\Big )}\;,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8f9ac53a02170fd927777530bb6eba9c1730a65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:40.142ex; height:7.843ex;" alt="{\displaystyle {\vec {C}}_{0}={\vec {E}}-{\frac {{\vec {E}}'}{|{\vec {E}}'|}}\;{\Big (}\int _{0}^{s}|{\vec {E}}'(w)|\;\mathrm {d} w+l_{0}\;{\Big )}\;,}" loading="lazy"></span></dd></dl>
<p>wobei <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 l_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l_{0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e26bea7ddadf1817f47b2d4ce7dc199185823989.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.748ex; height:2.509ex;" alt="{\displaystyle l_{0}}" loading="lazy"></span> eine Fadenverlängerung bedeutet (s. <a href="Evolvente#Evolvente_einer_parametrisierten_Kurve" title="Evolvente">Evolvente</a>).<br>
Mit <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 {\vec {E}}={\vec {c}}+\rho {\vec {n}}\;,\;{\vec {E}}'=\rho '{\vec {n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>c</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>+</mo>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mspace width="thickmathspace"></mspace>
<mo>,</mo>
<mspace width="thickmathspace"></mspace>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>′</mo>
</msup>
<mo>=</mo>
<msup>
<mi>ρ<!-- ρ --></mi>
<mo>′</mo>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}={\vec {c}}+\rho {\vec {n}}\;,\;{\vec {E}}'=\rho '{\vec {n}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f12f39a0eebd2ee671a64a9b0f1936c0f562974.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.699ex; height:3.676ex;" alt="{\displaystyle {\vec {E}}={\vec {c}}+\rho {\vec {n}}\;,\;{\vec {E}}'=\rho '{\vec {n}}}" loading="lazy"></span> und <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 \rho '>0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>ρ<!-- ρ --></mi>
<mo>′</mo>
</msup>
<mo>></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho '>0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5891e7a1e41cff3ca7d211cf9d6015baa4a50a80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.148ex; height:3.009ex;" alt="{\displaystyle \rho '>0}" loading="lazy"></span> ergibt sich
</p>
<dl><dd><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 {\vec {C}}_{0}={\vec {c}}+\rho {\vec {n}}-{\vec {n}}\;{\Big (}\int _{0}^{s}\rho '(w)\;\mathrm {d} w\;+l_{0}{\Big )}={\vec {c}}+(\rho (0)-l_{0})\;{\vec {n}}\;.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
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<mi>C</mi>
<mo stretchy="false">→<!-- → --></mo>
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</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>c</mi>
<mo stretchy="false">→<!-- → --></mo>
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<mo>+</mo>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
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<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
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<mspace width="thickmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo maxsize="1.623em" minsize="1.623em">(</mo>
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<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
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</msubsup>
<msup>
<mi>ρ<!-- ρ --></mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mspace width="thickmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>w</mi>
<mspace width="thickmathspace"></mspace>
<mo>+</mo>
<msub>
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo maxsize="1.623em" minsize="1.623em">)</mo>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>c</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<msub>
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mspace width="thickmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mspace width="thickmathspace"></mspace>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {C}}_{0}={\vec {c}}+\rho {\vec {n}}-{\vec {n}}\;{\Big (}\int _{0}^{s}\rho '(w)\;\mathrm {d} w\;+l_{0}{\Big )}={\vec {c}}+(\rho (0)-l_{0})\;{\vec {n}}\;.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb3344827635999b4b38929043977e670743ae44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:59.736ex; height:5.843ex;" alt="{\displaystyle {\vec {C}}_{0}={\vec {c}}+\rho {\vec {n}}-{\vec {n}}\;{\Big (}\int _{0}^{s}\rho '(w)\;\mathrm {d} w\;+l_{0}{\Big )}={\vec {c}}+(\rho (0)-l_{0})\;{\vec {n}}\;.}" loading="lazy"></span></dd></dl>
<p>D. h., für die Fadenverlängerung <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 l_{0}=\rho (0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l_{0}=\rho (0)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a73cd86d61a8cdd16aec83d9824d1f913e6b266.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.02ex; height:2.843ex;" alt="{\displaystyle l_{0}=\rho (0)}" loading="lazy"></span> erhält man die gegebene Kurve wieder.
</p>
<ul><li><i>Parallele Kurven</i> besitzen dieselbe Evolute.</li></ul>
<p><i>Beweis:</i> Eine zur gegebenen Kurve im Abstand <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 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" 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}" loading="lazy"></span> parallele Kurve besitzt die Parameterdarstellung <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 {\vec {c}}_{d}={\vec {c}}+d{\vec {n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>c</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>c</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>+</mo>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {c}}_{d}={\vec {c}}+d{\vec {n}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db7f0fddb1c09f65260a0d1c4bbc557e2aee1870.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.088ex; height:2.676ex;" alt="{\displaystyle {\vec {c}}_{d}={\vec {c}}+d{\vec {n}}}" loading="lazy"></span> und den Krümmungsradius (s. <a href="Parallelkurve#Parallelkurve_einer_parametrisierten_Kurve" title="Parallelkurve">Parallelkurve</a>) <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 \rho _{d}=\rho -d}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>ρ<!-- ρ --></mi>
<mo>−<!-- − --></mo>
<mi>d</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho _{d}=\rho -d}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4342f149b0b84dd462ee848886c96c023ab44b84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.651ex; height:2.676ex;" alt="{\displaystyle \rho _{d}=\rho -d}" loading="lazy"></span>. Die Evolute der Parallelkurve ist also <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 {\vec {E}}_{d}={\vec {c}}_{d}+\rho _{d}{\vec {n}}={\vec {c}}+d{\vec {n}}+(\rho -d){\vec {n}}={\vec {c}}+\rho {\vec {n}}={\vec {E}}\;.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>c</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
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</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>c</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
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</mrow>
<mo>+</mo>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
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<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>ρ<!-- ρ --></mi>
<mo>−<!-- − --></mo>
<mi>d</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
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<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>c</mi>
<mo stretchy="false">→<!-- → --></mo>
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<mo>+</mo>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
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</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mspace width="thickmathspace"></mspace>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}_{d}={\vec {c}}_{d}+\rho _{d}{\vec {n}}={\vec {c}}+d{\vec {n}}+(\rho -d){\vec {n}}={\vec {c}}+\rho {\vec {n}}={\vec {E}}\;.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84d83765d51bd3f38f3556f506cf59765afd1cff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.81ex; height:3.343ex;" alt="{\displaystyle {\vec {E}}_{d}={\vec {c}}_{d}+\rho _{d}{\vec {n}}={\vec {c}}+d{\vec {n}}+(\rho -d){\vec {n}}={\vec {c}}+\rho {\vec {n}}={\vec {E}}\;.}" loading="lazy"></span>
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Evolute_der_Normalparabel">Evolute der Normalparabel</h3></div>
<p>Die Normalparabel lässt sich durch die Parameterdarstellung <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 (t,t^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (t,t^{2})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/425e317965ea50f6d1f153d1b679c9c72f40a0c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.577ex; height:3.176ex;" alt="{\displaystyle (t,t^{2})}" loading="lazy"></span> beschreiben. Nach den obigen Formeln ergeben sich für die Evolute die folgenden Gleichungen:
</p>
<dl><dd><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=\cdots =-4t^{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>=</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>4</mn>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X=\cdots =-4t^{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7eebfebebb3e4d0c2ccb0c9ff0fe0487618b2fe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.765ex; height:2.843ex;" alt="{\displaystyle X=\cdots =-4t^{3}}" loading="lazy"></span></dd>
<dd><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 Y=\cdots ={\frac {1}{2}}+3t^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>=</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>+</mo>
<mn>3</mn>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y=\cdots ={\frac {1}{2}}+3t^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5657503ecdba5cf78637f0df40a05f14885f32f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.589ex; height:5.176ex;" alt="{\displaystyle Y=\cdots ={\frac {1}{2}}+3t^{2}}" loading="lazy"></span></dd></dl>
<p>Dies ist die Parameterdarstellung einer <a href="Neilsche_Parabel" title="Neilsche Parabel">Neilschen Parabel</a>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Evolute_einer_Ellipse">Evolute einer Ellipse</h3></div>
<p>Für die Ellipse mit der Parameterdarstellung <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 (a\cos t,b\sin t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>a</mi>
<mi>cos</mi>
<mo><!-- --></mo>
<mi>t</mi>
<mo>,</mo>
<mi>b</mi>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (a\cos t,b\sin t)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19e9cf86d51162f5d6f7a0fae0fcb26bce2b8739.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.265ex; height:2.843ex;" alt="{\displaystyle (a\cos t,b\sin t)}" loading="lazy"></span> ergibt sich:<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 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=\cdots ={\frac {a^{2}-b^{2}}{a}}\cos ^{3}t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>=</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mi>a</mi>
</mfrac>
</mrow>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X=\cdots ={\frac {a^{2}-b^{2}}{a}}\cos ^{3}t}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de22d5b8545682312c452545b3f0ef19e263b7ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.692ex; height:5.676ex;" alt="{\displaystyle X=\cdots ={\frac {a^{2}-b^{2}}{a}}\cos ^{3}t}" loading="lazy"></span></dd>
<dd><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 Y=\cdots ={\frac {b^{2}-a^{2}}{b}}\sin ^{3}t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>=</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mi>b</mi>
</mfrac>
</mrow>
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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</msup>
<mo><!-- --></mo>
<mi>t</mi>
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<annotation encoding="application/x-tex">{\displaystyle Y=\cdots ={\frac {b^{2}-a^{2}}{b}}\sin ^{3}t}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d057df3140465daa242580bbc71552f48fec9ec0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.23ex; height:5.843ex;" alt="{\displaystyle Y=\cdots ={\frac {b^{2}-a^{2}}{b}}\sin ^{3}t}" loading="lazy"></span></dd></dl>
<p>Diese Gleichungen beschreiben eine schiefe <a href="Astroide" title="Astroide">Astroide</a>. Elimination von <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 t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> liefert die implizite Darstellung
</p>
<ul><li><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 (aX)^{\tfrac {2}{3}}+(bY)^{\tfrac {2}{3}}=(a^{2}-b^{2})^{\tfrac {2}{3}}\ .}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">(</mo>
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<mo>+</mo>
<mo stretchy="false">(</mo>
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<mi>Y</mi>
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<mi>a</mi>
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<mn>2</mn>
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<mo>−<!-- − --></mo>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
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<mn>2</mn>
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<mtext> </mtext>
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<annotation encoding="application/x-tex">{\displaystyle (aX)^{\tfrac {2}{3}}+(bY)^{\tfrac {2}{3}}=(a^{2}-b^{2})^{\tfrac {2}{3}}\ .}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26daf15d7d262406306c8186ef68aae351a70ee5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.422ex; height:4.509ex;" alt="{\displaystyle (aX)^{\tfrac {2}{3}}+(bY)^{\tfrac {2}{3}}=(a^{2}-b^{2})^{\tfrac {2}{3}}\ .}" loading="lazy"></span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Evoluten_bekannter_Kurven">Evoluten bekannter Kurven</h2></div>
<ul><li>Zu einer <a href="Astroide" title="Astroide">Astroide</a>: wiederum eine Astroide (doppelt so groß)</li>
<li>Zu einer <a href="Ellipse" title="Ellipse">Ellipse</a>: eine schiefe Astroide</li>
<li>Zu einer <a href="Kardioide" title="Kardioide">Kardioide</a>: wiederum eine Kardioide (ein Drittel so groß)</li>
<li>Zu einem <a href="Kreis_(Geometrie)" class="mw-redirect" title="Kreis (Geometrie)">Kreis</a>: ein Punkt, nämlich dessen Mittelpunkt</li>
<li>Zu einer Deltoide: wiederum eine Deltoide (dreimal so groß)</li>
<li>Zu einer <a href="Zykloide" title="Zykloide">Zykloide</a>: eine kongruente Zykloide</li>
<li>Zu einer <a href="Epizykloide" title="Epizykloide">Epizykloide</a>: eine vergrößerte Epizykloide</li>
<li>Zu einer <a href="Hypozykloide" title="Hypozykloide">Hypozykloide</a>: eine ähnliche Hypozykloide</li>
<li>Zu einer <a href="Logarithmische_Spirale" title="Logarithmische Spirale">logarithmischen Spirale</a>: die gleiche logarithmische Spirale</li>
<li>Zu einer <a href="Nephroide" title="Nephroide">Nephroide</a>: wiederum eine Nephroide (halb so groß)</li>
<li>Zu einer <a href="Parabel_(Mathematik)" title="Parabel (Mathematik)">Parabel</a>: eine <a href="Neilsche_Parabel" title="Neilsche Parabel">Neilsche Parabel</a></li>
<li>Zu einer <a href="Traktrix" title="Traktrix">Traktrix</a>: eine <a href="Katenoide" class="mw-redirect" title="Katenoide">Katenoide</a> (Kettenlinie)</li></ul>
<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">R.Courant: <i>Vorlesungen über Differential- und Integralrechnung.</i> Band 1, Springer-Verlag, 1955, S. 268.</span>
</li>
</ol>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>K. Burg, H. Haf, F. Wille, A. Meister: <i>Vektoranalysis: Höhere Mathematik für Ingenieure, Naturwissenschaftler und …</i> Springer-Verlag, 2012, ISBN 3-8348-8346-8, S. 30.</li>
<li><i>Kleine Enzyklopädie Mathematik.</i> Harry Deutsch Verlag, 1977, ISBN 3-87144-323-9, S. 475.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</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"><span class="mw-default-size" typeof="mw:File"><span title="Commons"></span></span></div><b><span class=""><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Evolute?uselang=de"><span lang="en">Commons</span>: Evolute</a></span></b> – Sammlung von Bildern, Videos und Audiodateien</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/Evolute.html"><i>Evolute</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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