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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Euler-Rodrigues-Formel</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>In der <a href="Mathematik" title="Mathematik">Mathematik</a> und <a href="Mechanik" title="Mechanik">Mechanik</a> dient die <b>Euler-Rodrigues Formel</b> nach <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> und <a href="Olinde_Rodrigues" title="Olinde Rodrigues">Olinde Rodrigues</a> der Beschreibung einer Drehung in drei Dimensionen. Mit vier <b>Euler-Parametern</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 a,b,c,d}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>,</mo>
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<mi>c</mi>
<mo>,</mo>
<mi>d</mi>
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<annotation encoding="application/x-tex">{\displaystyle a,b,c,d}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd0b3d3b09ae6ae430f09c4b317742c56e8acace.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.552ex; height:2.509ex;" alt="{\displaystyle a,b,c,d}" loading="lazy"></span>, für 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 a^{2}+b^{2}+c^{2}+d^{2}=1}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
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<mn>2</mn>
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<mn>2</mn>
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<mo>=</mo>
<mn>1</mn>
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<annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be606d79367578df0853b6c6ee02407ded846146.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.451ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=1}" loading="lazy"></span> gilt, definiert
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q:={\begin{pmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2(bc-ad)&2(bd+ac)\\2(bc+ad)&a^{2}+c^{2}-b^{2}-d^{2}&2(cd-ab)\\2(bd-ac)&2(cd+ab)&a^{2}+d^{2}-b^{2}-c^{2}\end{pmatrix}}}">
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<mi>c</mi>
<mo stretchy="false">)</mo>
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<mtd>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mi>c</mi>
<mo>+</mo>
<mi>a</mi>
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<mo stretchy="false">)</mo>
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<mo>−<!-- − --></mo>
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<mo stretchy="false">(</mo>
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<mi>d</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
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<mn>2</mn>
<mo stretchy="false">(</mo>
<mi>b</mi>
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<mo>−<!-- − --></mo>
<mi>a</mi>
<mi>c</mi>
<mo stretchy="false">)</mo>
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<mn>2</mn>
<mo stretchy="false">(</mo>
<mi>c</mi>
<mi>d</mi>
<mo>+</mo>
<mi>a</mi>
<mi>b</mi>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle Q:={\begin{pmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2(bc-ad)&2(bd+ac)\\2(bc+ad)&a^{2}+c^{2}-b^{2}-d^{2}&2(cd-ab)\\2(bd-ac)&2(cd+ab)&a^{2}+d^{2}-b^{2}-c^{2}\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/722e105fb57fd263b882c85a304150ff829c865c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.13ex; margin-bottom: -0.208ex; width:66.618ex; height:9.843ex;" alt="{\displaystyle Q:={\begin{pmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2(bc-ad)&2(bd+ac)\\2(bc+ad)&a^{2}+c^{2}-b^{2}-d^{2}&2(cd-ab)\\2(bd-ac)&2(cd+ab)&a^{2}+d^{2}-b^{2}-c^{2}\end{pmatrix}}}" loading="lazy"></span></dd></dl>
<p>eine <a href="Drehmatrix" title="Drehmatrix">Drehmatrix</a>. Diese Formel basiert auf der <a href="Rodrigues-Formel" title="Rodrigues-Formel">Rodrigues-Formel</a>, benutzt aber eine andere Parametrisierung.
</p><p>Benutzt wird die Formel in <a href="Flugsimulation" title="Flugsimulation">Flugsimulatoren</a> und <a href="Computerspiel" title="Computerspiel">Computerspielen</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Symmetrie">Symmetrie</h3></div>
<p>Die 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 a,b,c,d}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
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<mo>,</mo>
<mi>d</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a,b,c,d}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd0b3d3b09ae6ae430f09c4b317742c56e8acace.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.552ex; height:2.509ex;" alt="{\displaystyle a,b,c,d}" 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 -a,-b,-c,-d}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mi>a</mi>
<mo>,</mo>
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<mi>b</mi>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mi>c</mi>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mi>d</mi>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle -a,-b,-c,-d}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a35db731b14d167e07b3ecde7115e30973dc3c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.784ex; height:2.509ex;" alt="{\displaystyle -a,-b,-c,-d}" loading="lazy"></span> ) beschreiben dieselbe Rotation, was daran liegt, dass sie in der Q-Matrix immer paarweise miteinander multipliziert werden und so die Minus-Zeichen neutralisiert werden. Von dieser Symmetrie abgesehen, definieren vier Parameter die Drehmatrix in eindeutiger Weise.
</p>
<div class="mw-heading mw-heading3"><h3 id="Vektorformulierung">Vektorformulierung</h3></div>
<p>Aus 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 b,c,d}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo>,</mo>
<mi>c</mi>
<mo>,</mo>
<mi>d</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b,c,d}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/731790d670b86ec5929ca924ab29faf20cb33414.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.288ex; height:2.509ex;" alt="{\displaystyle b,c,d}" loading="lazy"></span> kann ein <a href="Vektor" title="Vektor">Vektor</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 {\varphi }}=(b,c,d)^{\top }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
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<mi>φ<!-- φ --></mi>
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<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>b</mi>
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<mo>,</mo>
<mi>d</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">⊤<!-- ⊤ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle {\vec {\varphi }}=(b,c,d)^{\top }}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4b0de32c8bf646d9328e631c4009366e6c48b30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.227ex; height:3.176ex;" alt="{\displaystyle {\vec {\varphi }}=(b,c,d)^{\top }}" loading="lazy"></span> gebildet werden. Darin bezeichnet das hochgestellte <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 \top }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">⊤<!-- ⊤ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \top }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf12e436fef2365e76fcb1034a51179d8328bb33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }" loading="lazy"></span> die <a href="Transponierte_Matrix" title="Transponierte Matrix">transponierte Matrix</a>, sodass <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 {\varphi }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {\vec {\varphi }}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f82bcb624dfe10df9c55ed401628ed7ff7a02f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.843ex;" alt="{\displaystyle {\vec {\varphi }}}" loading="lazy"></span> ein Spaltenvektor ist. Dann gilt für alle <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}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {\vec {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>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q{\vec {x}}={\vec {x}}+2a{\vec {\varphi }}\times {\vec {x}}+2{\vec {\varphi }}\times ({\vec {\varphi }}\times {\vec {x}}).}">
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<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q{\vec {x}}={\vec {x}}+2a{\vec {\varphi }}\times {\vec {x}}+2{\vec {\varphi }}\times ({\vec {\varphi }}\times {\vec {x}}).}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c71a6848e95735bb78b2201484e4e749b48d076a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.029ex; height:2.843ex;" alt="{\displaystyle Q{\vec {x}}={\vec {x}}+2a{\vec {\varphi }}\times {\vec {x}}+2{\vec {\varphi }}\times ({\vec {\varphi }}\times {\vec {x}}).}" loading="lazy"></span></dd></dl>
<p>So motiviert sich die Bezeichnung 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 a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</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> als <i>skalarer Parameter</i> 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 b,c,d}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo>,</mo>
<mi>c</mi>
<mo>,</mo>
<mi>d</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b,c,d}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/731790d670b86ec5929ca924ab29faf20cb33414.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.288ex; height:2.509ex;" alt="{\displaystyle b,c,d}" loading="lazy"></span> als <i>Vektorparameter</i>. Mit der <a href="Kreuzprodukt#Kreuzproduktmatrix" title="Kreuzprodukt">Kreuzproduktmatrix</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\vec {\varphi }}]_{\times }={\begin{pmatrix}0&-d&c\\d&0&-b\\-c&b&0\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<msub>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>×<!-- × --></mo>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mo>−<!-- − --></mo>
<mi>d</mi>
</mtd>
<mtd>
<mi>c</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>d</mi>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mo>−<!-- − --></mo>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mi>c</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [{\vec {\varphi }}]_{\times }={\begin{pmatrix}0&-d&c\\d&0&-b\\-c&b&0\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6a3f483948c7325a8d57aaa164b0c405105b71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:25.531ex; height:9.176ex;" alt="{\displaystyle [{\vec {\varphi }}]_{\times }={\begin{pmatrix}0&-d&c\\d&0&-b\\-c&b&0\end{pmatrix}}}" loading="lazy"></span></dd></dl>
<p>zeigt sich
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=E_{3}+2a[{\vec {\varphi }}]_{\times }+2[{\vec {\varphi }}]_{\times }[{\vec {\varphi }}]_{\times }=(2a^{2}-1)E_{3}+2a[{\vec {\varphi }}]_{\times }+2{\vec {\varphi }}{\vec {\varphi }}^{\top }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
<mo>=</mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>+</mo>
<mn>2</mn>
<mi>a</mi>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<msub>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>×<!-- × --></mo>
</mrow>
</msub>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<msub>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>×<!-- × --></mo>
</mrow>
</msub>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<msub>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>×<!-- × --></mo>
</mrow>
</msub>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>+</mo>
<mn>2</mn>
<mi>a</mi>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<msub>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>×<!-- × --></mo>
</mrow>
</msub>
<mo>+</mo>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">⊤<!-- ⊤ --></mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q=E_{3}+2a[{\vec {\varphi }}]_{\times }+2[{\vec {\varphi }}]_{\times }[{\vec {\varphi }}]_{\times }=(2a^{2}-1)E_{3}+2a[{\vec {\varphi }}]_{\times }+2{\vec {\varphi }}{\vec {\varphi }}^{\top }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2af1949ca53fd0e0be4f2d64b453f7e5b752ebfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.153ex; height:3.343ex;" alt="{\displaystyle Q=E_{3}+2a[{\vec {\varphi }}]_{\times }+2[{\vec {\varphi }}]_{\times }[{\vec {\varphi }}]_{\times }=(2a^{2}-1)E_{3}+2a[{\vec {\varphi }}]_{\times }+2{\vec {\varphi }}{\vec {\varphi }}^{\top }}" loading="lazy"></span></dd></dl>
<p>Darin 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 E_{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/395f570db7ae98f5a9a0d7b8436ae8c61ce5ebdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{3}}" loading="lazy"></span> die <a href="Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</a>. Diese entsteht bei <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 {\varphi }}={\vec {0}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mn>0</mn>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {\varphi }}={\vec {0}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8277a972c021d4f6cb932bc03ce18c190eda60d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.781ex; height:3.343ex;" alt="{\displaystyle {\vec {\varphi }}={\vec {0}}}" loading="lazy"></span> mit den Euler-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,d)=(\pm 1,0,0,0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo>,</mo>
<mi>c</mi>
<mo>,</mo>
<mi>d</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mo>±<!-- ± --></mo>
<mn>1</mn>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (a,b,c,d)=(\pm 1,0,0,0)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4359af2484a35f23145530162586b685719d9397.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.829ex; height:2.843ex;" alt="{\displaystyle (a,b,c,d)=(\pm 1,0,0,0)}" loading="lazy"></span>. Bei <span style="white-space:nowrap">180°-Drehungen</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 a=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90d476e5e765a5d77bbcff32e4584579207ec7d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=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 {\varphi }}|=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |{\vec {\varphi }}|=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75f1aaa88a148e6a9788138a07a4ed2a70275e9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.075ex; height:2.843ex;" alt="{\displaystyle |{\vec {\varphi }}|=1}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Drehwinkel_und_Drehachse">Drehwinkel und Drehachse</h3></div>
<p>Jede Drehung in drei Dimensionen ist eindeutig bestimmt durch einen <a href="Winkel" title="Winkel">Drehwinkel</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 \phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"></span> und eine <a href="Drehachse" class="mw-redirect" title="Drehachse">Drehachse</a>, die durch einen <a href="Einheitsvektor" title="Einheitsvektor">Einheitsvektor</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 {e}}=(e_{x},e_{y},e_{z})^{\top }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>e</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">⊤<!-- ⊤ --></mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {e}}=(e_{x},e_{y},e_{z})^{\top }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f953ebfb1ecf3174fbeb563f7a709928cde64090.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.183ex; height:3.343ex;" alt="{\displaystyle {\vec {e}}=(e_{x},e_{y},e_{z})^{\top }}" 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 e_{x}^{2}+e_{y}^{2}+e_{z}^{2}=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{x}^{2}+e_{y}^{2}+e_{z}^{2}=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2427de88f065efd0d6bd9710da72c19c9c9e6c90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.473ex; height:3.176ex;" alt="{\displaystyle e_{x}^{2}+e_{y}^{2}+e_{z}^{2}=1}" loading="lazy"></span> definiert wird. Dann lauten die Euler-Parameter der Drehung:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=\cos(\phi /2)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a=\cos(\phi /2)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee1367d2815f2b82eeb3ac419dbb371fc57fcaf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.959ex; height:2.843ex;" alt="{\displaystyle a=\cos(\phi /2)}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=\sin(\phi /2)e_{x}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo>=</mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo stretchy="false">)</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b=\sin(\phi /2)e_{x}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a038506160c6df21a88114e555150d6c81406c66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.727ex; height:2.843ex;" alt="{\displaystyle b=\sin(\phi /2)e_{x}}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=\sin(\phi /2)e_{y}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
<mo>=</mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo stretchy="false">)</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c=\sin(\phi /2)e_{y}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d037c665f5e3dc6ee38cab7c3721afb02a29a5ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.613ex; height:3.009ex;" alt="{\displaystyle c=\sin(\phi /2)e_{y}}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=\sin(\phi /2)e_{z}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mo>=</mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo stretchy="false">)</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d=\sin(\phi /2)e_{z}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0461c7dc95a021f822ccad0015c4b97cafef6a67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.775ex; height:2.843ex;" alt="{\displaystyle d=\sin(\phi /2)e_{z}}" loading="lazy"></span></dd></dl>
<p>Wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"></span> um eine volle 360°-Drehung zunimmt, entstehen die Euler-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 (-a,-b,-c,-d)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>a</mi>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mi>b</mi>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mi>c</mi>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mi>d</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (-a,-b,-c,-d)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac2a7ee00c613e5209c241d54e19064486157295.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.594ex; height:2.843ex;" alt="{\displaystyle (-a,-b,-c,-d)}" loading="lazy"></span>, die – wie oben bereits bemerkt – dieselbe Drehung repräsentieren.
</p><p>Der Vektorparameter lautet hier also <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\varphi }}=\sin(\varphi /2){\hat {e}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>e</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {\varphi }}=\sin(\varphi /2){\hat {e}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/652c175e32569ea011b5882ea24e1b88c208533c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.42ex; height:2.843ex;" alt="{\displaystyle {\vec {\varphi }}=\sin(\varphi /2){\hat {e}}}" loading="lazy"></span>.
Mit diesen Parametern und den <a href="Formelsammlung_Trigonometrie#Doppelwinkelfunktionen" title="Formelsammlung Trigonometrie">Doppelwinkelfunktionen</a> entsteht die <a href="Rodrigues-Formel" title="Rodrigues-Formel">Rodrigues-Formel</a> für die Drehmatrix:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=E_{3}+\sin(\varphi )[{\hat {e}}]_{\times }+(1-\cos \phi )[{\hat {e}}]_{\times }[{\hat {e}}]_{\times }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
<mo>=</mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>e</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<msub>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>×<!-- × --></mo>
</mrow>
</msub>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>e</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<msub>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>×<!-- × --></mo>
</mrow>
</msub>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>e</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<msub>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>×<!-- × --></mo>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q=E_{3}+\sin(\varphi )[{\hat {e}}]_{\times }+(1-\cos \phi )[{\hat {e}}]_{\times }[{\hat {e}}]_{\times }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/991d49a4bf5ef7d1b131939beb43dbdd72b6e0df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.556ex; height:2.843ex;" alt="{\displaystyle Q=E_{3}+\sin(\varphi )[{\hat {e}}]_{\times }+(1-\cos \phi )[{\hat {e}}]_{\times }[{\hat {e}}]_{\times }}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Parameter_einer_Drehmatrix">Parameter einer Drehmatrix</h3></div>
<p>Ist die Drehmatrix <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 Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span> gegeben und sind die Euler-Parameter gesucht, dann werden sie wie folgt gewonnen.<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> Hat <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 Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span> nur positive Diagonalelemente, dann ist
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} (Q)=3a^{2}-b^{2}-c^{2}-d^{2}=4a^{2}-1\quad \rightarrow \quad a=\pm {\frac {1}{2}}{\sqrt {\operatorname {Sp} (Q)+1}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Sp</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>Q</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>3</mn>
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mn>4</mn>
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mspace width="1em"></mspace>
<mo stretchy="false">→<!-- → --></mo>
<mspace width="1em"></mspace>
<mi>a</mi>
<mo>=</mo>
<mo>±<!-- ± --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>Sp</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>Q</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (Q)=3a^{2}-b^{2}-c^{2}-d^{2}=4a^{2}-1\quad \rightarrow \quad a=\pm {\frac {1}{2}}{\sqrt {\operatorname {Sp} (Q)+1}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4382efbee2c927cceda1fd4bf9125d989ce4ee3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:67.185ex; height:5.176ex;" alt="{\displaystyle \operatorname {Sp} (Q)=3a^{2}-b^{2}-c^{2}-d^{2}=4a^{2}-1\quad \rightarrow \quad a=\pm {\frac {1}{2}}{\sqrt {\operatorname {Sp} (Q)+1}}}" loading="lazy"></span></dd></dl>
<p>Die restlichen Parameter entstehen aus
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{i}={\frac {Q_{kj}-Q_{jk}}{4a}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mn>4</mn>
<mi>a</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q_{i}={\frac {Q_{kj}-Q_{jk}}{4a}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/197cc72ab319275bc982bcd5aa1b31b89cf4758d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.821ex; height:5.676ex;" alt="{\displaystyle q_{i}={\frac {Q_{kj}-Q_{jk}}{4a}}}" loading="lazy"></span></dd></dl>
<p>mit
</p>
<dl><dd><table class="wikitable">
<tbody><tr>
<th>i
</th>
<td>1</td>
<td>2</td>
<td>3
</td></tr>
<tr>
<th><i>q</i><sub>i</sub>
</th>
<td><i>b</i></td>
<td><i>c</i></td>
<td><i>d</i>
</td></tr>
<tr>
<th>j
</th>
<td>2</td>
<td>3</td>
<td>1
</td></tr>
<tr>
<th>k
</th>
<td>3</td>
<td>1</td>
<td>2
</td></tr></tbody></table></dd></dl>
<p>Sind teilweise negative Diagonalelemente vorhanden, dann sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{ii}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q_{ii}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0dfb631b8f92ea53ed1ae6c423a447a9c2bbb465.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.205ex; height:2.509ex;" alt="{\displaystyle Q_{ii}}" loading="lazy"></span> das größte Diagonalelement und
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{i}={\frac {1}{2}}{\sqrt {1+2Q_{ii}-\operatorname {Sp} (Q)}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>i</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mi>Sp</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>Q</mi>
<mo stretchy="false">)</mo>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q_{i}={\frac {1}{2}}{\sqrt {1+2Q_{ii}-\operatorname {Sp} (Q)}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23560937440e985f42150b056d88a9ee3d38be9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.701ex; height:5.176ex;" alt="{\displaystyle q_{i}={\frac {1}{2}}{\sqrt {1+2Q_{ii}-\operatorname {Sp} (Q)}}}" loading="lazy"></span></dd></dl>
<p>Mit diesem Wert 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 j,k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j,k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d23e18a251a10a993e66d41e8dbcaf858ba4fa5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:3.23ex; height:2.509ex;" alt="{\displaystyle j,k}" loading="lazy"></span> aus obiger Tabelle ermittelt sich
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=\pm {\frac {Q_{kj}-Q_{jk}}{4q_{i}}},\quad q_{j}={\frac {Q_{ji}+Q_{ij}}{4q_{i}}},\quad q_{k}={\frac {Q_{ki}+Q_{ik}}{4q_{i}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<mo>±<!-- ± --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mn>4</mn>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
<mspace width="1em"></mspace>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mn>4</mn>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
<mspace width="1em"></mspace>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mi>i</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mn>4</mn>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a=\pm {\frac {Q_{kj}-Q_{jk}}{4q_{i}}},\quad q_{j}={\frac {Q_{ji}+Q_{ij}}{4q_{i}}},\quad q_{k}={\frac {Q_{ki}+Q_{ik}}{4q_{i}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ad5b29bee55300ca646bb9e0780faab96cc199b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.977ex; height:6.176ex;" alt="{\displaystyle a=\pm {\frac {Q_{kj}-Q_{jk}}{4q_{i}}},\quad q_{j}={\frac {Q_{ji}+Q_{ij}}{4q_{i}}},\quad q_{k}={\frac {Q_{ki}+Q_{ik}}{4q_{i}}}}" loading="lazy"></span></dd></dl>
<p>Berechnung der Drehmatrix einmal 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 a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</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> und einmal 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 -a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -a}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e0982b5868a66be1ed3ad7ef4bcd3d3db20f982.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.176ex;" alt="{\displaystyle -a}" loading="lazy"></span> und Vergleich mit der gegebenen Drehmatrix liefert schließlich das Vorzeichen 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 a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</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>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Verknüpfung_zweier_Rotationen"><span id="Verkn.C3.BCpfung_zweier_Rotationen"></span>Verknüpfung zweier Rotationen</h3></div>
<p>Die Verknüpfung zweier Rotationen ergibt wieder eine Rotation. Aus Euler-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_{1},b_{1},c_{1},d_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{1},b_{1},c_{1},d_{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3bb357c43ee176556a54d127928fb0f3cec97615.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.762ex; height:2.509ex;" alt="{\displaystyle a_{1},b_{1},c_{1},d_{1}}" loading="lazy"></span> für die erste Drehung <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 Q_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q_{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8ea6463cb36d8278ff71214fb4d13127039ae53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{1}}" 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 a_{2},b_{2},c_{2},d_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{2},b_{2},c_{2},d_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4dba5f5c84c4b94ecffb4719bb5e9b9ebeea0d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.762ex; height:2.509ex;" alt="{\displaystyle a_{2},b_{2},c_{2},d_{2}}" loading="lazy"></span> für die zweite Drehung <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 Q_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b86e8bff64d5e62fc8f45a35875e78bc9bef74a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{2}}" loading="lazy"></span> ergibt sich die kombinierte Drehung <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 Q_{2}Q_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q_{2}Q_{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02a9606983e22cbc95b3384f79ac9194794f302a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.785ex; height:2.509ex;" alt="{\displaystyle Q_{2}Q_{1}}" loading="lazy"></span> aus erster Drehung und anschließender zweiter Drehung aus den Euler-Parametern
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a=a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4663c9465b9aaa10c14a4b82d82d5a9c8b3d10d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.17ex; height:2.509ex;" alt="{\displaystyle a=a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2}}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=a_{1}b_{2}+b_{1}a_{2}-c_{1}d_{2}+d_{1}c_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo>=</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b=a_{1}b_{2}+b_{1}a_{2}-c_{1}d_{2}+d_{1}c_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3f2e8c2614f4c59400186b82ec3643196c705c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.937ex; height:2.509ex;" alt="{\displaystyle b=a_{1}b_{2}+b_{1}a_{2}-c_{1}d_{2}+d_{1}c_{2}}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=a_{1}c_{2}+c_{1}a_{2}-d_{1}b_{2}+b_{1}d_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
<mo>=</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c=a_{1}c_{2}+c_{1}a_{2}-d_{1}b_{2}+b_{1}d_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1fc3eb524381a15aabb8da04ed340eda0efc207e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.947ex; height:2.509ex;" alt="{\displaystyle c=a_{1}c_{2}+c_{1}a_{2}-d_{1}b_{2}+b_{1}d_{2}}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=a_{1}d_{2}+d_{1}a_{2}-b_{1}c_{2}+c_{1}b_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mo>=</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d=a_{1}d_{2}+d_{1}a_{2}-b_{1}c_{2}+c_{1}b_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d7d8767b1b67a795a48dd0880f8c99532375134.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.156ex; height:2.509ex;" alt="{\displaystyle d=a_{1}d_{2}+d_{1}a_{2}-b_{1}c_{2}+c_{1}b_{2}}" loading="lazy"></span>.</dd></dl>
<p>Auch hier gilt wieder <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^{2}+b^{2}+c^{2}+d^{2}=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be606d79367578df0853b6c6ee02407ded846146.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.451ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=1}" loading="lazy"></span>, was durch Einsetzen bestätigt werden kann. Letztere Identität hat über
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=1=(a_{1}^{2}+b_{1}^{2}+c_{1}^{2}+d_{1}^{2})(a_{2}^{2}+b_{2}^{2}+c_{2}^{2}+d_{2}^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mn>1</mn>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msubsup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<msubsup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=1=(a_{1}^{2}+b_{1}^{2}+c_{1}^{2}+d_{1}^{2})(a_{2}^{2}+b_{2}^{2}+c_{2}^{2}+d_{2}^{2})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55a18123ac4cea095a0f06bdef9e521e24e85050.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:62.549ex; height:3.343ex;" alt="{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=1=(a_{1}^{2}+b_{1}^{2}+c_{1}^{2}+d_{1}^{2})(a_{2}^{2}+b_{2}^{2}+c_{2}^{2}+d_{2}^{2})}" loading="lazy"></span></dd></dl>
<p>einen direkten Bezug zum <a href="Quadrate-Satz#Eulerscher_Vier-Quadrate_Satz" title="Quadrate-Satz">Euler’schen Vier-Quadrate-Satz</a> und den Quaternionen.
</p>
<div class="mw-heading mw-heading2"><h2 id="Verbindung_mit_anderen_Konstrukten">Verbindung mit anderen Konstrukten</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Quaternionen">Quaternionen</h3></div>
<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Quaternion#Drehungen_im_dreidimensionalen_Raum" title="Quaternion">„Drehungen im dreidimensionalen Raum“ im Artikel Quaternion</a></i></div>
<p>Die Euler-Parameter können als Komponenten einer Einheitsquaternion angesehen werden. Der 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 a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</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> ist ihr reeller Anteil 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 b,c,d}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo>,</mo>
<mi>c</mi>
<mo>,</mo>
<mi>d</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b,c,d}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/731790d670b86ec5929ca924ab29faf20cb33414.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.288ex; height:2.509ex;" alt="{\displaystyle b,c,d}" loading="lazy"></span> ihr imaginärer. Mit den Einheitsquaternionen <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 q_{1,2}=a_{1,2}+ib_{1,2}+jc_{1,2}+kd_{1,2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>i</mi>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>j</mi>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>k</mi>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q_{1,2}=a_{1,2}+ib_{1,2}+jc_{1,2}+kd_{1,2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa7ab4b109a7f4386de4bdf66dec315fc020c350.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.74ex; height:2.843ex;" alt="{\displaystyle q_{1,2}=a_{1,2}+ib_{1,2}+jc_{1,2}+kd_{1,2}}" loading="lazy"></span>, die aus den Euler-Parametern zweier Drehungen <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 Q_{1,2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q_{1,2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e08a5fc5ac1f92cc3f88eb8d5afb64b9a482e384.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.172ex; height:2.843ex;" alt="{\displaystyle Q_{1,2}}" loading="lazy"></span> bestehen, können die Euler-Parameter der kombinierten Drehung <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 Q_{2}Q_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q_{2}Q_{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02a9606983e22cbc95b3384f79ac9194794f302a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.785ex; height:2.509ex;" alt="{\displaystyle Q_{2}Q_{1}}" loading="lazy"></span> elegant mit dem Produkt der Quaternionen berechnet werden:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+ib+jc+kd=q_{1}q_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>+</mo>
<mi>i</mi>
<mi>b</mi>
<mo>+</mo>
<mi>j</mi>
<mi>c</mi>
<mo>+</mo>
<mi>k</mi>
<mi>d</mi>
<mo>=</mo>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a+ib+jc+kd=q_{1}q_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2bb94af215e1b8228aa6852db824a8b930859fdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.224ex; height:2.509ex;" alt="{\displaystyle a+ib+jc+kd=q_{1}q_{2}}" loading="lazy"></span></dd></dl>
<p>Hier sind <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i,j}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4cbf8bbc622154cda8208d6e339495fe16a1f9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.509ex;" alt="{\displaystyle i,j}" 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 k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> die komplex-imaginären Einheiten, die sich mit den <i>Hamilton-Regeln</i> <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 i^{2}=j^{2}=k^{2}=ijk=-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mi>j</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/899610c19f2ca4230e116b0465112141eaced989.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.471ex; height:3.009ex;" alt="{\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1}" loading="lazy"></span> nicht <a href="Kommutativgesetz" title="Kommutativgesetz">kommutativ</a> verknüpfen. Beispielsweise 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 jk=-kj=i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
<mi>k</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>k</mi>
<mi>j</mi>
<mo>=</mo>
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle jk=-kj=i}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2d8cca815f7a3b9b77b98a8afde65abd105f86f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:13.173ex; height:2.509ex;" alt="{\displaystyle jk=-kj=i}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Pauli-Matrizen">Pauli-Matrizen</h3></div>
<p>Die <a href="Unit%C3%A4re_Matrix" title="Unitäre Matrix">unitären</a> 2 × 2-Matrizen
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E_{2}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}=\sigma _{0},&\quad \sigma _{x}={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}=\mathrm {i} \sigma _{2}\\\sigma _{y}={\begin{pmatrix}0&\mathrm {i} \\\mathrm {i} &0\end{pmatrix}}=\mathrm {i} \sigma _{1},&\quad \sigma _{z}={\begin{pmatrix}\mathrm {i} &0\\0&-\mathrm {i} \end{pmatrix}}=\mathrm {i} \sigma _{3}\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
<mo>=</mo>
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
</mtd>
<mtd>
<mspace width="1em"></mspace>
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
</mrow>
<msub>
<mi>σ<!-- σ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E_{2}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}=\sigma _{0},&\quad \sigma _{x}={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}=\mathrm {i} \sigma _{2}\\\sigma _{y}={\begin{pmatrix}0&\mathrm {i} \\\mathrm {i} &0\end{pmatrix}}=\mathrm {i} \sigma _{1},&\quad \sigma _{z}={\begin{pmatrix}\mathrm {i} &0\\0&-\mathrm {i} \end{pmatrix}}=\mathrm {i} \sigma _{3}\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03b449348a656ac1c335d15f89e9ee643b43f0bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:46.497ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}E_{2}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}=\sigma _{0},&\quad \sigma _{x}={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}=\mathrm {i} \sigma _{2}\\\sigma _{y}={\begin{pmatrix}0&\mathrm {i} \\\mathrm {i} &0\end{pmatrix}}=\mathrm {i} \sigma _{1},&\quad \sigma _{z}={\begin{pmatrix}\mathrm {i} &0\\0&-\mathrm {i} \end{pmatrix}}=\mathrm {i} \sigma _{3}\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>mit der imaginären Einheit <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 \mathrm {i} ^{2}=-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
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<mo>=</mo>
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<annotation encoding="application/x-tex">{\displaystyle \mathrm {i} ^{2}=-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1903180d4ecaf05f4aef81a8ad389025f89c9d34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.77ex; height:2.843ex;" alt="{\displaystyle \mathrm {i} ^{2}=-1}" loading="lazy"></span> der <a href="Komplexe_Zahl" title="Komplexe Zahl">komplexen Zahlen</a> hängen mit den <a href="Pauli-Matrizen" title="Pauli-Matrizen">Pauli-Matrizen</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 \sigma _{0,1,2,3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>σ<!-- σ --></mi>
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<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mn>3</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{0,1,2,3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1efa091c01ddc04620cf39878300cd3e0516db05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.22ex; height:2.343ex;" alt="{\displaystyle \sigma _{0,1,2,3}}" loading="lazy"></span> zusammen, die im <a href="Standardmodell" title="Standardmodell">Standardmodell</a> der Elementarteilchenphysik und in der Quantenmechanik verwendet werden.
</p><p>Die Matrizen <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 \sigma _{x,y,z}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
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<mo>,</mo>
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<mo>,</mo>
<mi>z</mi>
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<annotation encoding="application/x-tex">{\displaystyle \sigma _{x,y,z}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/081d5c8414c5ff43d1f73fb79f98fc35af073ed0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.001ex; height:2.343ex;" alt="{\displaystyle \sigma _{x,y,z}}" loading="lazy"></span> transformieren sich ähnlich obiger Hamilton-Regeln der komplex-imaginären Einheiten der Quaternionen:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}^{2}=\sigma _{y}^{2}=\sigma _{z}^{2}=\sigma _{x}\sigma _{y}\sigma _{z}=-E_{2}}">
<semantics>
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<mi>σ<!-- σ --></mi>
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<mi>σ<!-- σ --></mi>
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<mi>z</mi>
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<mo>=</mo>
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<annotation encoding="application/x-tex">{\displaystyle \sigma _{x}^{2}=\sigma _{y}^{2}=\sigma _{z}^{2}=\sigma _{x}\sigma _{y}\sigma _{z}=-E_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94662a175c0d42b42e40d79e912c656bb2942dad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.446ex; height:3.176ex;" alt="{\displaystyle \sigma _{x}^{2}=\sigma _{y}^{2}=\sigma _{z}^{2}=\sigma _{x}\sigma _{y}\sigma _{z}=-E_{2}}" loading="lazy"></span></dd></dl>
<p>Entsprechend können diese unitären 2 × 2-Matrizen ebenfalls zur Beschreibung von Rotationen herangezogen werden. Details dazu findet sich bei <a href="Quaternion" title="Quaternion">Quaternion</a>, <a href="SU(2)" title="SU(2)">SU(2)</a> und <a href="Spin-Gruppe" title="Spin-Gruppe">Spin-Gruppe</a>.
</p><p>Die zu einer Rotation korrespondierende unitäre 2 × 2-Matrix lautet unter Verwendung der Euler-Parameter:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U={\begin{pmatrix}\ \ \,a+\mathrm {i} d&b+\mathrm {i} c\\-b+\mathrm {i} c&a-\mathrm {i} d\end{pmatrix}}=a\,E_{2}+b\,\sigma _{x}+c\,\sigma _{y}+d\,\sigma _{z}=a\,\sigma _{0}+\mathrm {i} b\,\sigma _{2}+\mathrm {i} c\,\sigma _{1}+\mathrm {i} d\,\sigma _{3}}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle U={\begin{pmatrix}\ \ \,a+\mathrm {i} d&b+\mathrm {i} c\\-b+\mathrm {i} c&a-\mathrm {i} d\end{pmatrix}}=a\,E_{2}+b\,\sigma _{x}+c\,\sigma _{y}+d\,\sigma _{z}=a\,\sigma _{0}+\mathrm {i} b\,\sigma _{2}+\mathrm {i} c\,\sigma _{1}+\mathrm {i} d\,\sigma _{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d2f5071d994152032351d4770d65d915d3e026a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:81.469ex; height:6.176ex;" alt="{\displaystyle U={\begin{pmatrix}\ \ \,a+\mathrm {i} d&b+\mathrm {i} c\\-b+\mathrm {i} c&a-\mathrm {i} d\end{pmatrix}}=a\,E_{2}+b\,\sigma _{x}+c\,\sigma _{y}+d\,\sigma _{z}=a\,\sigma _{0}+\mathrm {i} b\,\sigma _{2}+\mathrm {i} c\,\sigma _{1}+\mathrm {i} d\,\sigma _{3}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Spinor" title="Spinor">Spinor</a></li>
<li><a href="Orthogonaler_Tensor" title="Orthogonaler Tensor">Orthogonaler Tensor</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Axel Volkwein: <cite style="font-style:italic">Numerische Simulation von flexiblen Steinschlagschutzsystemen</cite>. Hrsg.: Institut für Baustatik und Konstruktion, <a href="Eidgen%C3%B6ssische_Technische_Hochschule_Z%C3%BCrich" class="mw-redirect" title="Eidgenössische Technische Hochschule Zürich">Eidgenössische Technische Hochschule Zürich</a>. vdf Hochschulverlag AG, 2004, ISBN 978-3-7281-2986-4 (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=eqDfstDy4DgC&pg=PA25#v=onepage">eingeschränkte Vorschau</a> in der Google-Buchsuche [abgerufen am 30. Juni 2017]).<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:Euler-Rodrigues-Formel&rft.au=Axel+Volkwein&rft.btitle=Numerische+Simulation+von+flexiblen+Steinschlagschutzsystemen&rft.date=2004&rft.genre=book&rft.isbn=9783728129864&rft.pub=vdf+Hochschulverlag+AG" style="display:none"> </span></span>
</li>
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