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

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Parameterform</b> oder <b>Punktrichtungsform</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> ist in der <a href="Mathematik" title="Mathematik">Mathematik</a> eine spezielle Form einer <a href="Geradengleichung" title="Geradengleichung">Geradengleichung</a> oder <a href="Ebenengleichung" title="Ebenengleichung">Ebenengleichung</a>. In der Parameterform wird eine <a href="Gerade" title="Gerade">Gerade</a> durch einen <a href="Ortsvektor" title="Ortsvektor">Ortsvektor</a> (<i>Stützvektor</i> genannt) und einen <b>Richtungsvektor</b> dargestellt. Jeder <a href="Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkt</a> der Geraden wird dann in Abhängigkeit von einem <a href="Parameter_(Mathematik)" title="Parameter (Mathematik)">Parameter</a> beschrieben, der angibt, wie viele Richtungsvektoren der Punkt vom Stützvektor entfernt ist. Eine Ebene wird durch einen Stützvektor und zwei Richtungsvektoren dargestellt. Jeder Punkt der Ebene wird dann in Abhängigkeit von zwei Parametern beschrieben. Bei der Parameterform handelt es sich um eine spezielle <a href="Parameterdarstellung" title="Parameterdarstellung">Parameterdarstellung</a>.

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<div class="mw-heading mw-heading2"><h2 id="Parameterform_einer_Geradengleichung">Parameterform einer Geradengleichung</h2></div>

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

<p>In der Parameterform wird eine Gerade in der Ebene durch einen Stützvektor <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 {p}}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span> und einen Richtungsvektor <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 {u}}\neq {\vec {0}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9045f0f35f4143342908df020a47b5a44d6c0f33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:3.343ex;" alt="{\displaystyle {\vec {u}}\neq {\vec {0}}}" loading="lazy"></span> beschrieben. Eine Gerade besteht dann aus denjenigen Punkten in der Ebene, deren <a href="Ortsvektor" title="Ortsvektor">Ortsvektoren</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}}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}}" loading="lazy"></span> die Gleichung

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<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}={\vec {p}}+s{\vec {u}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3eda5382a0f3eb91eb74432f2697f6978aa88c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.014ex; height:2.676ex;" alt="{\displaystyle {\vec {x}}={\vec {p}}+s{\vec {u}}}" 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 s\in \mathbb {R} }">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36efff902c6854b1196e79dec095b31e0c6a8ee9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.609ex; height:2.176ex;" alt="{\displaystyle s\in \mathbb {R} }" loading="lazy"></span></dd></dl>

<p>erfüllen. Der Stützvektor ist dabei der Ortsvektor eines beliebigen Punkts auf der Geraden, der auch als <i>Aufpunkt</i> bezeichnet wird. Der Richtungsvektor ist der Differenzvektor (Verbindungsvektor) zu einem beliebigen weiteren Punkt der Geraden. In der Parameterform werden die Punkte der Geraden in Abhängigkeit von dem Parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span> dargestellt. Jedem Wert 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 s}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span> entspricht genau ein Punkt der Geraden. Durchläuft der Parameter die <a href="Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a>, so erhält man alle Punkte der Geraden. Ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89c41e9cf70c5e5b56e2128a136985a75f90ba43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}" loading="lazy"></span> ein <a href="Einheitsvektor" title="Einheitsvektor">Einheitsvektor</a>, dann gibt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |s|}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ae65dea0cc836140252292ee9adf2d8b5102055.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.384ex; height:2.843ex;" alt="{\displaystyle |s|}" loading="lazy"></span> den <a href="Abstand" title="Abstand">Abstand</a> eines Punkts auf der Geraden vom Aufpunkt an.

</p><p>Durch jede Gleichung der Form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}={\vec {p}}+s{\vec {u}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}={\vec {p}}+s{\vec {u}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3eda5382a0f3eb91eb74432f2697f6978aa88c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.014ex; height:2.676ex;" alt="{\displaystyle {\vec {x}}={\vec {p}}+s{\vec {u}}}" loading="lazy"></span> ist genau eine Gerade festgelegt. Umgekehrt gibt es jedoch zu jeder Geraden <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}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> unendlich viele Parameterdarstellungen, da zum einen jeder Ortsvektor eines Geradenpunkts 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 g}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> als Stützvektor gewählt werden kann und zum anderen für jeden Richtungsvektor <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 {u}}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89c41e9cf70c5e5b56e2128a136985a75f90ba43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}" loading="lazy"></span> auch <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 k{\vec {u}}\ (k\neq 0)}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0efce587256a7d7affb7d0f6fa839c4cbd42cf9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.403ex; height:2.843ex;" alt="{\displaystyle k{\vec {u}}\ (k\neq 0)}" loading="lazy"></span> ein Richtungsvektor ist.<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>

</p><p>Legt man ein ebenes rechtwinkliges Koordinatensystem zugrunde und stellt die Vektoren mithilfe ihrer Komponenten dar, so liest sich die Geradengleichung als

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}p_{1}\\p_{2}\end{pmatrix}}+s{\begin{pmatrix}u_{1}\\u_{2}\end{pmatrix}}={\begin{pmatrix}p_{1}+su_{1}\\p_{2}+su_{2}\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10749325e4adad25b35373d40e94ac974f26786f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.348ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}p_{1}\\p_{2}\end{pmatrix}}+s{\begin{pmatrix}u_{1}\\u_{2}\end{pmatrix}}={\begin{pmatrix}p_{1}+su_{1}\\p_{2}+su_{2}\end{pmatrix}}}" 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 s\in \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle s\in \mathbb {R} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36efff902c6854b1196e79dec095b31e0c6a8ee9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.609ex; height:2.176ex;" alt="{\displaystyle s\in \mathbb {R} }" loading="lazy"></span>.</dd></dl>

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

<p>Im Bild oben ist der Stützvektor <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 {p}}={\begin{pmatrix}2\\2\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}={\begin{pmatrix}2\\2\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15f0953dab7120ff24b9d39bf433af3b3c634d17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:9.848ex; height:6.176ex;" alt="{\displaystyle {\vec {p}}={\begin{pmatrix}2\\2\end{pmatrix}}}" loading="lazy"></span> und der Richtungsvektor <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 {u}}={\begin{pmatrix}2\\-1\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {u}}={\begin{pmatrix}2\\-1\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/531d377c520c32b999ba48d789bbd4f415f6fde6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.571ex; height:6.176ex;" alt="{\displaystyle {\vec {u}}={\begin{pmatrix}2\\-1\end{pmatrix}}}" loading="lazy"></span>, man erhält als Geradengleichung

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}2\\2\end{pmatrix}}+s{\begin{pmatrix}2\\-1\end{pmatrix}}={\begin{pmatrix}2+2s\\2-s\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5befb05011f0991bb472901db840624d2c307749.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.591ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}2\\2\end{pmatrix}}+s{\begin{pmatrix}2\\-1\end{pmatrix}}={\begin{pmatrix}2+2s\\2-s\end{pmatrix}}}" loading="lazy"></span>.</dd></dl>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span> ergibt dann einen Geradenpunkt, zum Beispiel liefert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=0}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7903b8069a44c70f6f96511675bdd9a4ff200ed7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=0}" loading="lazy"></span> den Geradenpunkt <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 {\begin{pmatrix}2\\2\end{pmatrix}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fbe898a2eef9b1feb7612f7ab3b539d958a5b3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:5.335ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}2\\2\end{pmatrix}}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bac386d8f227fb823cede9b3e33d706cad3ed306.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=1}" loading="lazy"></span> den Geradenpunkt <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 {\begin{pmatrix}4\\1\end{pmatrix}}}">

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}4\\1\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dab805fcac8a53da23e8957fb1ee173031bc24cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:5.335ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}4\\1\end{pmatrix}}}" loading="lazy"></span>.

</p>

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

<div class="mw-heading mw-heading4"><h4 id="Aus_der_Zweipunkteform">Aus der Zweipunkteform</h4></div>

<p>Aus der <a href="Zweipunkteform" title="Zweipunkteform">Zweipunkteform</a> einer Geradengleichung lässt sich ein Richtungsvektor der Geraden als Differenzvektor zwischen den Ortsvektoren <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 {p}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {q}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a4e063f8ee7dae2488859c45a4e645db5148085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.309ex; height:2.676ex;" alt="{\displaystyle {\vec {q}}}" loading="lazy"></span> der beiden Punkte erhalten, das heißt

</p>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30b22fd36fb77096a4bf26e30b515e893e9533fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.902ex; height:2.676ex;" alt="{\displaystyle {\vec {u}}={\vec {q}}-{\vec {p}}}" loading="lazy"></span>.</dd></dl>

<p>Als Stützvektor <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 {p}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span> kann der Ortsvektor eines der Punkte verwendet werden.

</p>

<div class="mw-heading mw-heading4"><h4 id="Aus_der_Normalenform">Aus der Normalenform</h4></div>

<p>Aus der <a href="Normalenform#Normalenform_einer_Geradengleichung" title="Normalenform">Normalenform einer Geradengleichung</a> kann ein Richtungsvektor der Geraden bestimmt werden, indem die beiden Komponenten des <a href="Normalenvektor" title="Normalenvektor">Normalenvektors</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 {n}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49569db585c1b6306d5ffd91161775f67235fae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.343ex;" alt="{\displaystyle {\vec {n}}}" loading="lazy"></span> der Geraden vertauscht werden und bei einer der beiden Komponenten das <a href="Vorzeichen_(Zahl)" title="Vorzeichen (Zahl)">Vorzeichen</a> geändert wird, das heißt

</p>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {u}}={\begin{pmatrix}-n_{2}\\n_{1}\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e47cd5abd8ce0831b195e0f52176fd1006a5728e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.858ex; height:6.176ex;" alt="{\displaystyle {\vec {u}}={\begin{pmatrix}-n_{2}\\n_{1}\end{pmatrix}}}" loading="lazy"></span>.</dd></dl>

<p>Der Stützvektor <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 {p}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span> kann aus der Normalenform übernommen werden.

</p>

<div class="mw-heading mw-heading4"><h4 id="Aus_der_Normalform">Aus der Normalform</h4></div>

<p>Aus der Normalform <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=mx+n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ae774e96bc6f9f2eb75c360415fe3096dd3e8e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.859ex; height:2.343ex;" alt="{\displaystyle y=mx+n}" loading="lazy"></span> kann mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}=(1,m)^{T}}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {u}}=(1,m)^{T}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c09ff23344a07efc0d6ad5211dc546653481ad9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.864ex; height:3.176ex;" alt="{\displaystyle {\vec {u}}=(1,m)^{T}}" loading="lazy"></span>ein Richtungsvektor gewählt werden. Aber auch jeder Vektor <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,rm)^{T}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c56379261b523fbc8cbcfa1cba4b7595ba9e2ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.37ex; height:3.176ex;" alt="{\displaystyle (r,rm)^{T}}" 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 r\neq 0}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/034cc599221cc81da7ebd4c9090e1a988809b475.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.31ex; height:2.676ex;" alt="{\displaystyle r\neq 0}" loading="lazy"></span> ist ein Richtungsvektor. Als Stützvektor bietet sich <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 {p}}=(0,n)^{T}}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}=(0,n)^{T}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5f3d51a111425b76bcaec74ed45e41b1ff099ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:11.303ex; height:3.176ex;" alt="{\displaystyle {\vec {p}}=(0,n)^{T}}" loading="lazy"></span>an.

</p>

<div class="mw-heading mw-heading4"><h4 id="Aus_der_allgemeinen_Koordinatenform">Aus der allgemeinen Koordinatenform</h4></div>

<p>Aus der <a href="Allgemeine_Koordinatenform#Allgemeine_Koordinatenform_einer_Geradengleichung" title="Allgemeine Koordinatenform">allgemeinen Koordinatenform</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 ax+by=c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ax+by=c}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a39c048e6c827a005e417be0e1bd5ac314e6fcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.658ex; height:2.509ex;" alt="{\displaystyle ax+by=c}" loading="lazy"></span> lässt sich ein Normalenvektor der Gerade direkt als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}=(a,b)^{T}}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}=(a,b)^{T}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d77b0daabd4d4a97099645684cf60f839e04b79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.953ex; height:3.176ex;" alt="{\displaystyle {\vec {n}}=(a,b)^{T}}" loading="lazy"></span> ablesen und damit ein Richtungsvektor der Gerade analog zur Normalenform über

</p>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {u}}={\begin{pmatrix}-b\\a\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70724ad3a8f975068d56f6e5db0f4b4c221bf1e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.406ex; height:6.176ex;" alt="{\displaystyle {\vec {u}}={\begin{pmatrix}-b\\a\end{pmatrix}}}" loading="lazy"></span></dd></dl>

<p>ermitteln. Einen Stützvektor der Gerade erhält man, je nachdem ob <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" 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 b}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span> ungleich null ist, durch Wahl von

</p>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}={\begin{pmatrix}c/a\\0\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a7c5faf3483ff44333c59401ed3fa77cd59fb05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:12.085ex; height:6.176ex;" alt="{\displaystyle {\vec {p}}={\begin{pmatrix}c/a\\0\end{pmatrix}}}" 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 {\vec {p}}={\begin{pmatrix}0\\c/b\end{pmatrix}}}">

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}={\begin{pmatrix}0\\c/b\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5449b95103fe83ca7668f76d043c94f97357dc5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:11.853ex; height:6.176ex;" alt="{\displaystyle {\vec {p}}={\begin{pmatrix}0\\c/b\end{pmatrix}}}" loading="lazy"></span>.</dd></dl>

<p>Analog lassen sich auf diese Weise auch aus der <a href="Achsenabschnittsform" title="Achsenabschnittsform">Achsenabschnittsform</a> und der <a href="Hessesche_Normalform" title="Hessesche Normalform">hesseschen Normalform</a> ein Stützvektor und ein Richtungsvektor berechnen.

</p>

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

<p>Allgemein lassen sich durch die Parameterform nicht nur Geraden in der Ebene, sondern auch Geraden im drei- oder höherdimensionalen Raum beschreiben. Im <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-dimensionalen <a href="Euklidischer_Raum" title="Euklidischer Raum">euklidischen Raum</a> besteht eine Gerade entsprechend aus denjenigen Punkten, deren Ortsvektoren <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}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}}" loading="lazy"></span> die Gleichung

</p>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}={\vec {p}}+s{\vec {u}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3eda5382a0f3eb91eb74432f2697f6978aa88c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.014ex; height:2.676ex;" alt="{\displaystyle {\vec {x}}={\vec {p}}+s{\vec {u}}}" 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 s\in \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\in \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36efff902c6854b1196e79dec095b31e0c6a8ee9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.609ex; height:2.176ex;" alt="{\displaystyle s\in \mathbb {R} }" loading="lazy"></span></dd></dl>

<p>erfüllen. Es wird dabei lediglich 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 n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-komponentigen statt zweikomponentigen Vektoren gerechnet.

</p>

<div class="mw-heading mw-heading2"><h2 id="Parameterform_einer_Ebenengleichung">Parameterform einer Ebenengleichung</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Darstellung_2">Darstellung</h3></div>

<p>In der Parameterform wird eine Ebene im dreidimensionalen Raum durch einen Stützvektor <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 {p}}}">

<mover>

</mover>

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</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span> und zwei Richtungsvektoren <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 {u}}\neq {\vec {0}}}">

<mover>

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<mover>

</mover>

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</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9045f0f35f4143342908df020a47b5a44d6c0f33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:3.343ex;" alt="{\displaystyle {\vec {u}}\neq {\vec {0}}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\neq {\vec {0}}}">

<mover>

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<mover>

</mover>

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</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6fe47c74132950e340324614acc1003970733de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.436ex; height:3.343ex;" alt="{\displaystyle {\vec {v}}\neq {\vec {0}}}" loading="lazy"></span> beschrieben. Eine Ebene besteht dann aus denjenigen Punkten im Raum, deren Ortsvektoren <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}}}">

<mover>

</mover>

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</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}}" loading="lazy"></span> die Gleichung

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<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}={\vec {p}}+s{\vec {u}}+t{\vec {v}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ddc14f5146956e317bb1fdcb1334edbe4a005d21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.869ex; height:2.676ex;" alt="{\displaystyle {\vec {x}}={\vec {p}}+s{\vec {u}}+t{\vec {v}}}" 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 s,t\in \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle s,t\in \mathbb {R} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bae1d32304f0a658e915eadc6c9c983c51bb2692.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.483ex; height:2.509ex;" alt="{\displaystyle s,t\in \mathbb {R} }" loading="lazy"></span></dd></dl>

<p>erfüllen. Der Stützvektor ist dabei der Ortsvektor eines beliebigen Punkts in der Ebene, der wiederum als <i>Aufpunkt</i> bezeichnet wird. Die beiden Richtungsvektoren, hier auch <i>Spannvektoren</i> genannt, müssen in der Ebene liegen und ungleich dem <a href="Nullvektor" title="Nullvektor">Nullvektor</a> sein. Sie dürfen auch nicht <a href="Kollinearit%C3%A4t" title="Kollinearität">kollinear</a> sein, das heißt <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 {u}}}">

<mover>

</mover>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89c41e9cf70c5e5b56e2128a136985a75f90ba43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}" loading="lazy"></span> darf sich nicht als <a href="Vielfaches" title="Vielfaches">Vielfaches</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 {\vec {v}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85820588abd7333ef4d0c56539cb31c20e730753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}" loading="lazy"></span> schreiben lassen und umgekehrt. In der Parameterform werden die Punkte der Ebene in Abhängigkeit von den zwei Parametern <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> dargestellt. Jedem Wertepaar dieser Parameter entspricht dann genau ein Punkt der Ebene. Die Richtungsvektoren spannen somit ein affines Koordinatensystem auf, wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s,t)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4a80873677b13851457efa3447c2f412c91e425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle (s,t)}" loading="lazy"></span> die <a href="Affine_Koordinaten" title="Affine Koordinaten">affinen Koordinaten</a> eines Punkts der Ebene sind.

</p>

<div class="mw-heading mw-heading3"><h3 id="Beispiel_2">Beispiel</h3></div>

<p>Ausgeschrieben lautet die Parameterform einer Ebenengleichung

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\end{pmatrix}}+s{\begin{pmatrix}u_{1}\\u_{2}\\u_{3}\end{pmatrix}}+t{\begin{pmatrix}v_{1}\\v_{2}\\v_{3}\end{pmatrix}}={\begin{pmatrix}p_{1}+su_{1}+tv_{1}\\p_{2}+su_{2}+tv_{2}\\p_{3}+su_{3}+tv_{3}\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bce3e845258a32e49f8f3f48df670477937ccaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:61.473ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\end{pmatrix}}+s{\begin{pmatrix}u_{1}\\u_{2}\\u_{3}\end{pmatrix}}+t{\begin{pmatrix}v_{1}\\v_{2}\\v_{3}\end{pmatrix}}={\begin{pmatrix}p_{1}+su_{1}+tv_{1}\\p_{2}+su_{2}+tv_{2}\\p_{3}+su_{3}+tv_{3}\end{pmatrix}}}" loading="lazy"></span></dd></dl>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s,t\in \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bae1d32304f0a658e915eadc6c9c983c51bb2692.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.483ex; height:2.509ex;" alt="{\displaystyle s,t\in \mathbb {R} }" loading="lazy"></span>. Ist beispielsweise der Stützvektor <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 {p}}=(3,2,1)^{T}}">

<mover>

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<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}=(3,2,1)^{T}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/362363079d17311e8184f06af7f55cb34e417450.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.267ex; height:3.176ex;" alt="{\displaystyle {\vec {p}}=(3,2,1)^{T}}" loading="lazy"></span> und sind die Richtungsvektoren <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 {u}}=(2,-1,0)^{T}}">

<mover>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {u}}=(2,-1,0)^{T}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52f11ac48b1728d62e474678ca35b89db86e2b57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.99ex; height:3.176ex;" alt="{\displaystyle {\vec {u}}=(2,-1,0)^{T}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}=(-1,0,2)^{T}}">

<mover>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=(-1,0,2)^{T}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf54130b8212c8cc9c3555ae5fd91834be13abc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.836ex; height:3.176ex;" alt="{\displaystyle {\vec {v}}=(-1,0,2)^{T}}" loading="lazy"></span>, so erhält man als Ebenengleichung

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}3\\2\\1\end{pmatrix}}+s{\begin{pmatrix}2\\-1\\0\end{pmatrix}}+t{\begin{pmatrix}-1\\0\\2\end{pmatrix}}={\begin{pmatrix}3+2s-t\\2-s\\1+2t\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21ee5b76a35520fe3b8bb21a1b061c319cf63b37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:57.323ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}3\\2\\1\end{pmatrix}}+s{\begin{pmatrix}2\\-1\\0\end{pmatrix}}+t{\begin{pmatrix}-1\\0\\2\end{pmatrix}}={\begin{pmatrix}3+2s-t\\2-s\\1+2t\end{pmatrix}}}" loading="lazy"></span>.</dd></dl>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4a80873677b13851457efa3447c2f412c91e425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle (s,t)}" loading="lazy"></span>, beispielsweise <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 (0,0)}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d630d3e781a53b0a3559ae7e5b45f9479a3141a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}" loading="lazy"></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 (1,2)}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d0c72462a85992b95f90cc4a69048c9d83e8cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (1,2)}" loading="lazy"></span>, ergibt dann einen Ebenenpunkt.

</p>

<div class="mw-heading mw-heading3"><h3 id="Berechnung_2">Berechnung</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Aus_der_Dreipunkteform">Aus der Dreipunkteform</h4></div>

<p>Aus der <a href="Dreipunkteform" title="Dreipunkteform">Dreipunkteform</a> einer Ebenengleichung lassen sich zwei Richtungsvektoren der Ebene als Differenzvektoren zwischen den Ortsvektoren <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 {p}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" 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 {\vec {q}}}">

<mover>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a4e063f8ee7dae2488859c45a4e645db5148085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.309ex; height:2.676ex;" alt="{\displaystyle {\vec {q}}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}">

<mover>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aec3c9ce13b53e9e24c98e7cce4212627884c91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}" loading="lazy"></span> jeweils zweier Punkte erhalten, also

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30b22fd36fb77096a4bf26e30b515e893e9533fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.902ex; height:2.676ex;" alt="{\displaystyle {\vec {u}}={\vec {q}}-{\vec {p}}}" loading="lazy"></span>   und   <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}={\vec {r}}-{\vec {p}}}">

<mover>

</mover>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}={\vec {r}}-{\vec {p}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e283e794778455c495a6a0e93b087afd8603f5b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.662ex; height:2.676ex;" alt="{\displaystyle {\vec {v}}={\vec {r}}-{\vec {p}}}" loading="lazy"></span>.</dd></dl>

<p>Als Stützvektor <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 {p}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span> kann der Ortsvektor eines der Punkte verwendet werden.

</p>

<div class="mw-heading mw-heading4"><h4 id="Aus_der_Normalenform_2">Aus der Normalenform</h4></div>

<p>Aus der <a href="Normalenform#Normalenform_einer_Ebenengleichung" title="Normalenform">Normalenform einer Ebenengleichung</a> können aus dem Normalenvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49569db585c1b6306d5ffd91161775f67235fae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.343ex;" alt="{\displaystyle {\vec {n}}}" loading="lazy"></span> zwei Richtungsvektoren der Ebene durch Setzen von

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<annotation encoding="application/x-tex">{\displaystyle {\vec {u}}={\begin{pmatrix}-n_{2}\\n_{1}\\0\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8adbe39b6cb40f0f2112f4d5de78f529a9abb72a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.504ex; height:9.176ex;" alt="{\displaystyle {\vec {u}}={\begin{pmatrix}-n_{2}\\n_{1}\\0\end{pmatrix}}}" loading="lazy"></span>   und   <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}={\begin{pmatrix}0\\-n_{3}\\n_{2}\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}={\begin{pmatrix}0\\-n_{3}\\n_{2}\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38e578bf12a120a838ec7bd0b230e9fae6aaf00c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.349ex; height:9.176ex;" alt="{\displaystyle {\vec {v}}={\begin{pmatrix}0\\-n_{3}\\n_{2}\end{pmatrix}}}" loading="lazy"></span></dd></dl>

<p>bestimmt werden. Sollte einer dieser beiden Vektoren gleich dem <a href="Nullvektor" title="Nullvektor">Nullvektor</a> sein, kann stattdessen der Vektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-n_{3},0,n_{1})^{T}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle (-n_{3},0,n_{1})^{T}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a1a2ddaacca6b24dc13551fa5cf83dcb3580ef8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.135ex; height:3.176ex;" alt="{\displaystyle (-n_{3},0,n_{1})^{T}}" loading="lazy"></span> gewählt werden. Der Stützvektor <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 {p}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span> kann aus der Normalenform übernommen werden.

</p>

<div class="mw-heading mw-heading4"><h4 id="Aus_der_allgemeinen_Koordinatenform_2">Aus der allgemeinen Koordinatenform</h4></div>

<p>Aus der <a href="Koordinatenform#Allgemeine_Koordinatenform_einer_Ebenengleichung" class="mw-redirect" title="Koordinatenform">allgemeinen Koordinatenform einer Ebenengleichung</a> mit den Parametern <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,c}">

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f13f068df656c1b1911ae9f81628c49a6181194d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"></span> lässt sich ein Normalenvektor der Ebene als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}=(a,b,c)^{T}}">

<mover>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}=(a,b,c)^{T}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f9d07214a3146b1a4272127318a7df088be9efe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.994ex; height:3.176ex;" alt="{\displaystyle {\vec {n}}=(a,b,c)^{T}}" loading="lazy"></span> ablesen und damit zwei Richtungsvektoren der Ebene über

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<annotation encoding="application/x-tex">{\displaystyle {\vec {u}}={\begin{pmatrix}-b\\a\\0\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf0ed8973b460d870f146e5b28b76fafcaadf5c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:12.052ex; height:9.176ex;" alt="{\displaystyle {\vec {u}}={\begin{pmatrix}-b\\a\\0\end{pmatrix}}}" loading="lazy"></span>   und   <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}={\begin{pmatrix}0\\-c\\b\end{pmatrix}}}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}={\begin{pmatrix}0\\-c\\b\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fee5d60c0994b2ea2851730677039034c18be55a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.907ex; height:9.176ex;" alt="{\displaystyle {\vec {v}}={\begin{pmatrix}0\\-c\\b\end{pmatrix}}}" loading="lazy"></span></dd></dl>

<p>ermitteln. Sollte einer dieser beiden Vektoren gleich dem Nullvektor sein, kann stattdessen der Vektor <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,0,a)^{T}}">

<msup>

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</mstyle>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b9adc714fc9ffbba08e82ea0a26c22d6c1f4290.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.474ex; height:3.176ex;" alt="{\displaystyle (-c,0,a)^{T}}" loading="lazy"></span> gewählt werden. Einen Stützvektor erhält man, je nachdem, welche der Zahlen <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,c}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f13f068df656c1b1911ae9f81628c49a6181194d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}" loading="lazy"></span> ungleich null ist, durch Wahl von

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<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}={\begin{pmatrix}d/a\\0\\0\end{pmatrix}},~{\vec {p}}={\begin{pmatrix}0\\d/b\\0\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86361c08d94274feb34732efc6c2bbfb03f276e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; margin-left: -0.089ex; width:27.172ex; height:9.509ex;" alt="{\displaystyle {\vec {p}}={\begin{pmatrix}d/a\\0\\0\end{pmatrix}},~{\vec {p}}={\begin{pmatrix}0\\d/b\\0\end{pmatrix}}}" 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 {\vec {p}}={\begin{pmatrix}0\\0\\d/c\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}={\begin{pmatrix}0\\0\\d/c\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/664c61b67cc226298033d75665fa7df6015cc01f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; margin-left: -0.089ex; width:12.717ex; height:9.509ex;" alt="{\displaystyle {\vec {p}}={\begin{pmatrix}0\\0\\d/c\end{pmatrix}}}" loading="lazy"></span>.</dd></dl>

<p>Analog lassen sich auf diese Weise auch aus der <a href="Achsenabschnittsform" title="Achsenabschnittsform">Achsenabschnittsform</a> und der <a href="Hessesche_Normalform" title="Hessesche Normalform">hesseschen Normalform</a> ein Stützvektor und ein beziehungsweise zwei Richtungsvektoren berechnen.

</p>

<div class="mw-heading mw-heading3"><h3 id="Verallgemeinerung_2">Verallgemeinerung</h3></div>

<p>Allgemein lassen sich durch die Parameterform nicht nur Ebenen im dreidimensionalen Raum, sondern auch in höherdimensionalen Räumen beschreiben. Im <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-dimensionalen euklidischen Raum besteht eine Ebene entsprechend aus denjenigen Punkten, deren Ortsvektoren <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}}}">

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

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<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}={\vec {p}}+s{\vec {u}}+t{\vec {v}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ddc14f5146956e317bb1fdcb1334edbe4a005d21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.869ex; height:2.676ex;" alt="{\displaystyle {\vec {x}}={\vec {p}}+s{\vec {u}}+t{\vec {v}}}" 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 s,t\in \mathbb {R} }">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s,t\in \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bae1d32304f0a658e915eadc6c9c983c51bb2692.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.483ex; height:2.509ex;" alt="{\displaystyle s,t\in \mathbb {R} }" loading="lazy"></span></dd></dl>

<p>erfüllen. Es wird dabei lediglich 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 n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-komponentigen statt dreikomponentigen Vektoren gerechnet.

</p>

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

<ul><li>Steffen Goebbels, Stefan Ritter: <cite style="font-style:italic">Mathematik verstehen und anwenden: Differenzial- und Integralrechnung, Lineare Algebra</cite>. 4. Auflage. Springer, 2023, ISBN 978-3-662-68366-8, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>522<span style="display:inline-block;width:.2em"> </span>ff</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:Parameterform&rft.au=Steffen+Goebbels%2C+Stefan+Ritter&rft.btitle=Mathematik+verstehen+und+anwenden%3A+Differenzial-+und+Integralrechnung%2C+Lineare+Algebra&rft.date=2023&rft.edition=4.&rft.genre=book&rft.isbn=9783662683668&rft.pages=522+ff.&rft.pub=Springer" style="display:none"> </span></li></ul>

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

<ul><li><span class="cite"><a rel="nofollow" class="external text" href="https://de.serlo.org/mathe/1903/ebene-von-normalform-in-parameterform-umwandeln"><i>Ebene von Normalform in Parameterform umwandeln.</i></a> In: <i>Serlo.</i><span class="Abrufdatum"> Abgerufen am 23. Februar 2014</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%3AParameterform&rft.title=Ebene+von+Normalform+in+Parameterform+umwandeln&rft.description=Ebene+von+Normalform+in+Parameterform+umwandeln&rft.identifier=https%3A%2F%2Fde.serlo.org%2Fmathe%2F1903%2Febene-von-normalform-in-parameterform-umwandeln"> </span></li>

<li><span class="cite"><a rel="nofollow" class="external text" href="https://de.serlo.org/mathe/1899/ebene-von-koordinatenform-in-parameterform-umwandeln"><i>Ebene von Koordinatenform in Parameterform umwandeln.</i></a> In: <i>Serlo.</i><span class="Abrufdatum"> Abgerufen am 23. Februar 2014</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%3AParameterform&rft.title=Ebene+von+Koordinatenform+in+Parameterform+umwandeln&rft.description=Ebene+von+Koordinatenform+in+Parameterform+umwandeln&rft.identifier=https%3A%2F%2Fde.serlo.org%2Fmathe%2F1899%2Febene-von-koordinatenform-in-parameterform-umwandeln"> </span></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">Steffen Goebbels, Stefan Ritter: <cite style="font-style:italic">Mathematik verstehen und anwenden</cite>. 3. Auflage. Springer, Berlin, Heidelberg 2018, ISBN 978-3-662-57393-8, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>453<span style="display:inline-block;width:.2em"> </span>f</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:Parameterform&rft.au=Steffen+Goebbels%2C+Stefan+Ritter&rft.btitle=Mathematik+verstehen+und+anwenden&rft.date=2018&rft.edition=3.&rft.genre=book&rft.isbn=9783662573938&rft.pages=453+f.&rft.place=Berlin%2C+Heidelberg&rft.pub=Springer" 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"><cite style="font-style:italic">Analytische Geometrie: Leistungskurs</cite>. Volk und Wissen, Berlin 2000, ISBN 978-3-06-001173-5, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>67</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:Parameterform&rft.btitle=Analytische+Geometrie%3A+Leistungskurs&rft.date=2000&rft.genre=book&rft.isbn=9783060011735&rft.pages=67&rft.place=Berlin&rft.pub=Volk+und+Wissen" style="display:none"> </span></span>

</li>

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