GermanWikipedia - /page/entry.mu

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<title>Neilsche Parabel</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Neilsche Parabel</h1>

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Die Neil’sche Parabel (nach dem englischen Mathematiker <a href="William_Neile" title="William Neile">William Neile</a> benannt) oder semikubische Parabel<a href="#cite_note-1">[1]</a> ist eine spezielle <a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">ebene</a> <a href="Algebraische_Kurve" title="Algebraische Kurve">algebraische Kurve</a>, die durch eine Gleichung der Form

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<annotation encoding="application/x-tex">{\displaystyle \quad y^{2}-a^{2}x^{3}\;=\,0\;,\;a>0\;,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe634d996505c39763850695b5a0b5c64b3eeb64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.446ex; height:3.009ex;" alt="{\displaystyle \quad y^{2}-a^{2}x^{3}\;=\,0\;,\;a>0\;,}" loading="lazy"></li></ul>

beschrieben werden kann. Auflösen nach <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"> ergibt die explizite Form

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<annotation encoding="application/x-tex">{\displaystyle \quad y=\pm ax^{\frac {3}{2}}\;,\;x\geq 0\;,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c15aeae81a1aa22c81edaf315a850976eacdb29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.887ex; height:3.843ex;" alt="{\displaystyle \quad y=\pm ax^{\frac {3}{2}}\;,\;x\geq 0\;,}" loading="lazy"></li></ul>

die Anlass für die Bezeichnung semikubische Parabel liefert.

(Eine gewöhnliche Parabel kann durch eine Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=ax^{2}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5d52a44a3db46af9ddcfabec595af948aff95b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.868ex; height:3.009ex;" alt="{\displaystyle y=ax^{2}}" loading="lazy"> beschrieben werden.)

Löst man (A) nach <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"> auf, so erhält man die Gleichung

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<annotation encoding="application/x-tex">{\displaystyle \quad x={\Big (}{\frac {y}{a}}{\Big )}^{\color {red}{\frac {2}{3}}}\;,\quad y\in \mathbb {R} \;.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d880f426d1726f07caeccbdc21daeca0f8b3c40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.296ex; height:5.843ex;" alt="{\displaystyle \quad x={\Big (}{\frac {y}{a}}{\Big )}^{\color {red}{\frac {2}{3}}}\;,\quad y\in \mathbb {R} \;.}" loading="lazy"></li></ul>

Mit Hilfe der ersten Gleichung erkennt man, dass

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<annotation encoding="application/x-tex">{\displaystyle \quad x=t^{2}\;,\quad y=at^{3}\;,\quad t\in \mathbb {R} \;,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f3c381e5af59188344c7f72143de785b4377df6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.676ex; height:3.009ex;" alt="{\displaystyle \quad x=t^{2}\;,\quad y=at^{3}\;,\quad t\in \mathbb {R} \;,}" loading="lazy"></li></ul>

eine Parameterdarstellung der Neilschen Parabel ist.

William Neile hatte erstmals die <a href="L%C3%A4nge_(Mathematik)" title="Länge (Mathematik)">Bogenlänge</a> dieser <a href="Kurve_(Mathematik)" title="Kurve (Mathematik)">Kurve</a> berechnet, die sog. <a href="L%C3%A4nge_(Mathematik)" title="Länge (Mathematik)">Rektifizierung</a>, und dies 1657 bekannt gemacht<a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a>. Aufgrund der Probleme bei der Rektifizierung von <a href="Ellipse#Umfang" title="Ellipse">Ellipsen</a> und Parabeln vermutete man zu dieser Zeit, dass der Kreis und die Gerade die einzigen rektifizierbaren algebraischen Kurven seien.

Die Neil’sche Parabel ist <a href="Rationale_Variet%C3%A4t" class="mw-redirect" title="Rationale Varietät">rational</a>, es existiert also eine <a href="Rationale_Abbildung" title="Rationale Abbildung">rationale Abbildung</a> mit einer inversen rationalen Abbildung, die die Neil'sche Parabel auf die <a href="Projektive_Gerade" title="Projektive Gerade">projektive Gerade</a> abbildet.

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften_einer_Neilschen_Parabel">Eigenschaften einer Neilschen Parabel</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Ähnlichkeit">Ähnlichkeit</h3></div>

<ul><li>Jede Neilsche Parabel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t^{2},at^{3})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c8b3bfd0ae1bc03b11fe5ca9b1bf2d0f801b5d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.861ex; height:3.176ex;" alt="{\displaystyle (t^{2},at^{3})}" loading="lazy"> ist zur Neilschen Einheitsparabel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u^{2},u^{3})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34755850b37cfb994c64aaf5708c31bf851049c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.611ex; height:3.176ex;" alt="{\displaystyle (u^{2},u^{3})}" loading="lazy"> ähnlich.</li></ul>

Beweis: Die <a href="%C3%84hnlichkeitsabbildung" title="Ähnlichkeitsabbildung">Ähnlichkeitsabbildung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\rightarrow (a^{2}x,a^{2}y)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dccc08c551d9cc2890defc870835e406a55c9e51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.839ex; height:3.176ex;" alt="{\displaystyle (x,y)\rightarrow (a^{2}x,a^{2}y)}" loading="lazy"> (Streckung am Ursprung) führt die Neilsche Parabel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t^{2},at^{3})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c8b3bfd0ae1bc03b11fe5ca9b1bf2d0f801b5d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.861ex; height:3.176ex;" alt="{\displaystyle (t^{2},at^{3})}" loading="lazy"> in die Kurve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((at)^{2},(at)^{3})=(u^{2},u^{3})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0077a08d7aeb39cfac126b3e7dbdefeb43c0fc85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.419ex; height:3.176ex;" alt="{\displaystyle ((at)^{2},(at)^{3})=(u^{2},u^{3})}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=at}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/445eb29891dc864ffd48c5fe02bd59a475b26493.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.498ex; height:2.009ex;" alt="{\displaystyle u=at}" loading="lazy"> über.

<div class="mw-heading mw-heading3"><h3 id="Singularität">Singularität</h3></div>

<ul><li>Die Parameterdarstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t^{2},at^{3})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c8b3bfd0ae1bc03b11fe5ca9b1bf2d0f801b5d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.861ex; height:3.176ex;" alt="{\displaystyle (t^{2},at^{3})}" loading="lazy"> ist überall außer im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}">

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<annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d630d3e781a53b0a3559ae7e5b45f9479a3141a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}" loading="lazy"> <a href="Regul%C3%A4re_Kurve" class="mw-redirect" title="Reguläre Kurve">regulär</a>. Die Kurve besitzt im Nullpunkt eine <a href="Singularit%C3%A4t_(Mathematik)" class="mw-redirect" title="Singularität (Mathematik)">Singularität</a> (Spitze).</li></ul>

Der Beweis folgt aus dem Tangentenvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2t,3t^{2})^{T}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79f4930c1d6426a40da41fd5a7ccebe28a2162e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.291ex; height:3.176ex;" alt="{\displaystyle (2t,3t^{2})^{T}}" loading="lazy">. Nur für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy"> ergibt sich der Nullvektor.

<div class="mw-heading mw-heading3"><h3 id="Tangenten">Tangenten</h3></div>

Für die Neilsche Einheitsparabel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;y=\pm x^{\frac {3}{2}}\;}">

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<annotation encoding="application/x-tex">{\displaystyle \;y=\pm x^{\frac {3}{2}}\;}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10c0f0c4a1aff06997ff5e7a3ab02ba27758d7f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.418ex; height:3.843ex;" alt="{\displaystyle \;y=\pm x^{\frac {3}{2}}\;}" loading="lazy"> ergibt sich durch Differentiation die Gleichung der Tangente in einem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0})}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle y={\frac {\sqrt {x_{0}}}{2}}{\Big (}3x-x_{0}{\Big )}\;.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a0978d9baee48fa3ab7745685cf9c5dc6d79179.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.194ex; height:5.843ex;" alt="{\displaystyle y={\frac {\sqrt {x_{0}}}{2}}{\Big (}3x-x_{0}{\Big )}\;.}" loading="lazy"></li></ul>

Diese Tangente schneidet die Kurve in genau einem weiteren Punkt des unteren Astes mit den Koordinaten<a href="#cite_note-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\Big (}{\frac {x_{0}}{4}},-{\frac {y_{0}}{8}}{\Big )}\;.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36e10c41d6a3c27671ad2a0562d75d8ea3c73e8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.159ex; height:4.843ex;" alt="{\displaystyle {\Big (}{\frac {x_{0}}{4}},-{\frac {y_{0}}{8}}{\Big )}\;.}" loading="lazy"></li></ul>

(Beim Nachrechnen sollte man berücksichtigen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29c296094af9a1c665425debeac5eaab99a37a04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0})}" loading="lazy"> ein doppelter Schnittpunkt der Tangente mit der Kurve ist.)

<div class="mw-heading mw-heading3"><h3 id="Bogenlänge">Bogenlänge</h3></div>

Um die Bogenlänge einer parametrisierten Kurve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x(t),y(t))}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7b80868600db89fef84e4d41317b7c8a1e0d047.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.626ex; height:2.843ex;" alt="{\displaystyle (x(t),y(t))}" loading="lazy"> zu bestimmen, muss man das unbestimmte Integral <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int {\sqrt {x'(t)^{2}+y'(t)^{2}}}\;dt}">

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<annotation encoding="application/x-tex">{\displaystyle \int {\sqrt {x'(t)^{2}+y'(t)^{2}}}\;dt}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle (t^{2},at^{3}),\;0\leq t\leq b\;,}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{b}{\sqrt {x'(t)^{2}+y'(t)^{2}}}\;dt=\int _{0}^{b}t{\sqrt {4+9a^{2}t^{2}}}\;dt=\cdots ={\Big [}{\frac {1}{27a^{2}}}(4+9a^{2}t^{2})^{\frac {3}{2}}{\Big ]}_{0}^{b}\;.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fda246771d5b9e35a4ff441dad3a50b74ab4352a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:76.809ex; height:6.343ex;" alt="{\displaystyle \int _{0}^{b}{\sqrt {x'(t)^{2}+y'(t)^{2}}}\;dt=\int _{0}^{b}t{\sqrt {4+9a^{2}t^{2}}}\;dt=\cdots ={\Big [}{\frac {1}{27a^{2}}}(4+9a^{2}t^{2})^{\frac {3}{2}}{\Big ]}_{0}^{b}\;.}" loading="lazy"></dd></dl>

(Das Integral lässt sich mit Hilfe der <a href="Integration_durch_Substitution" title="Integration durch Substitution">Substitution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=4+9a^{2}t^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u=4+9a^{2}t^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d35a2dc0ce5704565b825c5f255a9f268f7cb878.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.771ex; height:2.843ex;" alt="{\displaystyle u=4+9a^{2}t^{2}}" loading="lazy"> lösen.)

Beispiel: Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6104442ed30596ef4d7795d3186273f68d796ea4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=1}" loading="lazy"> (Neilsche Einheitsparabel) und die obere Grenze <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=2}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32584049ed5f72969777f89d69b74ee462875e82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=2}" loading="lazy">, d. h. bis zum Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (4,8)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (4,8)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c34b45045069c6301c7aec705b6392d3eb37aea4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (4,8)}" loading="lazy">, ist die Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9{,}073}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 9{,}073}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b4cb2952302c6e405e94d781f3e64f5e4e228f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.297ex; height:2.509ex;" alt="{\displaystyle 9{,}073}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Evolute_der_Einheitsparabel">Evolute der Einheitsparabel</h3></div>

<ul><li>Die <a href="Evolute#Evolute_der_Normalparabel" title="Evolute">Evolute der Parabel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t^{2},t)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50c0ea1cf80f9bdfb13d5d98a8b0041800cefa2e.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^{2},t)}" loading="lazy"> ist eine in x-Richtung um 1/2 verschobene Neilsche Parabel: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {1}{2}}+t^{2},{\frac {4}{{\sqrt {3}}^{3}}}t^{3}\right)\;.}">

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

<msup>

<msqrt>

</msqrt>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\frac {1}{2}}+t^{2},{\frac {4}{{\sqrt {3}}^{3}}}t^{3}\right)\;.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4621fb80842b8d491cc6bb13fab8c20423ef7198.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.01ex; height:7.509ex;" alt="{\displaystyle \left({\frac {1}{2}}+t^{2},{\frac {4}{{\sqrt {3}}^{3}}}t^{3}\right)\;.}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Polarkoordinaten">Polarkoordinaten</h3></div>

Um die Darstellung der Neilschen Parabel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t^{2},at^{3})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c8b3bfd0ae1bc03b11fe5ca9b1bf2d0f801b5d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.861ex; height:3.176ex;" alt="{\displaystyle (t^{2},at^{3})}" loading="lazy"> in Polarkoordinaten zu finden, schneidet man die Ursprungsgerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=mx}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55d00c656a79888b762b248a648aa987c729f3c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.624ex; height:2.009ex;" alt="{\displaystyle y=mx}" loading="lazy"> mit der Kurve. Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\neq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce13986af1c35b143f1e634e932044b2cb83a92e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.301ex; height:2.676ex;" alt="{\displaystyle m\neq 0}" loading="lazy"> gibt es einen vom Nullpunkt (Spitze) verschiedenen Punkt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {m^{2}}{a^{2}}},{\frac {m^{3}}{a^{2}}}\right)}">

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\frac {m^{2}}{a^{2}}},{\frac {m^{3}}{a^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/291e6de77813fbd64fd485b77c88da12e9406e36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.317ex; height:6.343ex;" alt="{\displaystyle \left({\frac {m^{2}}{a^{2}}},{\frac {m^{3}}{a^{2}}}\right)}" loading="lazy">. Der Abstand dieses Punktes zum Nullpunkt ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {m^{2}}{a^{2}}}{\sqrt {1+m^{2}}}}">

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {m^{2}}{a^{2}}}{\sqrt {1+m^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5830e44edbae9a8e79b21674a2aa88a12c424e7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.352ex; height:6.009ex;" alt="{\displaystyle {\frac {m^{2}}{a^{2}}}{\sqrt {1+m^{2}}}}" loading="lazy">. Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=\tan \varphi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0536b25b33d6632e7af55a59586c42f33a17707.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.406ex; height:2.509ex;" alt="{\displaystyle m=\tan \varphi }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec ^{2}\varphi =1+\tan ^{2}\varphi }">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97597275654667a7b2b5faa268de6ad79f56a655.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.365ex; height:3.176ex;" alt="{\displaystyle \sec ^{2}\varphi =1+\tan ^{2}\varphi }" loading="lazy"> ergibt sich<a href="#cite_note-5">[5]</a>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r={\Big (}{\frac {\tan \varphi }{a}}{\Big )}^{2}\cdot \sec \varphi \;,\quad -\pi /2<\varphi <\pi /2\ .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e56809c60d94c378674dfff68a506602031986e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.715ex; height:5.176ex;" alt="{\displaystyle r={\Big (}{\frac {\tan \varphi }{a}}{\Big )}^{2}\cdot \sec \varphi \;,\quad -\pi /2<\varphi <\pi /2\ .}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Projektive_Äquivalenz_zur_kubischen_Parabel">Projektive Äquivalenz zur kubischen Parabel</h3></div>

Bildet man die Neilsche Einheitsparabel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t^{2},t^{3})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd2588177761b3973d7e625987242fdcddfdac43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:3.176ex;" alt="{\displaystyle (t^{2},t^{3})}" loading="lazy"> mit der projektiven Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\rightarrow ({\tfrac {x}{y}},{\tfrac {1}{y}})}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y)\rightarrow ({\tfrac {x}{y}},{\tfrac {1}{y}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7659634a82eaeae57436528cfcaa97dd5faac6ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.22ex; height:3.676ex;" alt="{\displaystyle (x,y)\rightarrow ({\tfrac {x}{y}},{\tfrac {1}{y}})}" loading="lazy"> (involutorische <a href="Projektive_Perspektivit%C3%A4t" class="mw-redirect" title="Projektive Perspektivität">Perspektivität</a> mit der Achse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10f53b404b1fdd041a589f1f2425e45a2edba110.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=1}" loading="lazy"> und Zentrum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,-1)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6982e7eb515f8a46400ff23c38a6f1430ae97278.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (0,-1)}" loading="lazy"> ) ab, so erhält man die Kurve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {1}{t}},{\frac {1}{t^{3}}}\right)}">

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\frac {1}{t}},{\frac {1}{t^{3}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1775916f3572a160fa5f509ea7e0a04de74cece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.184ex; height:6.176ex;" alt="{\displaystyle \left({\frac {1}{t}},{\frac {1}{t^{3}}}\right)}" loading="lazy">, also die <a href="Kubische_Parabel" class="mw-redirect" title="Kubische Parabel">kubische Parabel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x^{3}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a75ec71227fe7827ab9f8e423856f8540b32772.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.638ex; height:3.009ex;" alt="{\displaystyle y=x^{3}}" loading="lazy">. Die Spitze (Nullpunkt) der Neilschen Parabel wird mit dem Fernpunkt der y-Achse vertauscht.

Diese Eigenschaft lässt sich auch an der Darstellung der Neilschen Parabel in <a href="Homogene_Koordinaten" title="Homogene Koordinaten">homogenen Koordinaten</a> erkennen: Ersetzt man in (A) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\tfrac {x_{1}}{x_{3}}},\;y={\tfrac {x_{2}}{x_{3}}}}">

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x={\tfrac {x_{1}}{x_{3}}},\;y={\tfrac {x_{2}}{x_{3}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0dfcfbf67f82d4bb10cb2ad51125a51b91785cab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.577ex; height:3.676ex;" alt="{\displaystyle x={\tfrac {x_{1}}{x_{3}}},\;y={\tfrac {x_{2}}{x_{3}}}}" loading="lazy"> (die Ferngerade hat die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{3}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{3}=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55547972561662fda8d82465866be57f729271c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle x_{3}=0}" loading="lazy">) und multipliziert mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{3}^{3}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aeeab5f868407718f52a08eb53b5738fbc4fe73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.384ex; height:3.176ex;" alt="{\displaystyle x_{3}^{3}}" loading="lazy">, erhält man die Kurvengleichung

<ul><li>in homogenen Koordinaten: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad x_{3}x_{2}^{2}-x_{1}^{3}=0\;.}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \quad x_{3}x_{2}^{2}-x_{1}^{3}=0\;.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d0e2380530df0080ab300c026d509a216f17c5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.868ex; height:3.176ex;" alt="{\displaystyle \quad x_{3}x_{2}^{2}-x_{1}^{3}=0\;.}" loading="lazy"></li></ul>

Wählt man nun die Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\color {red}2}=0}">

<msub>

</mstyle>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{\color {red}2}=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/330d2309f31693ecc4252076917c0b37a31a5548.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle x_{\color {red}2}=0}" loading="lazy"> als Ferngerade und setzt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\tfrac {x_{1}}{x_{2}}},\;y={\tfrac {x_{3}}{x_{2}}}}">

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x={\tfrac {x_{1}}{x_{2}}},\;y={\tfrac {x_{3}}{x_{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3bed8da023695e50decc0d2d46f2ecdb2425fa17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.577ex; height:3.676ex;" alt="{\displaystyle x={\tfrac {x_{1}}{x_{2}}},\;y={\tfrac {x_{3}}{x_{2}}}}" loading="lazy">, erhält man die (affine) Kurve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x^{3}\;.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=x^{3}\;.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3379dd793a8366358d114ffcabc8a928e17fd812.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.93ex; height:3.009ex;" alt="{\displaystyle y=x^{3}\;.}" loading="lazy">

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

<ul><li><a href="Johann_August_Pein" title="Johann August Pein">August Pein</a>: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten , 1875, <a rel="nofollow" class="external text" href="https://books.google.de/books?hl=de&lr=&id=wFxaAAAAYAAJ&oi=fnd&pg=PA9&dq=neilsche+parabel&ots=fAht0kQVES&sig=CscgUFFbNiUO-svYa7T0qAeYA6s#v=onepage&q=neilsche%20parabel&f=false">Dissertation</a></li>

<li><a rel="nofollow" class="external text" href="https://books.google.com/books?id=JrslMKTgSZwC&pg=PA148">Clifford A. Pickover: The Length of Neile's Semicubical Parabola</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Parabola_nodata" title="Parabola nodata">Parabola nodata</a></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> Walter Gellert, <a href="Herbert_K%C3%A4stner" title="Herbert Kästner">Herbert Kästner</a>, Siegfried Neuber (Hrsg.): Lexikon der Mathematik, VEB Bibliographisches Institut Leipzig, 1979. S 461, rationale Kurve.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> <a href="Johann_August_Pein" title="Johann August Pein">August Pein</a>: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten , S. 2

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Clifford A. Pickover: The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publishing Company, 2009, ISBN 9781402757969, S. 148

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> <a href="Johann_August_Pein" title="Johann August Pein">August Pein</a>: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten , S. 26

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> <a href="Johann_August_Pein" title="Johann August Pein">August Pein</a>: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten , S. 10

</li>

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