GermanWikipedia - /page/entry.mu

<!DOCTYPE html>

<title>Hyperboloid</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Hyperboloid</span></h1>

</header>

</div>

<p>Ein <b>Hyperboloid</b> ist im einfachsten Fall eine Fläche, die durch <a href="Drehung" title="Drehung">Rotation</a> einer <a href="Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbel</a> um eine ihrer Achsen entsteht (<a href="Rotationsfl%C3%A4che" title="Rotationsfläche">Rotationsfläche</a>).

</p>

<ul><li>Bei Rotation einer Hyperbel um ihre <i>Neben</i>achse entsteht ein <b>einschaliges Hyperboloid.</b> Es besteht aus einem zusammenhängenden Flächenstück.</li>

<li>Bei Rotation einer Hyperbel um ihre <i>Haupt</i>achse entsteht ein <b>zweischaliges Hyperboloid.</b> Es besteht aus zwei getrennten Flächenstücken.</li></ul>

<p>Beide <a href="Fl%C3%A4che_(Mathematik)" title="Fläche (Mathematik)">Flächen</a> lassen sich durch eine <a href="Quadratische_Gleichung" title="Quadratische Gleichung">quadratische Gleichung</a> – analog zu den <a href="Gleichung" title="Gleichung">Gleichungen</a> von <a href="Ellipse" title="Ellipse">Ellipse</a> und <a href="Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbel</a> – beschreiben. Sie sind deshalb Spezialfälle von <a href="Quadrik" title="Quadrik">Quadriken</a> (z. B. <a href="Kugel" title="Kugel">Kugel</a>, <a href="Kegel_(Geometrie)" title="Kegel (Geometrie)">Kegel</a>, <a href="Paraboloid" title="Paraboloid">Paraboloid</a>) und werden typischerweise von <a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebenen</a> in <a href="Kegelschnitt" title="Kegelschnitt">Kegelschnitten</a> geschnitten.

</p><p>Ein wesentlicher Unterschied zwischen einem einschaligen und einem zweischaligen Hyperboloid ist, dass das einschalige Hyperboloid <a href="Gerade" title="Gerade">Geraden</a> enthält, es also eine <a href="Regelfl%C3%A4che" title="Regelfläche">Regelfläche</a> ist, das zweischalige nicht.

</p><p>Diese Eigenschaft macht das einschalige Hyperboloid für <a href="Architekt" title="Architekt">Architekten</a> und <a href="Bauingenieurwesen" title="Bauingenieurwesen">Bauingenieure</a> interessant, da sich einschalige Hyperboloide leicht aus <a href="Gerade" title="Gerade">Geraden</a> modellieren lassen. Einige <a href="K%C3%BChlturm" title="Kühlturm">Kühltürme</a> haben die Form eines einschaligen Hyperboloids. Auch im <a href="Maschinenbau" title="Maschinenbau">Maschinenbau</a> finden einschalige Hyperboloide Verwendung bei Hyperboloidgetrieben,<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><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> Einschalige Hyperboloide spielen auch in der <a href="Synthetische_Geometrie" title="Synthetische Geometrie">synthetischen Geometrie</a> eine Rolle: Eine <a href="Minkowski-Ebene" title="Minkowski-Ebene">Minkowski-Ebene</a> ist die <a href="Geometrie" title="Geometrie">Geometrie</a> der ebenen Schnitte eines einschaligen Hyperboloids. Während das einschalige Hyperboloid von <a href="Tangentialebene" title="Tangentialebene">Tangentialebenen</a> in zwei sich schneidenden Geraden geschnitten wird (siehe unten), hat ein zweischaliges Hyperboloid mit Tangentialebenen immer nur einen <a href="Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkt</a> gemeinsam und ist deshalb <a href="Geometrisch" class="mw-redirect" title="Geometrisch">geometrisch</a> mehr mit einer <a href="Kugel" title="Kugel">Kugel</a> verwandt.

</p>

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

<div class="mw-heading mw-heading3"><h3 id="Einschaliges_Einheitshyperboloid">Einschaliges Einheitshyperboloid</h3></div>

<p>Lässt man die <a href="Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbel</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 x^{2}-z^{2}=1}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7344cbb0eefe0a4229f09e5f3744137c5e9ea775.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.63ex; height:2.843ex;" alt="{\displaystyle x^{2}-z^{2}=1}" loading="lazy"></span> in der x-z-<a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a> um die z-Achse rotieren (siehe Abbildung), so erhält man das einschalige Einheits-Hyperboloid mit der <a href="Gleichung" title="Gleichung">Gleichung</a>

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce3230389c1b59d5d70229db8c940598eca9d42b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.285ex; height:3.009ex;" alt="{\displaystyle H_{1}\colon \ x^{2}+y^{2}-z^{2}=1}" loading="lazy"></span>.</dd></dl>

<p>Bei der <a href="Drehung" title="Drehung">Rotation</a> wird <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^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf0bf28fd28f45d07e1ceb909ce333c18c558c93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}" loading="lazy"></span> durch <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^{2}+y^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f455605b597282c27d7cf2238821bc331479a7e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.439ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}}" loading="lazy"></span> ersetzt.

</p><p>Das einschalige Einheits-Hyperboloid ergibt sich durch <a href="Drehung" title="Drehung">Rotation</a> des <a href="Funktionsgraph" title="Funktionsgraph">Graphen</a> der <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</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 f(z)={\sqrt {z^{2}+1}}}">

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a81bd71f8f4f3119b7518c70fabba7ab819d2006.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.746ex; height:3.509ex;" alt="{\displaystyle f(z)={\sqrt {z^{2}+1}}}" loading="lazy"></span> um 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}">

</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. Für die <a href="Ableitung_(Mathematik)" class="mw-redirect" title="Ableitung (Mathematik)">Ableitung</a> gilt <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 f'(z)={\frac {z}{\sqrt {z^{2}+1}}}}">

<msup>

</msup>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f'(z)={\frac {z}{\sqrt {z^{2}+1}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/917d7e10fd0d8e5746855fe10596ab7349271500.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.308ex; height:6.009ex;" alt="{\displaystyle f'(z)={\frac {z}{\sqrt {z^{2}+1}}}}" loading="lazy"></span>. Das <a href="Volumen" title="Volumen">Volumen</a> und die Oberfläche für ein einschalige Einheits-Hyperboloid mit der Höhe <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 h}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> ergeben sich nach den <a href="Guldinsche_Regeln" class="mw-redirect" title="Guldinsche Regeln">Guldinschen Regeln</a> mithilfe von <a href="Integralrechnung" title="Integralrechnung">Integralen</a>.

</p>

<div class="mw-heading mw-heading4"><h4 id="Volumen">Volumen</h4></div>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=\pi \int _{0}^{h}(f(z))^{2}\ \mathrm {d} z=\pi \int _{0}^{h}z^{2}+1\ \mathrm {d} z=\pi \left({\frac {h^{3}}{3}}+h\right)={\frac {\pi }{3}}\left(h^{3}+3h\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a3c713dcdcc444f4cc5050640def38629b24043.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:69.371ex; height:6.509ex;" alt="{\displaystyle V=\pi \int _{0}^{h}(f(z))^{2}\ \mathrm {d} z=\pi \int _{0}^{h}z^{2}+1\ \mathrm {d} z=\pi \left({\frac {h^{3}}{3}}+h\right)={\frac {\pi }{3}}\left(h^{3}+3h\right)}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Oberfläche"><span id="Oberfl.C3.A4che"></span>Oberfläche</h4></div>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

<msqrt>

<msup>

<mrow>

<mrow>

<msup>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A&=2\pi \int _{0}^{h}f(z){\sqrt {1+\left(f'(z)\right)^{2}}}\ \mathrm {d} z\\&=2\pi \int _{0}^{h}{\sqrt {z^{2}+1}}{\sqrt {1+\left({\frac {z}{\sqrt {z^{2}+1}}}\right)^{2}}}\ \mathrm {d} z\\&=2\pi \int _{0}^{h}{\sqrt {2z^{2}+1}}\ \mathrm {d} z\\&=2\pi \left({\frac {1}{2}}z{\sqrt {2z^{2}+1}}+{\frac {1}{2{\sqrt {2}}}}\ln \left({\sqrt {2}}z+{\sqrt {2z^{2}+1}}\right){\Big |}_{z=0}^{z=h}\right)\\&=\pi \left(h{\sqrt {2h^{2}+1}}+{\frac {1}{\sqrt {2}}}\ln \left({\sqrt {2}}h+{\sqrt {2h^{2}+1}}\right)\right)\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0920812d45797187defcc6c46deaf17b738328c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -16.575ex; margin-bottom: -0.263ex; width:58.19ex; height:34.843ex;" alt="{\displaystyle {\begin{aligned}A&=2\pi \int _{0}^{h}f(z){\sqrt {1+\left(f'(z)\right)^{2}}}\ \mathrm {d} z\\&=2\pi \int _{0}^{h}{\sqrt {z^{2}+1}}{\sqrt {1+\left({\frac {z}{\sqrt {z^{2}+1}}}\right)^{2}}}\ \mathrm {d} z\\&=2\pi \int _{0}^{h}{\sqrt {2z^{2}+1}}\ \mathrm {d} z\\&=2\pi \left({\frac {1}{2}}z{\sqrt {2z^{2}+1}}+{\frac {1}{2{\sqrt {2}}}}\ln \left({\sqrt {2}}z+{\sqrt {2z^{2}+1}}\right){\Big |}_{z=0}^{z=h}\right)\\&=\pi \left(h{\sqrt {2h^{2}+1}}+{\frac {1}{\sqrt {2}}}\ln \left({\sqrt {2}}h+{\sqrt {2h^{2}+1}}\right)\right)\end{aligned}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Parameterdarstellung">Parameterdarstellung</h4></div>

<p>Offensichtlich ist jeder Höhenschnitt mit einer <a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</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 z=z_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f8e63a3f2769739a78c3c24a091f778fdc72dfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.322ex; height:2.009ex;" alt="{\displaystyle z=z_{0}}" loading="lazy"></span> ein <a href="Kreis" title="Kreis">Kreis</a> mit <a href="Radius" title="Radius">Radius</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 {\sqrt {1+z_{0}^{2}}}}">

<msqrt>

</mrow>

</mrow>

</msubsup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {1+z_{0}^{2}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34b12e67b4d7aac0ae7aa8dd999aa37688a8064f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.471ex; height:4.843ex;" alt="{\displaystyle {\sqrt {1+z_{0}^{2}}}}" loading="lazy"></span>. Der Schnitt der Ebene <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=1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee42176e76ae6b56d68c42ced807e08b962a2b54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}" loading="lazy"></span> liefert die beiden Schnittgeraden <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 (1,t,\pm t)^{\top },t\in \mathbb {R} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1,t,\pm t)^{\top },t\in \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ba04915050928efb04c97bfdd6564798909f158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.43ex; height:3.176ex;" alt="{\displaystyle (1,t,\pm t)^{\top },t\in \mathbb {R} }" loading="lazy"></span>. Durch Rotation dieser Geraden erhält man <a href="Parameterdarstellung" title="Parameterdarstellung">Parameterdarstellungen</a> aller Geraden auf dem Hyperboloid:

</p>

</mrow>

</mrow>

</msubsup>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{\alpha }^{\pm }:{\vec {x}}(t)={\begin{pmatrix}\cos \alpha \\\sin \alpha \\0\end{pmatrix}}+t\cdot {\begin{pmatrix}-\sin \alpha \\\cos \alpha \\\pm 1\end{pmatrix}}\ ,\quad t\in \mathbb {R} ,\ 0\leq \alpha \leq 2\pi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa02a83e4f5eb5ab196a5157cc4fe5bd9eddb422.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:60.802ex; height:9.176ex;" alt="{\displaystyle g_{\alpha }^{\pm }:{\vec {x}}(t)={\begin{pmatrix}\cos \alpha \\\sin \alpha \\0\end{pmatrix}}+t\cdot {\begin{pmatrix}-\sin \alpha \\\cos \alpha \\\pm 1\end{pmatrix}}\ ,\quad t\in \mathbb {R} ,\ 0\leq \alpha \leq 2\pi }" loading="lazy"></span></dd></dl>

<p>Das einschalige Hyperboloid <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 H_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d4d9a872a55b209f2eb7cc23a71e5e1541bd1f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{1}}" loading="lazy"></span> lässt sich also auch durch <a href="Drehung" title="Drehung">Rotation</a> der <a href="Gerade" title="Gerade">Geraden</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 g_{0}^{+}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0d96b7cb6da2dadd6ed3a709b836ddb86730458.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.629ex; height:3.176ex;" alt="{\displaystyle g_{0}^{+}}" loading="lazy"></span> oder <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 g_{0}^{-}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd432ed4844ce649f82f39d4bf438ebad4e3b60f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.629ex; height:3.176ex;" alt="{\displaystyle g_{0}^{-}}" loading="lazy"></span> (<a href="Windschiefe" title="Windschiefe">windschief</a> zur <a href="Rotationsachse" title="Rotationsachse">Rotationsachse</a>) erzeugen (siehe Abbildung). Diese Aussage wird in der Literatur als <i>Satz von <a href="Christopher_Wren" title="Christopher Wren">Wren</a></i> bezeichnet.<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>

</p>

<div class="mw-heading mw-heading4"><h4 id="Tangentialebenen">Tangentialebenen</h4></div>

<p>Die <a href="Gleichung" title="Gleichung">Gleichung</a> der <a href="Tangentialebene" title="Tangentialebene">Tangentialebene</a> einer implizit durch <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 f(x,y,z)=0}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d9dc9c0f7052aaefb6f7194bb0d9e419086a4bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.99ex; height:2.843ex;" alt="{\displaystyle f(x,y,z)=0}" loading="lazy"></span> gegebenen <a href="Fl%C3%A4che_(Mathematik)" title="Fläche (Mathematik)">Fläche</a> in einem <a href="Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkt</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 {x}}_{0}=(x_{0},y_{0},z_{0})}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/084f678fa4504a98882e9b6e5529a3a83957174c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.072ex; height:2.843ex;" alt="{\displaystyle {\vec {x}}_{0}=(x_{0},y_{0},z_{0})}" loading="lazy"></span> 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 f_{x}({\vec {x}}_{0})(x-x_{0})+f_{y}({\vec {x}}_{0})(y-y_{0})+f_{z}({\vec {x}}_{0})(z-z_{0})=0}">

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{x}({\vec {x}}_{0})(x-x_{0})+f_{y}({\vec {x}}_{0})(y-y_{0})+f_{z}({\vec {x}}_{0})(z-z_{0})=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76e41671b2d1483858691a74948c2c8202b90252.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.398ex; height:3.009ex;" alt="{\displaystyle f_{x}({\vec {x}}_{0})(x-x_{0})+f_{y}({\vec {x}}_{0})(y-y_{0})+f_{z}({\vec {x}}_{0})(z-z_{0})=0}" loading="lazy"></span>.

</p><p>Für <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 H_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d4d9a872a55b209f2eb7cc23a71e5e1541bd1f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{1}}" loading="lazy"></span> ergibt sich

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}x+y_{0}y-z_{0}z-1=0\ .}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c88440cd9bf4d7eb71d1d7a9b169e4e60cb008d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.458ex; height:2.509ex;" alt="{\displaystyle x_{0}x+y_{0}y-z_{0}z-1=0\ .}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Ebene_Schnitte">Ebene Schnitte</h4></div>

<ul><li>Ebenen mit einer Neigung kleiner 1 (1 ist die Neigung der <a href="Gerade" title="Gerade">Geraden</a> auf dem Hyperboloid) schneiden <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 H_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d4d9a872a55b209f2eb7cc23a71e5e1541bd1f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{1}}" loading="lazy"></span> in einer <i><a href="Ellipse" title="Ellipse">Ellipse</a>,</i></li>

<li>Ebenen mit einer Neigung gleich 1 durch den <a href="Koordinatenursprung" class="mw-redirect" title="Koordinatenursprung">Koordinatenursprung</a> schneiden <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 H_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d4d9a872a55b209f2eb7cc23a71e5e1541bd1f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{1}}" loading="lazy"></span> in einem <i>parallelen Geradenpaar,</i></li>

<li>Ebenen mit einer Neigung gleich 1 nicht durch den Koordinatenursprung schneiden <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 H_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d4d9a872a55b209f2eb7cc23a71e5e1541bd1f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{1}}" loading="lazy"></span> in einer <i><a href="Parabel_(Mathematik)" title="Parabel (Mathematik)">Parabel</a>,</i></li>

<li>Tangentialebenen schneiden <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 H_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d4d9a872a55b209f2eb7cc23a71e5e1541bd1f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{1}}" loading="lazy"></span> in einem sich <i>schneidenden Geradenpaar,</i></li>

<li>Ebenen mit einer Neigung größer 1, die keine <a href="Tangentialebene" title="Tangentialebene">Tangentialebenen</a> sind, schneiden <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 H_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d4d9a872a55b209f2eb7cc23a71e5e1541bd1f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{1}}" loading="lazy"></span> in einer <i><a href="Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbel</a>.</i><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></li></ul>

<p>Eine <a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a>, die eine Hyperboloid-Gerade <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 e}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"></span> enthält, ist entweder eine <a href="Tangentialebene" title="Tangentialebene">Tangentialebene</a> und enthält damit eine zweite <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 e}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"></span> schneidende Hyperboloid-Gerade oder enthält eine zu <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 e}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"></span> parallele Hyperboloid-Gerade und ist damit Tangentialebene in einem Fernpunkt.

</p>

<div class="mw-heading mw-heading4"><h4 id="Affine_Bilder">Affine Bilder</h4></div>

<p>Analog wie eine beliebige <a href="Ellipse" title="Ellipse">Ellipse</a> als affines Bild des <a href="Einheitskreis" title="Einheitskreis">Einheitskreises</a> aufgefasst werden kann, ist ein <i>beliebiges</i> einschaliges Hyperboloid das <a href="Affine_Abbildung" title="Affine Abbildung">affine Bild</a> des Einheitshyperboloids <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 H_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d4d9a872a55b209f2eb7cc23a71e5e1541bd1f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{1}}" loading="lazy"></span>. Die einfachsten affinen Bilder erhält man durch Skalierung der Koordinatenachsen

</p>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1\ ,\ a,b,c>0.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5f011979e8fcd2726457dd2a066d11999dec62d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:31.598ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1\ ,\ a,b,c>0.}" loading="lazy"></span></dd></dl>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}" loading="lazy"></span> sind die Höhenschnitte <a href="Kreis" title="Kreis">Kreise</a>. Andernfalls sind es <a href="Ellipse" title="Ellipse">Ellipsen</a>. Ein solches Hyperboloid nennt man <i>einschaliges Rotationshyperboloid</i>. Dass ein beliebiges einschaliges Hyperboloid auch immer Kreise enthält, wird in <a href="Kreisschnittebene" title="Kreisschnittebene">Kreisschnittebene</a> gezeigt.

</p><p>Da ein beliebiges einschaliges Hyperboloid <a href="Gerade" title="Gerade">Geraden</a> enthält, ist es eine <a href="Regelfl%C3%A4che" title="Regelfläche">Regelfläche</a>. Da jede <a href="Tangentialebene" title="Tangentialebene">Tangentialebene</a> eines einschaligen Hyperboloids in der Nähe seines <a href="Ber%C3%BChrung_(Mathematik)" title="Berührung (Mathematik)">Berührpunktes</a> die <a href="Fl%C3%A4che_(Mathematik)" title="Fläche (Mathematik)">Fläche</a> schneidet, hat es eine negative <a href="Gau%C3%9Fsche_Kr%C3%BCmmung" title="Gaußsche Krümmung">Gaußsche Krümmung</a> und ist deswegen nicht abwickelbar, im Gegensatz zu den Regelflächen <a href="Kegel_(Geometrie)" title="Kegel (Geometrie)">Kegel</a> und <a href="Zylinder_(Geometrie)" title="Zylinder (Geometrie)">Zylinder</a>, die die Gaußsche Krümmung 0 haben. Aus der üblichen <a href="Parameterdarstellung" title="Parameterdarstellung">Parameterdarstellung</a> einer <a href="Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbel</a> mit <a href="Hyperbelfunktion" title="Hyperbelfunktion">Hyperbelfunktionen</a> erhält man die folgende Parameterdarstellung des Hyperboloids <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 {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}-{\tfrac {z^{2}}{c^{2}}}=1:}">

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}-{\tfrac {z^{2}}{c^{2}}}=1:}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63a7195c9457edd1bfc301f60389122cb53c6295.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:18.769ex; height:4.843ex;" alt="{\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}-{\tfrac {z^{2}}{c^{2}}}=1:}" loading="lazy"></span>

</p>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}(s,t)={\begin{pmatrix}a\cosh s\cos t\\b\cosh s\sin t\\c\sinh s\end{pmatrix}},\quad s\in \mathbb {R} ,\ 0\leq t\leq 2\pi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/957a47c000666f4a82c9c9e845e2f5ac1ca2351e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:47.517ex; height:9.176ex;" alt="{\displaystyle {\vec {x}}(s,t)={\begin{pmatrix}a\cosh s\cos t\\b\cosh s\sin t\\c\sinh s\end{pmatrix}},\quad s\in \mathbb {R} ,\ 0\leq t\leq 2\pi }" loading="lazy"></span></dd></dl>

<p>Die Oberfläche kann durch Rotation einer Geraden erhalten werden. Die Gerade mit der Parametergleichung

</p>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}(u)={\begin{pmatrix}r\\0\\0\end{pmatrix}}+u{\begin{pmatrix}0\\\cos(\gamma )\\\sin(\gamma )\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3693fdbd08cdfdd6fa1e34eec65e62430acc00a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:28.719ex; height:9.509ex;" alt="{\displaystyle {\vec {x}}(u)={\begin{pmatrix}r\\0\\0\end{pmatrix}}+u{\begin{pmatrix}0\\\cos(\gamma )\\\sin(\gamma )\end{pmatrix}}}" loading="lazy"></span></dd></dl>

<p>ist parallel zur y-z-Ebene, hat den 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 r}">

</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> zur z-Achse und den Steigungswinkel <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 \gamma }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"></span> gegenüber der x-y-Ebene (siehe Bild).

</p><p>Lässt man diese Gerade um die z-Achse <a href="Drehmatrix" title="Drehmatrix">rotieren</a>, erhält man eine Fläche mit der Parametergleichung

</p>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}(u,v)={\begin{pmatrix}r\cos(v)\\r\sin(v)\\0\end{pmatrix}}+u{\begin{pmatrix}-\cos(\gamma )\sin(v)\\\cos(\gamma )\cos(v)\\\sin(\gamma )\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01ca3b6a4a87297ccf3fd527559c11a9f72ac8ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:45.577ex; height:9.843ex;" alt="{\displaystyle {\vec {x}}(u,v)={\begin{pmatrix}r\cos(v)\\r\sin(v)\\0\end{pmatrix}}+u{\begin{pmatrix}-\cos(\gamma )\sin(v)\\\cos(\gamma )\cos(v)\\\sin(\gamma )\end{pmatrix}}}" loading="lazy"></span>.</dd></dl>

<p>Man rechnet nach, dass im Fall <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 \;0<\gamma <\pi /2\;}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;0<\gamma <\pi /2\;}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6480bc168ec4332c95a3447e1b7710c4a226dcb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.569ex; height:2.843ex;" alt="{\displaystyle \;0<\gamma <\pi /2\;}" loading="lazy"></span> die Koordinaten der Flächenpunkte die obige Gleichung eines Rotationshyperboloids 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 \;c=r\tan \gamma \;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;c=r\tan \gamma \;}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40a5f82bb9becb96f37378187e653afbaa5ccf52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.84ex; height:2.509ex;" alt="{\displaystyle \;c=r\tan \gamma \;}" loading="lazy"></span> erfüllt. Außerdem erkennt man: die Gerade mit dem Steigungswinkel <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 -\gamma }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0cc7b8fd4d10c93e59759b25344321128de3718.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.07ex; height:2.509ex;" alt="{\displaystyle -\gamma }" loading="lazy"></span> erzeugt dasselbe Hyperboloid (s. Bild). Durch jeden Punkt des Hyperboloids gehen also zwei Geraden (Stangen), was die Stabilität eines Modells erheblich steigert.<br>

(Im Fall <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 \gamma =0}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a5e84cac32e896a80a89f8cd1917c2defcf4108.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.523ex; height:2.676ex;" alt="{\displaystyle \gamma =0}" loading="lazy"></span> liegt die Gerade in der x-y-Ebene und überstreicht das Äußere des Kreises mit der Gleichung <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^{2}+y^{2}=r^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37dd4f282df84a83620f71dc52345122e0e3a514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.64ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=r^{2}}" loading="lazy"></span>. Falls <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 \gamma =\pi /2}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/982a2789fde8a2fa65fd04c5e1bfec4dfecffc73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.018ex; height:2.843ex;" alt="{\displaystyle \gamma =\pi /2}" loading="lazy"></span> ist, entsteht ein Zylinder 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}">

</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>.)

</p>

<div class="mw-heading mw-heading4"><h4 id="Homogene_Koordinaten">Homogene Koordinaten</h4></div>

<p>Führt man <a href="Homogene_Koordinaten" title="Homogene Koordinaten">homogene Koordinaten</a> so ein, dass die <a href="Fernebene" class="mw-redirect" title="Fernebene">Fernebene</a> durch die <a href="Gleichung" title="Gleichung">Gleichung</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 x_{4}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/efc78587303e5b6b94128da81327c072b4644939.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_{4}=0}" loading="lazy"></span> beschrieben wird, muss man <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={\tfrac {x_{1}}{x_{4}}},y={\tfrac {x_{2}}{x_{4}}},z={\tfrac {x_{3}}{x_{4}}}}">

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

<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_{4}}},y={\tfrac {x_{2}}{x_{4}}},z={\tfrac {x_{3}}{x_{4}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e999906a873940a877fa99a6ccc4bc1795d9a20d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.76ex; height:3.676ex;" alt="{\displaystyle x={\tfrac {x_{1}}{x_{4}}},y={\tfrac {x_{2}}{x_{4}}},z={\tfrac {x_{3}}{x_{4}}}}" loading="lazy"></span> setzen. Nach Beseitigung des Nenners erhält man die homogene Beschreibung 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 H_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d4d9a872a55b209f2eb7cc23a71e5e1541bd1f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{1}}" loading="lazy"></span> durch die Gleichung:

</p>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1}^{2}+x_{2}^{2}-x_{3}^{2}-x_{4}^{2}=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/475a95c2a27331d828f52e1c07b4c1c275dbf7f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.318ex; height:3.176ex;" alt="{\displaystyle x_{1}^{2}+x_{2}^{2}-x_{3}^{2}-x_{4}^{2}=0}" loading="lazy"></span>.</dd></dl>

<p>Der Schnitt des Hyperboloids mit der <a href="Fernebene" class="mw-redirect" title="Fernebene">Fernebene</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 x_{4}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/efc78587303e5b6b94128da81327c072b4644939.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_{4}=0}" loading="lazy"></span> ist ein <a href="Kreis" title="Kreis">Kreis</a>.<br>Die Umformung zu <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_{1}-x_{3})(x_{1}+x_{3})+(x_{2}-x_{4})(x_{2}+x_{4})=0}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{1}-x_{3})(x_{1}+x_{3})+(x_{2}-x_{4})(x_{2}+x_{4})=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cef864e95bcf4c9761b8110e4e2ed8d71170619d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.771ex; height:2.843ex;" alt="{\displaystyle (x_{1}-x_{3})(x_{1}+x_{3})+(x_{2}-x_{4})(x_{2}+x_{4})=0}" loading="lazy"></span> und anschließende Einführung neuer <a href="Koordinatensystem" title="Koordinatensystem">Koordinaten</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 u_{1}=x_{1}-x_{2},\;u_{2}=x_{1}+x_{3},\;u_{3}=x_{2}-x_{4},\;u_{4}=x_{2}+x_{4}\;}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{1}=x_{1}-x_{2},\;u_{2}=x_{1}+x_{3},\;u_{3}=x_{2}-x_{4},\;u_{4}=x_{2}+x_{4}\;}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78be18c6daa7dd585285dda4e8fdd9e674c9881b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:58.045ex; height:2.343ex;" alt="{\displaystyle u_{1}=x_{1}-x_{2},\;u_{2}=x_{1}+x_{3},\;u_{3}=x_{2}-x_{4},\;u_{4}=x_{2}+x_{4}\;}" loading="lazy"></span> liefert die Beschreibung des einschaligen Hyperboloids in <a href="Homogene_Koordinaten" title="Homogene Koordinaten">homogenen Koordinaten</a> durch die <a href="Gleichung" title="Gleichung">Gleichung</a>

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{1}u_{2}+u_{3}u_{4}=0\ .}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24d0deb7e505d0d7c1ba841ec2a331e0e1792b22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.864ex; height:2.509ex;" alt="{\displaystyle u_{1}u_{2}+u_{3}u_{4}=0\ .}" loading="lazy"></span></dd></dl>

<p>In den neuen <a href="Koordinatensystem" title="Koordinatensystem">Koordinaten</a> schneidet die <a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</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 u_{4}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f2a71f9e7c05fd0b648fad09b3515e2ae5552c9.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 u_{4}=0}" loading="lazy"></span> das Hyperboloid in zwei <a href="Gerade" title="Gerade">Geraden</a>.<br>Führt man jetzt wieder <a href="Affine_Koordinaten" title="Affine Koordinaten">affine Koordinaten</a> durch <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={\tfrac {u_{1}}{u_{4}}},y={\tfrac {u_{2}}{u_{4}}},z={\tfrac {u_{3}}{u_{4}}}}">

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

<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 {u_{1}}{u_{4}}},y={\tfrac {u_{2}}{u_{4}}},z={\tfrac {u_{3}}{u_{4}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/354f76808940b542a76c548502cfb5a433ad50f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.76ex; height:3.676ex;" alt="{\displaystyle x={\tfrac {u_{1}}{u_{4}}},y={\tfrac {u_{2}}{u_{4}}},z={\tfrac {u_{3}}{u_{4}}}}" loading="lazy"></span> ein, erhält man die <a href="Gleichung" title="Gleichung">Gleichung</a> eines <a href="Hyperbolisches_Paraboloid" class="mw-redirect" title="Hyperbolisches Paraboloid">hyperbolischen Paraboloids</a>:

</p>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e98fe7c3b188da8361721aaeab358c8ed2d228f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.707ex; height:2.343ex;" alt="{\displaystyle z=-xy\ .}" loading="lazy"></span></dd></dl>

<p>Dies zeigt: Ein einschaliges Hyperboloid ist <i>projektiv</i> äquivalent zu einem <a href="Hyperbolisches_Paraboloid" class="mw-redirect" title="Hyperbolisches Paraboloid">hyperbolischen Paraboloid</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Zweischaliges_Hyperboloid">Zweischaliges Hyperboloid</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Zweischaliges_Einheitshyperboloid">Zweischaliges Einheitshyperboloid</h4></div>

<p>Lässt man die <a href="Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbel</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 -x^{2}+z^{2}=1}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/231bb5c06c44f80667434d0dc351b2767374e4af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.438ex; height:2.843ex;" alt="{\displaystyle -x^{2}+z^{2}=1}" loading="lazy"></span> 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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>-<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}">

</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>-<a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a> um 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}">

</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 rotieren (siehe Abbildung), so erhält man das zweischalige Einheits-Hyperboloid mit der <a href="Gleichung" title="Gleichung">Gleichung</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 -x^{2}-y^{2}+z^{2}=1}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49fe41f2ad3fbe55016b96f9bfeac26ff573cb70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.493ex; height:3.009ex;" alt="{\displaystyle -x^{2}-y^{2}+z^{2}=1}" loading="lazy"></span> oder in üblicher Form

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f958c5cda7b34cc5ac8503a213c8e8daf1ca92f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.093ex; height:3.009ex;" alt="{\displaystyle H_{2}\colon \ x^{2}+y^{2}-z^{2}=-1}" loading="lazy"></span>.</dd></dl>

<p>Der Schnitt der <a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</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 z=z_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f8e63a3f2769739a78c3c24a091f778fdc72dfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.322ex; height:2.009ex;" alt="{\displaystyle z=z_{0}}" loading="lazy"></span> 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 H_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}" loading="lazy"></span> ist ein <a href="Kreis" title="Kreis">Kreis</a> (falls <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_{0}^{2}>1}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{0}^{2}>1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04d1d7251ab2f14e688b43987d9fc7121e0bd589.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.405ex; height:3.176ex;" alt="{\displaystyle z_{0}^{2}>1}" loading="lazy"></span>) oder ein <a href="Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkt</a> (falls <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_{0}=\pm 1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{0}=\pm 1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b95cba5bce56db5070787cb6ee2612b548d39c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.204ex; height:2.509ex;" alt="{\displaystyle z_{0}=\pm 1}" loading="lazy"></span>) oder leer (falls <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_{0}^{2}<1}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{0}^{2}<1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60a5b1a98fd52d2aca6424a195ba4d5da45cdcc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.405ex; height:3.176ex;" alt="{\displaystyle z_{0}^{2}<1}" loading="lazy"></span>). <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 H_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}" loading="lazy"></span> besteht aus zwei Teilen, entsprechend den zwei Teilen der <a href="Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbel</a>.

</p><p>Das zweischalige Einheits-Hyperboloid ergibt sich durch <a href="Drehung" title="Drehung">Rotation</a> des <a href="Funktionsgraph" title="Funktionsgraph">Graphen</a> der <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</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 f(z)={\sqrt {z^{2}-1}}}">

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e11626a8e404fe23aeb918bf244c8d099afb5b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.746ex; height:3.509ex;" alt="{\displaystyle f(z)={\sqrt {z^{2}-1}}}" loading="lazy"></span> um 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}">

</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. Für die <a href="Ableitung_(Mathematik)" class="mw-redirect" title="Ableitung (Mathematik)">Ableitung</a> gilt <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 f'(z)={\frac {z}{\sqrt {z^{2}-1}}}}">

<msup>

</msup>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f'(z)={\frac {z}{\sqrt {z^{2}-1}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9071467a1e073121318e94e66ed1c32b38897cd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.308ex; height:6.009ex;" alt="{\displaystyle f'(z)={\frac {z}{\sqrt {z^{2}-1}}}}" loading="lazy"></span>. Das <a href="Volumen" title="Volumen">Volumen</a> und die Oberfläche für ein zweischalige Einheits-Hyperboloid mit der Höhe <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 h-1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98e6bd19265b6d8d6687ba7ecffeb872e0504cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.342ex; height:2.343ex;" alt="{\displaystyle h-1}" loading="lazy"></span> ergeben sich nach den <a href="Guldinsche_Regeln" class="mw-redirect" title="Guldinsche Regeln">Guldinschen Regeln</a> mithilfe von <a href="Integralrechnung" title="Integralrechnung">Integralen</a>.

</p>

<div class="mw-heading mw-heading4"><h4 id="Volumen_2">Volumen</h4></div>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=\pi \int _{1}^{h}(f(z))^{2}\ \mathrm {d} z=\pi \int _{1}^{h}z^{2}-1\ \mathrm {d} z=\pi \left({\frac {h^{3}}{3}}-h+{\frac {2}{3}}\right)={\frac {\pi }{3}}\left(h^{3}-3h+2\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b27c32ed6e8a5bc1221e856c5678153bf601f591.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:78.213ex; height:6.509ex;" alt="{\displaystyle V=\pi \int _{1}^{h}(f(z))^{2}\ \mathrm {d} z=\pi \int _{1}^{h}z^{2}-1\ \mathrm {d} z=\pi \left({\frac {h^{3}}{3}}-h+{\frac {2}{3}}\right)={\frac {\pi }{3}}\left(h^{3}-3h+2\right)}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Oberfläche_2"><span id="Oberfl.C3.A4che_2"></span>Oberfläche</h4></div>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

<msqrt>

<msup>

<mrow>

<mrow>

<msup>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A&=2\pi \int _{0}^{h}f(z){\sqrt {1+\left(f'(z)\right)^{2}}}\ \mathrm {d} z\\&=2\pi \int _{0}^{h}{\sqrt {z^{2}-1}}{\sqrt {1+\left({\frac {z}{\sqrt {z^{2}-1}}}\right)^{2}}}\ \mathrm {d} z\\&=2\pi \int _{0}^{h}{\sqrt {2z^{2}-1}}\ \mathrm {d} z\\&=2\pi \left({\frac {1}{2}}z{\sqrt {2z^{2}-1}}-{\frac {\sqrt {2}}{2}}\ln \left({\sqrt {2}}z+{\sqrt {2z^{2}-1}}\right){\Big |}_{z=0}^{z=h}\right)\\&=\pi \left(h{\sqrt {2h^{2}-1}}-{\sqrt {2}}\ln \left({\sqrt {2}}h+{\sqrt {2h^{2}-1}}\right)\right)\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df1fe968b5e31bc5eb841c9e6a49c22185f08d55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -16.005ex; width:57.027ex; height:33.176ex;" alt="{\displaystyle {\begin{aligned}A&=2\pi \int _{0}^{h}f(z){\sqrt {1+\left(f'(z)\right)^{2}}}\ \mathrm {d} z\\&=2\pi \int _{0}^{h}{\sqrt {z^{2}-1}}{\sqrt {1+\left({\frac {z}{\sqrt {z^{2}-1}}}\right)^{2}}}\ \mathrm {d} z\\&=2\pi \int _{0}^{h}{\sqrt {2z^{2}-1}}\ \mathrm {d} z\\&=2\pi \left({\frac {1}{2}}z{\sqrt {2z^{2}-1}}-{\frac {\sqrt {2}}{2}}\ln \left({\sqrt {2}}z+{\sqrt {2z^{2}-1}}\right){\Big |}_{z=0}^{z=h}\right)\\&=\pi \left(h{\sqrt {2h^{2}-1}}-{\sqrt {2}}\ln \left({\sqrt {2}}h+{\sqrt {2h^{2}-1}}\right)\right)\end{aligned}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Tangentialebenen_2">Tangentialebenen</h4></div>

<p>Die <a href="Tangentialebene" title="Tangentialebene">Tangentialebene</a> 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 H_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}" loading="lazy"></span> in einem <a href="Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkt</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 (x_{0},y_{0},z_{0})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39177ddeeb9f9a393b664e522bc8e3bf0face153.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.59ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0},z_{0})}" loading="lazy"></span> hat die <a href="Gleichung" title="Gleichung">Gleichung</a> (siehe oben)

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}x+y_{0}y-z_{0}z+1=0\ .}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6251b36917d87d7ca3648099675a12ad9f5e55d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.458ex; height:2.509ex;" alt="{\displaystyle x_{0}x+y_{0}y-z_{0}z+1=0\ .}" loading="lazy"></span></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Ebene_Schnitte_2">Ebene Schnitte</h4></div>

<ul><li>Ebenen mit einer Neigung kleiner 1 (Neigung der <a href="Asymptote" title="Asymptote">Asymptoten</a> der erzeugenden <a href="Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbel</a>) schneiden <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 H_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}" loading="lazy"></span> entweder in einer <i><a href="Ellipse" title="Ellipse">Ellipse</a></i> oder in einem <i><a href="Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkt</a></i> oder nicht,</li>

<li>Ebenen mit einer Neigung gleich 1 und durch den <a href="Koordinatenursprung" class="mw-redirect" title="Koordinatenursprung">Koordinatenursprung</a> schneiden <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 H_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}" loading="lazy"></span> nicht,</li>

<li>Ebenen mit einer Neigung gleich 1 und nicht durch den Koordinatenursprung schneiden <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 H_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}" loading="lazy"></span> in einer <i><a href="Parabel_(Mathematik)" title="Parabel (Mathematik)">Parabel</a>,</i></li>

<li>Ebenen mit einer Neigung größer 1 schneiden <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 H_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}" loading="lazy"></span> in einer <i><a href="Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbel</a>.</i><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Affine_Bilder_2">Affine Bilder</h4></div>

<p>Ein <i>beliebiges</i> zweischaliges Hyperboloid ist das <a href="Affine_Abbildung" title="Affine Abbildung">affine Bild</a> des Einheitshyperboloids <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 H_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}" loading="lazy"></span>. Die einfachsten affinen Bilder erhält man durch Skalierung der Koordinatenachsen:

</p>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=-1\ ,\ a,b,c>0.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd7afc1fb89c3d9fc19ff5bb51b298ec0d4a3e1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:33.407ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=-1\ ,\ a,b,c>0.}" loading="lazy"></span></dd></dl>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}" loading="lazy"></span> sind die Höhenschnitte <a href="Kreis" title="Kreis">Kreise</a>. Andern falls sind es <a href="Ellipse" title="Ellipse">Ellipsen</a>. Ein solches Hyperboloid nennt man <i>zweischaliges Rotationshyperboloid</i>. Dass ein beliebiges zweischaliges Hyperboloid auch immer Kreise enthält, wird in <a href="Kreisschnittebene" title="Kreisschnittebene">Kreisschnittebene</a> gezeigt.

</p><p>Für ein zweischaliges Hyperboloid <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 {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}-{\tfrac {z^{2}}{c^{2}}}=-1}">

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}-{\tfrac {z^{2}}{c^{2}}}=-1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f35b53c65a3571c87152d2ce37568b09ec1c4b91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:19.285ex; height:4.843ex;" alt="{\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}-{\tfrac {z^{2}}{c^{2}}}=-1}" loading="lazy"></span> ergibt sich die folgende <a href="Parameterdarstellung" title="Parameterdarstellung">Parameterdarstellung</a>:

</p>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}(s,t)={\begin{pmatrix}a\sinh s\cos t\\b\sinh s\sin t\\\pm \,c\cosh s\end{pmatrix}},\quad s\in \mathbb {R} ,\ 0\leq t\leq 2\pi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a04f8c86ba575ba7635d2d9d99d7f55a7723d975.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:47.261ex; height:9.176ex;" alt="{\displaystyle {\vec {x}}(s,t)={\begin{pmatrix}a\sinh s\cos t\\b\sinh s\sin t\\\pm \,c\cosh s\end{pmatrix}},\quad s\in \mathbb {R} ,\ 0\leq t\leq 2\pi }" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Homogene_Koordinaten_2">Homogene Koordinaten</h4></div>

<p>Führt man wie bei <b><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 H_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d4d9a872a55b209f2eb7cc23a71e5e1541bd1f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{1}}" loading="lazy"></span></b> <a href="Homogene_Koordinaten" title="Homogene Koordinaten">homogene Koordinaten</a> ein, erhält man die homogene Beschreibung von <b><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 H_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}" loading="lazy"></span></b> durch die <a href="Gleichung" title="Gleichung">Gleichung</a>:

</p>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1}^{2}+x_{2}^{2}-x_{3}^{2}+x_{4}^{2}=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ca4e7706534cef12e2006136fcce1fea49bfc24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.318ex; height:3.176ex;" alt="{\displaystyle x_{1}^{2}+x_{2}^{2}-x_{3}^{2}+x_{4}^{2}=0}" loading="lazy"></span>.</dd></dl>

<p>Vertauscht man die <a href="Koordinatensystem" title="Koordinatensystem">Koordinaten</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 x_{3},x_{4}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{3},x_{4}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a42452f484dd7662299d0ee3a522e44ac25d8646.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.802ex; height:2.009ex;" alt="{\displaystyle x_{3},x_{4}}" loading="lazy"></span> und kehrt wieder zu <a href="Affine_Koordinaten" title="Affine Koordinaten">affinen Koordinaten</a> zurück, ergibt sich die <a href="Gleichung" title="Gleichung">Gleichung</a> der <a href="Einheitskugel" title="Einheitskugel">Einheitskugel</a>:

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e43859ae677372fd13413fd5edbdc3d11e57d87f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.912ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}+z^{2}=1\ .}" loading="lazy"></span></dd></dl>

<p>Dies zeigt: Ein zweischaliges Hyperboloid ist <i>projektiv</i> äquivalent zu einer <a href="Kugel" title="Kugel">Kugel</a>.

</p>

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

<p>Wie <a href="Ellipse" title="Ellipse">Ellipsen</a> und <a href="Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbeln</a> haben auch Hyperboloide Scheitel und Nebenscheitel und <a href="Symmetrie_(Geometrie)" title="Symmetrie (Geometrie)">Symmetrien</a>. Die Hyperboloide <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 {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1,\quad {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=-1\ }">

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1,\quad {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=-1\ }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff213ad2811d5de370c41e848b5840d4f50c4d1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:44.132ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1,\quad {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=-1\ }" loading="lazy"></span>sind offensichtlich

</p>

<ul><li><a href="Punktsymmetrisch" class="mw-redirect" title="Punktsymmetrisch">punktsymmetrisch</a> zum <a href="Koordinatenursprung" class="mw-redirect" title="Koordinatenursprung">Koordinatenursprung</a>,</li>

<li><a href="Symmetrie_(Geometrie)" title="Symmetrie (Geometrie)">symmetrisch</a> zu den <a href="Koordinatenebene" title="Koordinatenebene">Koordinatenebenen</a> sowie</li>

<li><a href="Rotationssymmetrisch" class="mw-redirect" title="Rotationssymmetrisch">rotationssymmetrisch</a> zur z-Achse und symmetrisch zu jeder <a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a> durch die z-Achse, falls <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=b}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}" loading="lazy"></span> ist.</li></ul>

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

<p>Den <a href="Doppelkegel" class="mw-redirect" title="Doppelkegel">Doppelkegel</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 x^{2}+y^{2}-z^{2}=0}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc134db1287ee8472bd9245403648e30d2aa727b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.685ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}-z^{2}=0}" loading="lazy"></span> kann man als <a href="Grenzfl%C3%A4che" title="Grenzfläche">Grenzfläche</a> zwischen den Scharen von einschaligen bzw. zweischaligen Hyperboloiden <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^{2}+y^{2}-z^{2}=c^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e418b8b30da149a6b5803f25b78e8d675c08f36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.584ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}-z^{2}=c^{2}}" 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 x^{2}+y^{2}-z^{2}=-c^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07174e15331cd5804ad5fa997ae751b195b08d53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.392ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}-z^{2}=-c^{2}}" loading="lazy"></span> auffassen. Er entsteht durch <a href="Drehung" title="Drehung">Rotation</a> der gemeinsamen <a href="Asymptote" title="Asymptote">Asymptoten</a> der Erzeuger-Hyperbeln.

</p>

<div class="mw-heading mw-heading2"><h2 id="Gemeinsame_Parameterdarstellung">Gemeinsame Parameterdarstellung</h2></div>

<p>Es gibt verschiedene Möglichkeiten. Hyperboloide zu <a href="Parametrisieren" class="mw-redirect" title="Parametrisieren">parametrisieren</a>. Eine einfache Möglichkeit, das einschalige und zweischalige Hyperboloid und den <a href="Kegel_(Geometrie)" title="Kegel (Geometrie)">Kegel</a> zu parametrisieren, ist:

</p>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}(s,t)={\begin{pmatrix}a\ {\sqrt {s^{2}+d}}\ \cos t\\b\ {\sqrt {s^{2}+d}}\ \sin t\\cs\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5ac690c28f568e6173224b3221260040fee4b07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:29.661ex; height:10.509ex;" alt="{\displaystyle {\vec {x}}(s,t)={\begin{pmatrix}a\ {\sqrt {s^{2}+d}}\ \cos t\\b\ {\sqrt {s^{2}+d}}\ \sin t\\cs\end{pmatrix}}}" loading="lazy"></span></dd></dl>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/958b426c5ace91d4fb7f5a3becd7b21dba288d50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.176ex;" alt="{\displaystyle d=1}" loading="lazy"></span> ergibt sich ein einschaliges, für <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=-1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d51c156019d41a34e4ab088ef479980c6c6e5614.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.285ex; height:2.343ex;" alt="{\displaystyle d=-1}" loading="lazy"></span> ein zweischaliges Hyperboloid und für <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=0}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c87f7389ad2498c0f93551ec4fc92a882548484f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.176ex;" alt="{\displaystyle d=0}" loading="lazy"></span> ein <a href="Doppelkegel" class="mw-redirect" title="Doppelkegel">Doppelkegel</a>.

</p>

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

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

<p>Die Form des Rotationshyperboloids wird unter anderem im <a href="Bauwesen" title="Bauwesen">Bauwesen</a> bei <a href="Hyperboloidkonstruktion" title="Hyperboloidkonstruktion">Hyperboloidkonstruktionen</a> angewendet. Den ersten Turm der Welt in dieser Form baute <a href="Wladimir_Grigorjewitsch_Schuchow" title="Wladimir Grigorjewitsch Schuchow">Wladimir Schuchow</a> für die <a href="Allrussische_Industrie-_und_Handwerksausstellung_1896" title="Allrussische Industrie- und Handwerksausstellung 1896">Allrussische Industrie- und Handwerksausstellung 1896</a>.

</p><p>Der Architekt <a href="Antoni_Gaud%C3%AD" title="Antoni Gaudí">Antoni Gaudí</a> verwendete die Form als gestalterisches Konstruktionsprinzip. Auch das Kunstwerk <a href="Mae_West_(Kunstwerk)" title="Mae West (Kunstwerk)">Mae West</a> in <a href="M%C3%BCnchen" title="München">München</a> ist ein 52 Meter hoher <a href="Rotationshyperboloid" class="mw-redirect" title="Rotationshyperboloid">Rotationshyperboloid</a> aus <a href="Kohlenstofffaserverst%C3%A4rkter_Kunststoff" title="Kohlenstofffaserverstärkter Kunststoff">kohlenstofffaserverstärktem Kunststoff</a>.

</p>

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

<p>In Teleskopen, die nach der <a href="Cassegrain-Teleskop" title="Cassegrain-Teleskop">Cassegrain-Bauweise</a> aufgebaut sind, gibt es einen konvex-hyperbolischen Fangspiegel.

</p>

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

<ul><li><a href="HP-Schale" title="HP-Schale">HP-Schale</a></li>

<li><a href="Ellipsoid" title="Ellipsoid">Ellipsoid</a></li>

<li><a href="Paraboloid" title="Paraboloid">Paraboloid</a></li>

<li><a href="Rotationsparaboloid" class="mw-redirect" title="Rotationsparaboloid">Rotationsparaboloid</a></li>

<li><a href="Zylinder_(Geometrie)" title="Zylinder (Geometrie)">Zylinder</a></li>

<li><a href="Kegel_(Geometrie)" title="Kegel (Geometrie)">Kegel</a></li>

<li><a href="Konfokale_Quadriken" class="mw-redirect" title="Konfokale Quadriken">Konfokale Quadriken</a></li>

<li><a href="NIGRES-Stromleitungsmast_an_der_Oka" title="NIGRES-Stromleitungsmast an der Oka">NIGRES-Stromleitungsmast an der Oka</a></li></ul>

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

<ul><li><a href="Albrecht_Beutelspacher" title="Albrecht Beutelspacher">Albrecht Beutelspacher</a>, Ute Rosenbaum: <cite style="font-style:italic">Projektive Geometrie. Von den Grundlagen bis zu den Anwendungen</cite> (= <cite style="font-style:italic">Vieweg Studium: Aufbaukurs Mathematik</cite>). 2., durchgesehene und erweiterte Auflage. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X (<a rel="nofollow" class="external text" href="http://d-nb.info/972794298/04">online</a> [abgerufen am 1. April 2012]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hyperboloid&rft.au=Albrecht+Beutelspacher%2C+Ute+Rosenbaum&rft.btitle=Projektive+Geometrie.+Von+den+Grundlagen+bis+zu+den+Anwendungen&rft.date=2004&rft.edition=2.%2C+durchgesehene+und+erweiterte&rft.genre=book&rft.isbn=352817241X&rft.place=Wiesbaden&rft.pub=Vieweg&rft.series=Vieweg+Studium%3A+Aufbaukurs+Mathematik" style="display:none"> </span></li>

<li>Burkard Polster: <cite style="font-style:italic">A geometrical picture book</cite>. 1. Auflage. Springer, New York / Berlin / Heidelberg 1998, ISBN 0-387-98437-2.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hyperboloid&rft.au=Burkard+Polster&rft.btitle=A+geometrical+picture+book&rft.date=1998&rft.edition=1.&rft.genre=book&rft.isbn=0387984372&rft.place=New+York+%2F+Berlin+%2F+Heidelberg&rft.pub=Springer" style="display:none"> </span></li>

<li>Hermann Schaal: <cite style="font-style:italic">Lineare Algebra und analytische Geometrie</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>III</span>. Vieweg, 1980, ISBN 3-528-13057-1.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hyperboloid&rft.au=Hermann+Schaal&rft.btitle=Lineare+Algebra+und+analytische+Geometrie&rft.date=1980&rft.genre=book&rft.isbn=3528130571&rft.pub=Vieweg&rft.volume=III" style="display:none"> </span></li>

<li>Günter Scheja, Uwe Storch: <cite style="font-style:italic">Lehrbuch der Algebra. Unter Einschluß der linearen Algebra</cite>. 2., überarb. und erw. Auflage. Teubner, Stuttgart 1994, ISBN 3-519-12203-0.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hyperboloid&rft.au=G%C3%BCnter+Scheja%2C+Uwe+Storch&rft.btitle=Lehrbuch+der+Algebra.+Unter+Einschlu%C3%9F+der+linearen+Algebra&rft.date=1994&rft.edition=2.%2C+%C3%BCberarb.+und+erw.&rft.genre=book&rft.isbn=3519122030&rft.place=Stuttgart&rft.pub=Teubner" style="display:none"> </span></li>

<li>Uwe Storch, Hartmut Wiebe: <cite style="font-style:italic">Lehrbuch der Mathematik</cite>. 2., überarb. und erw. Auflage. BI-Wissenschafts-Verlag, 1999, ISBN 3-411-14101-8.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hyperboloid&rft.au=Uwe+Storch%2C+Hartmut+Wiebe&rft.btitle=Lehrbuch+der+Mathematik&rft.date=1999&rft.edition=2.%2C+%C3%BCberarb.+und+erw.&rft.genre=book&rft.isbn=3411141018&rft.pub=BI-Wissenschafts-Verlag" style="display:none"> </span></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:Hyperboloid?uselang=de"><span lang="en">Commons</span>: Hyperboloid</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/Hyperboloid.html"><i>Hyperboloid</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li>

/* start https://de.wikipedia.org/ */

.mw-parser-output .webarchiv-memento a{color:inherit}

/* end https://de.wikipedia.org/ */

</style><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100805010909/http://www.exopas.com/beta/faces/pages/gallery/animated.jsp"><i>Animiertes Hyperboloid bei EXOPAS.</i></a> (<span class="webarchiv-memento"><a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a></span> vom 5. August 2010 im <i><a href="Internet_Archive" title="Internet Archive">Internet Archive</a></i>).</li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">W. Steinhilper (Hrsg.): <cite style="font-style:italic">Konstruktionselemente des Maschinenbaus 2</cite>. Springer-Verlag, 2006, ISBN 3-540-29629-8, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>374</span> (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=E7ojBAAAQBAJ&printsec=frontcover&pg=374#v=onepage&q&f=false">google.de</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hyperboloid&rft.btitle=Konstruktionselemente+des+Maschinenbaus+2&rft.date=2006&rft.genre=book&rft.isbn=3540296298&rft.pages=374&rft.pub=Springer-Verlag" style="display:none"> </span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170918204037/http://modellsammlung.uni-goettingen.de/index.php?lang=de&r=4&sr=14&m=61"><i>Modellsammlung d. Uni Göttingen: Hyperboloidgetriebe.</i></a> Archiviert vom <style data-mw-deduplicate="TemplateStyles:r250917974">

/* start https://de.wikipedia.org/ */

.mw-parser-output .dewiki-iconexternal>a{background-position:center right!important;background-repeat:no-repeat!important}body.skin-minerva .mw-parser-output .dewiki-iconexternal>a{background-image:url("./_mw_/OOjs_UI_icon_external-link-ltr-progressive.svg")!important;background-size:10px!important;padding-right:13px!important}body.skin-timeless .mw-parser-output .dewiki-iconexternal>a,body.skin-monobook .mw-parser-output .dewiki-iconexternal>a{background-image:url("./_mw_/MediaWiki_external_link_icon.svg")!important;padding-right:13px!important}body.skin-vector .mw-parser-output .dewiki-iconexternal>a{background-image:url("./_mw_/Link.ernal-small-ltr-progressive.svg")!important;background-size:0.857em!important;padding-right:1em!important}

/* end https://de.wikipedia.org/ */

</style><span class="dewiki-iconexternal"><a class="external text" href="https://redirecter.toolforge.org/?url=http%3A%2F%2Fmodellsammlung.uni-goettingen.de%2Findex.php%3Flang%3Dde%26r%3D4%26sr%3D14%26m%3D61">Original</a></span> (nicht mehr online verfügbar) am <span style="white-space:nowrap;">18. September 2017</span><span>;</span><span class="Abrufdatum"> abgerufen am 3. April 2023</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AHyperboloid&rft.title=Modellsammlung+d.+Uni+G%C3%B6ttingen%3A+Hyperboloidgetriebe&rft.description=Modellsammlung+d.+Uni+G%C3%B6ttingen%3A+Hyperboloidgetriebe&rft.identifier=https%3A%2F%2Fweb.archive.org%2Fweb%2F20170918204037%2Fhttp%3A%2F%2Fmodellsammlung.uni-goettingen.de%2Findex.php%3Flang%3Dde%26r%3D4%26sr%3D14%26m%3D61&rft.source=http://modellsammlung.uni-goettingen.de/index.php?lang=de&r=4&sr=14&m=61"> </span></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">K. Strubecker: <cite style="font-style:italic">Vorlesungen der Darstellenden Geometrie</cite>. Vandenhoeck & Ruprecht, Göttingen 1967, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>218</span> (<a rel="nofollow" class="external text" href="https://people.math.harvard.edu/~knill/history/darstellend/Strubecker.pdf">harvard.edu</a> [PDF; <span style="white-space:nowrap">12,5<span style="display:inline-block;width:.2em"> </span>MB</span>]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hyperboloid&rft.au=K.+Strubecker&rft.btitle=Vorlesungen+der+Darstellenden+Geometrie&rft.date=1967&rft.genre=book&rft.pages=218&rft.place=G%C3%B6ttingen&rft.pub=Vandenhoeck+%26+Ruprecht" style="display:none"> </span></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><i><a rel="nofollow" class="external text" href="http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf">CDKG: Computerunterstützte Darstellende und Konstruktive Geometrie.</a></i> TU Darmstadt (PDF; 3,4 MB), S. 116.</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><i><a rel="nofollow" class="external text" href="http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf">CDKG: Computerunterstützte Darstellende und Konstruktive Geometrie.</a></i> TU Darmstadt (PDF; 3,4 MB), S. 122.</span>

</li>

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

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-12-19" href="https://de.wikipedia.org/wiki/?title=Hyperboloid&oldid=262556606">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>