ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

Micron Document

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<title>Formule d'Euler–Rodrigues</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Formule d'Euler–Rodrigues</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="fr" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr">En <a href="Math%C3%A9matiques" title="Mathématiques">mathématiques</a> et en <a href="M%C3%A9canique_(science)" title="Mécanique (science)">mécanique</a>, la formule d'Euler – Rodrigues est une formule générale pour les rotations vectorielles en dimension trois, faisant intervenir quatre paramètres.

Elle est ainsi nommée en référence à <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a><a href="#cite_note-1">[1]</a> et <a href="Olinde_Rodrigues" title="Olinde Rodrigues">Olinde Rodrigues</a><a href="#cite_note-2">[2]</a>.

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

<div class="mw-heading mw-heading3"><h3 id="Formulation_matricielle">Formulation matricielle</h3></div>

La matrice générale d'une <a href="Rotation_vectorielle" title="Rotation vectorielle">rotation</a> de l'espace vectoriel euclidien de dimension trois dans une <a href="Base_orthonorm%C3%A9e" title="Base orthonormée">base orthonormée</a> directe s'écrit

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a1239dc8c3f9ca8bc9703a31510a364e3a053af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.13ex; margin-bottom: -0.208ex; width:60.069ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}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{bmatrix}}}" loading="lazy"></dd></dl>

où <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c,d}">

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

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</math><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"> sont quatre paramètres réels, dits d'Euler-Rodrigues, vérifiant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=1}">

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

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</math><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">.

C'est donc aussi la formule générale d'une <a href="Matrice_orthogonale" title="Matrice orthogonale">matrice orthogonale</a> positive d'ordre trois.

<div class="mw-heading mw-heading3"><h3 id="Formulation_vectorielle">Formulation vectorielle</h3></div>

Si on note <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\omega }}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4e066a68ceb355e3314fb2b97f1c0c421ca6074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:2.343ex;" alt="{\displaystyle {\vec {\omega }}}" loading="lazy"> le vecteur de coordonnées (b, c, d) dans la base orthonormée, la formule précédente est l'écriture matricielle de la formule :

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<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}'=r({\vec {x}})={\vec {x}}+2a({\vec {\omega }}\land {\vec {x}})+2\left({\vec {\omega }}\land ({\vec {\omega }}\land {\vec {x}})\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86104585fc2a7af94e7c5777fdde40b47c7b90d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.524ex; height:3.176ex;" alt="{\displaystyle {\vec {x}}'=r({\vec {x}})={\vec {x}}+2a({\vec {\omega }}\land {\vec {x}})+2\left({\vec {\omega }}\land ({\vec {\omega }}\land {\vec {x}})\right)}" loading="lazy">.

C'est la raison pour laquelle le paramètre a est appelé le paramètre scalaire, et le triplet (b, c, d) le paramètre vectoriel .

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

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

Les paramètres (a, b, c, d) et (−a, −b, −c, −d) décrivent la même rotation. En dehors de cette symétrie, chaque quadruplet de paramètres décrit une rotation unique.

<div class="mw-heading mw-heading3"><h3 id="Composition_des_rotations">Composition des rotations</h3></div>

Soit (a1, b1, c1, d1) et (a2, b2, c2, d2) les paramètres d'Euler-Rodrigues de deux rotations. Les paramètres de la rotation composée (rotation 1 puis rotation 2) sont les suivants:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a&=a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2};\\b&=a_{1}b_{2}+b_{1}a_{2}-c_{1}d_{2}+d_{1}c_{2};\\c&=a_{1}c_{2}+c_{1}a_{2}-d_{1}b_{2}+b_{1}d_{2};\\d&=a_{1}d_{2}+d_{1}a_{2}-b_{1}c_{2}+c_{1}b_{2}.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4dcaafe679216a8c4a1b862bf3fc967be8286b04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:31.568ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}a&=a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2};\\b&=a_{1}b_{2}+b_{1}a_{2}-c_{1}d_{2}+d_{1}c_{2};\\c&=a_{1}c_{2}+c_{1}a_{2}-d_{1}b_{2}+b_{1}d_{2};\\d&=a_{1}d_{2}+d_{1}a_{2}-b_{1}c_{2}+c_{1}b_{2}.\end{aligned}}}" loading="lazy"></dd></dl>

Il est simple, bien que fastidieux, de vérifier que a2 + b2 + c2 + d2 = 1 . Il s'agit essentiellement de <a href="Identit%C3%A9_des_quatre_carr%C3%A9s_d'Euler" title="Identité des quatre carrés d'Euler">l'identité des quatre carrés d'Euler</a>, également utilisée par Rodrigues.

<div class="mw-heading mw-heading2"><h2 id="Liaison_avec_l'angle_et_l'axe_de_rotation">Liaison avec l'angle et l'axe de rotation</h2></div>

Toute rotation vectorielle en dimension trois est uniquement déterminée par son axe de rotation (dirigé par un <a href="Vecteur_unitaire" title="Vecteur unitaire">vecteur unitaire</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}}">

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</math><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"> de coordonnées <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n_{1},n_{2},n_{3})}">

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

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a&=\cos {\frac {\theta }{2}};\\b&=n_{1}\sin {\frac {\theta }{2}};\\c&=n_{2}\sin {\frac {\theta }{2}};\\d&=n_{3}\sin {\frac {\theta }{2}}.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7cf47b983c13a2f42eff93ee31060b9a77e4d72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:13.804ex; height:21.843ex;" alt="{\displaystyle {\begin{aligned}a&=\cos {\frac {\theta }{2}};\\b&=n_{1}\sin {\frac {\theta }{2}};\\c&=n_{2}\sin {\frac {\theta }{2}};\\d&=n_{3}\sin {\frac {\theta }{2}}.\end{aligned}}}" loading="lazy"></dd></dl>

Autrement dit, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\omega }}=\sin {\frac {\theta }{2}}{\vec {n}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}=\sin {\frac {\theta }{2}}{\vec {n}}}</annotation>

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Notons que si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> est augmenté d'une rotation complète de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy">, les arguments des sinus et cosinus n'augmentent que de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy">. Les paramètres résultants sont les opposés des valeurs originales, (−a, −b, −c, −d) ; ils représentent la même rotation.

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

<ul><li>La transformation identique (rotation nulle, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =0}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a7bc6e34b53e0e8a8815159c356b1acccf7ea24.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 \theta =0}" loading="lazy">) correspond à des valeurs de paramètres (a, b, c, d) = (±1, 0, 0, 0) .</li></ul>

<ul><li>Les rotations de 180 degrés (demi-tours, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\pi }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab4db588619489e27efb50a1d0d5aa016c49ce15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.521ex; height:2.176ex;" alt="{\displaystyle \theta =\pi }" loading="lazy">) autour de n'importe quel axe sont obtenues pour a = 0 , ce qui donne la matrice générale d'un demi-tour autour de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}b^{2}-c^{2}-d^{2}&2bc&2bd\\2bc&c^{2}-b^{2}-d^{2}&2(cd)\\2bd&2cd&d^{2}-b^{2}-c^{2}\end{bmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e28da60914aae7ee14f99128c7d32bca85d0564.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:44.695ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}b^{2}-c^{2}-d^{2}&2bc&2bd\\2bc&c^{2}-b^{2}-d^{2}&2(cd)\\2bd&2cd&d^{2}-b^{2}-c^{2}\end{bmatrix}}}" loading="lazy">.</center>

<ul><li>La matrice des <a href="Angles_d'Euler" title="Angles d'Euler">angles d'Euler</a> <center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}\cos \psi \cos \varphi -\sin \psi \cos \theta \sin \varphi &-\cos \psi \sin \varphi -\sin \psi \cos \theta \cos \varphi &\sin \psi \sin \theta \\\sin \psi \cos \varphi +\cos \psi \cos \theta \sin \varphi &-\sin \psi \sin \varphi +\cos \psi \cos \theta \cos \varphi &-\cos \psi \sin \theta \\\sin \theta \sin \varphi &\sin \theta \cos \varphi &\cos \theta \end{bmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}\cos \psi \cos \varphi -\sin \psi \cos \theta \sin \varphi &-\cos \psi \sin \varphi -\sin \psi \cos \theta \cos \varphi &\sin \psi \sin \theta \\\sin \psi \cos \varphi +\cos \psi \cos \theta \sin \varphi &-\sin \psi \sin \varphi +\cos \psi \cos \theta \cos \varphi &-\cos \psi \sin \theta \\\sin \theta \sin \varphi &\sin \theta \cos \varphi &\cos \theta \end{bmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44d1306985359083791ec753fb1b6e1395e818d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:83.739ex; height:9.509ex;" alt="{\displaystyle A={\begin{bmatrix}\cos \psi \cos \varphi -\sin \psi \cos \theta \sin \varphi &-\cos \psi \sin \varphi -\sin \psi \cos \theta \cos \varphi &\sin \psi \sin \theta \\\sin \psi \cos \varphi +\cos \psi \cos \theta \sin \varphi &-\sin \psi \sin \varphi +\cos \psi \cos \theta \cos \varphi &-\cos \psi \sin \theta \\\sin \theta \sin \varphi &\sin \theta \cos \varphi &\cos \theta \end{bmatrix}}}" loading="lazy"></center>est égale à la matrice d'Euler-Rodrigues avec <center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b,c,d)=(\cos {\frac {\theta }{2}}\cos {\frac {\psi +\varphi }{2}},\sin {\frac {\theta }{2}}\sin {\frac {\psi +\varphi }{2}},\sin {\frac {\theta }{2}}\cos {\frac {\psi -\varphi }{2}},\cos {\frac {\theta }{2}}\sin {\frac {\psi -\varphi }{2}})}">

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a,b,c,d)=(\cos {\frac {\theta }{2}}\cos {\frac {\psi +\varphi }{2}},\sin {\frac {\theta }{2}}\sin {\frac {\psi +\varphi }{2}},\sin {\frac {\theta }{2}}\cos {\frac {\psi -\varphi }{2}},\cos {\frac {\theta }{2}}\sin {\frac {\psi -\varphi }{2}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c281644d6daafea0437df5920dacaccc48eff1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:80.717ex; height:5.343ex;" alt="{\displaystyle (a,b,c,d)=(\cos {\frac {\theta }{2}}\cos {\frac {\psi +\varphi }{2}},\sin {\frac {\theta }{2}}\sin {\frac {\psi +\varphi }{2}},\sin {\frac {\theta }{2}}\cos {\frac {\psi -\varphi }{2}},\cos {\frac {\theta }{2}}\sin {\frac {\psi -\varphi }{2}})}" loading="lazy">.</center></li>

</mstyle>

</mrow>

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

</semantics>

</math><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"> sont des entiers non tous nuls, la matrice <center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{a^{2}+b^{2}+c^{2}+d^{2}}}{\begin{bmatrix}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{bmatrix}}}">

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a^{2}+b^{2}+c^{2}+d^{2}}}{\begin{bmatrix}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{bmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74f027d6c845412540d4f76aeb0003e33fbae531.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.13ex; margin-bottom: -0.208ex; width:78.095ex; height:9.843ex;" alt="{\displaystyle {\frac {1}{a^{2}+b^{2}+c^{2}+d^{2}}}{\begin{bmatrix}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{bmatrix}}}" loading="lazy"> </center>est une matrice de rotation à coefficients rationnels.</li></ul>

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

<div class="mw-heading mw-heading3"><h3 id="1)_Formule_de_rotation_d'Olinde_Rodrigues">1) Formule de rotation d'Olinde Rodrigues</h3></div>

Théorème : si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> est la rotation d'angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> autour de <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><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"> (unitaire) , l'image d'un vecteur <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>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r({\vec {x}})={\vec {x}}+\sin \theta ({\vec {n}}\land {\vec {x}})+(1-\cos \theta )({\vec {n}}\land ({\vec {n}}\land {\vec {x}}))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e364153ad437c965827d3f0cbc1c97375de1330.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.05ex; height:2.843ex;" alt="{\displaystyle r({\vec {x}})={\vec {x}}+\sin \theta ({\vec {n}}\land {\vec {x}})+(1-\cos \theta )({\vec {n}}\land ({\vec {n}}\land {\vec {x}}))}" loading="lazy"></div>

Démonstration : Le vecteur <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><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"> se décompose suivant le plan P orthogonal à <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><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"> et la droite engendrée par <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><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"> en <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}={\vec {x}}_{1}+{\vec {x}}_{2}}">

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95c2b3e69a40da616c3a44a604bb22da97e8ff00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.036ex; height:2.676ex;" alt="{\displaystyle {\vec {x}}={\vec {x}}_{1}+{\vec {x}}_{2}}" loading="lazy">. Or, si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {y}}_{1}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c071261ae54982d0a6bc5230df347c5fc8bdfff9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.319ex; height:2.843ex;" alt="{\displaystyle {\vec {y}}_{1}}" loading="lazy"> est le vecteur directement orthogonal à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}_{1}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c78435b0d85b92bd9ee899b3318b3fa57395252d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle {\vec {x}}_{1}}" loading="lazy"> dans P, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r({\vec {x}})=\cos \theta {\vec {x}}_{1}+\sin \theta {\vec {y}}_{1}+{\vec {x}}_{2}}">

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r({\vec {x}})=\cos \theta {\vec {x}}_{1}+\sin \theta {\vec {y}}_{1}+{\vec {x}}_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c409b4d9e1366c798938b208294d6c189e81663c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.975ex; height:2.843ex;" alt="{\displaystyle r({\vec {x}})=\cos \theta {\vec {x}}_{1}+\sin \theta {\vec {y}}_{1}+{\vec {x}}_{2}}" loading="lazy">, donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r({\vec {x}})-{\vec {x}}=(\cos \theta -1){\vec {x}}_{1}+\sin \theta {\vec {y}}_{1}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r({\vec {x}})-{\vec {x}}=(\cos \theta -1){\vec {x}}_{1}+\sin \theta {\vec {y}}_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1db27fef4615f62061ce9ee0eb222b1f14f27317.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.733ex; height:2.843ex;" alt="{\displaystyle r({\vec {x}})-{\vec {x}}=(\cos \theta -1){\vec {x}}_{1}+\sin \theta {\vec {y}}_{1}}" loading="lazy">.

Or, avec un dessin, on peut se convaincre que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}_{1}=-{\vec {n}}\land ({\vec {n}}\land {\vec {x}})}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}_{1}=-{\vec {n}}\land ({\vec {n}}\land {\vec {x}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/502570b9ebb939018ab3e35c400c129ce412420e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.384ex; height:2.843ex;" alt="{\displaystyle {\vec {x}}_{1}=-{\vec {n}}\land ({\vec {n}}\land {\vec {x}})}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {y}}_{1}={\vec {n}}\land {\vec {x}}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb40cbcff55ea74ef7b84007890c620988c4c25d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.724ex; height:2.843ex;" alt="{\displaystyle {\vec {y}}_{1}={\vec {n}}\land {\vec {x}}}" loading="lazy">, d'où la formule énoncée.

Attention, il ne faut pas confondre cette formule d'Olinde Rodrigues avec <a href="Formule_de_Rodrigues" title="Formule de Rodrigues">cette autre</a>, concernant les <a href="Suite_de_polyn%C3%B4mes_orthogonaux" title="Suite de polynômes orthogonaux">polynômes orthogonaux</a>.

<div class="mw-heading mw-heading3"><h3 id="2)_Obtention_de_la_formule_vectorielle">2) Obtention de la formule vectorielle</h3></div>

La formule d'Olinde Rodrigues s'écrit aussi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r({\vec {x}})={\vec {x}}+2\sin {\theta \over 2}\cos {\theta \over 2}({\vec {n}}\land {\vec {x}})+2\sin ^{2}{\theta \over 2}({\vec {n}}\land ({\vec {n}}\land {\vec {x}}))}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r({\vec {x}})={\vec {x}}+2\sin {\theta \over 2}\cos {\theta \over 2}({\vec {n}}\land {\vec {x}})+2\sin ^{2}{\theta \over 2}({\vec {n}}\land ({\vec {n}}\land {\vec {x}}))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/213b7a9e49ec71addb43eb5cb5bec6419a0cc57c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.836ex; height:5.343ex;" alt="{\displaystyle r({\vec {x}})={\vec {x}}+2\sin {\theta \over 2}\cos {\theta \over 2}({\vec {n}}\land {\vec {x}})+2\sin ^{2}{\theta \over 2}({\vec {n}}\land ({\vec {n}}\land {\vec {x}}))}" loading="lazy"> , et en posant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=\cos {\frac {\theta }{2}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=\cos {\frac {\theta }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f1048c8ef48d3f3aebc50ff5bd661a202c970fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.825ex; height:5.343ex;" alt="{\displaystyle a=\cos {\frac {\theta }{2}}}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\omega }}=\sin {\frac {\theta }{2}}{\vec {n}}}">

<mover>

</mover>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}=\sin {\frac {\theta }{2}}{\vec {n}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0070e1095afbb4b8a50839711ecc4d1dbafb659d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.18ex; height:5.343ex;" alt="{\displaystyle {\vec {\omega }}=\sin {\frac {\theta }{2}}{\vec {n}}}" loading="lazy">, on obtient bien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r({\vec {x}})={\vec {x}}+2a({\vec {\omega }}\land {\vec {x}})+2\left({\vec {\omega }}\land ({\vec {\omega }}\land {\vec {x}})\right)}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r({\vec {x}})={\vec {x}}+2a({\vec {\omega }}\land {\vec {x}})+2\left({\vec {\omega }}\land ({\vec {\omega }}\land {\vec {x}})\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b24257595e2094c011e464727c3a695b1658e8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.411ex; height:2.843ex;" alt="{\displaystyle r({\vec {x}})={\vec {x}}+2a({\vec {\omega }}\land {\vec {x}})+2\left({\vec {\omega }}\land ({\vec {\omega }}\land {\vec {x}})\right)}" loading="lazy">.

La formule matricielle s'obtient alors en passant aux coordonnées.

<div class="mw-heading mw-heading3"><h3 id="3)_Variante_matricielle_directe">3) Variante matricielle directe</h3></div>

La formule d'Olinde Rodrigues donne la matrice de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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

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<annotation encoding="application/x-tex">{\displaystyle \left[{\begin{array}{ccc}n_{1}^{2}\left(1-\cos \theta \right)+\cos \theta &n_{1}n_{2}\left(1-\cos \theta \right)-n_{3}\sin \theta &n_{1}n_{3}\left(1-\cos \theta \right)+n_{2}\sin \theta \\n_{1}n_{2}\left(1-\cos \theta \right)+n_{3}\sin \theta &n_{2}^{2}\left(1-\cos \theta \right)+\cos \theta &n_{2}n_{3}\left(1-\cos \theta \right)-n_{1}\sin \theta \\n_{1}n_{3}\left(1-\cos \theta \right)-n_{2}\sin \theta &n_{2}n_{3}\left(1-\cos \theta \right)+n_{1}\sin \theta &n_{3}^{2}\left(1-\cos \theta \right)+\cos \theta \end{array}}\right]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59cf0eef3f5fa47b4b00751fdec335a82204760d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:85.584ex; height:10.176ex;" alt="{\displaystyle \left[{\begin{array}{ccc}n_{1}^{2}\left(1-\cos \theta \right)+\cos \theta &n_{1}n_{2}\left(1-\cos \theta \right)-n_{3}\sin \theta &n_{1}n_{3}\left(1-\cos \theta \right)+n_{2}\sin \theta \\n_{1}n_{2}\left(1-\cos \theta \right)+n_{3}\sin \theta &n_{2}^{2}\left(1-\cos \theta \right)+\cos \theta &n_{2}n_{3}\left(1-\cos \theta \right)-n_{1}\sin \theta \\n_{1}n_{3}\left(1-\cos \theta \right)-n_{2}\sin \theta &n_{2}n_{3}\left(1-\cos \theta \right)+n_{1}\sin \theta &n_{3}^{2}\left(1-\cos \theta \right)+\cos \theta \end{array}}\right]}" loading="lazy">

qui donne elle-même la matrice d'Euler-Rodrigues en utilisant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b,c,d)=(\cos {\frac {\theta }{2}},n_{1}\sin {\frac {\theta }{2}},n_{2}\sin {\frac {\theta }{2}},n_{3}\sin {\frac {\theta }{2}})}">

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<annotation encoding="application/x-tex">{\displaystyle (a,b,c,d)=(\cos {\frac {\theta }{2}},n_{1}\sin {\frac {\theta }{2}},n_{2}\sin {\frac {\theta }{2}},n_{3}\sin {\frac {\theta }{2}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a40f4e8db50d5ce3ca76e81e0186fdf7394a120.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.1ex; height:5.343ex;" alt="{\displaystyle (a,b,c,d)=(\cos {\frac {\theta }{2}},n_{1}\sin {\frac {\theta }{2}},n_{2}\sin {\frac {\theta }{2}},n_{3}\sin {\frac {\theta }{2}})}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Connexion_avec_les_quaternions">Connexion avec les quaternions</h2></div>

Les paramètres d'Euler-Rodrigues peuvent être considérés comme les coefficients d'un <a href="Quaternion" title="Quaternion">quaternion</a>

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<annotation encoding="application/x-tex">{\displaystyle q=a+bi+cj+dk,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14ac6c006e6b50f6b6258d426cde09ef37eee222.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.758ex; height:2.509ex;" alt="{\displaystyle q=a+bi+cj+dk,}" loading="lazy">

dont le paramètre scalaire a est la <a href="Partie_r%C3%A9elle" title="Partie réelle">partie réelle</a>, et les paramètres vectoriels b, c, d les parties imaginaires. Il est unitaire puisque

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<annotation encoding="application/x-tex">{\displaystyle \left\|q\right\|^{2}=a^{2}+b^{2}+c^{2}+d^{2}=1.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7d1dcaa20cb1f1d51bafc1f4bca95251de36e49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.645ex; height:3.343ex;" alt="{\displaystyle \left\|q\right\|^{2}=a^{2}+b^{2}+c^{2}+d^{2}=1.}" loading="lazy"></dd></dl>

Plus important encore, les relations ci-dessus pour la composition des rotations sont précisément les relations pour la multiplication des quaternions. En d'autres termes, le groupe de quaternions unitaires muni de la multiplication, modulo le signe moins, est isomorphe au groupe des rotations muni de la composition.

<div class="mw-heading mw-heading2"><h2 id="Connexion_avec_les_matrices_de_spin_SU(2)">Connexion avec les matrices de spin SU(2)</h2></div>

Le <a href="Groupe_de_Lie" title="Groupe de Lie">groupe de Lie</a> <a href="Groupe_sp%C3%A9cial_unitaire" title="Groupe spécial unitaire">SU(2)</a> peut être utilisé pour représenter des rotations tridimensionnelles par des matrices 2 × 2 . La matrice de SU(2) correspondant à une rotation, en fonction de ses paramètres d'Euler-Rodrigues, est

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<annotation encoding="application/x-tex">{\displaystyle U={\begin{pmatrix}\ \ \,a+di&b+ci\\-b+ci&a-di\end{pmatrix}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd722252a7b40a8be9039e0531aa2a979e1af015.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.749ex; height:6.176ex;" alt="{\displaystyle U={\begin{pmatrix}\ \ \,a+di&b+ci\\-b+ci&a-di\end{pmatrix}}.}" loading="lazy"></dd></dl>

Ce qui peut s'écrire :

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U&=a\ {\begin{pmatrix}1&0\\0&1\end{pmatrix}}+b\ {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}+c\ {\begin{pmatrix}0&i\\i&0\end{pmatrix}}+d\ {\begin{pmatrix}i&0\\0&-i\end{pmatrix}}\\&=a\,I+ic\,\sigma _{x}+ib\,\sigma _{y}+id\,\sigma _{z},\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a8918de3cebafd645a07a21111b9dd5b69cbcdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:59.463ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}U&=a\ {\begin{pmatrix}1&0\\0&1\end{pmatrix}}+b\ {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}+c\ {\begin{pmatrix}0&i\\i&0\end{pmatrix}}+d\ {\begin{pmatrix}i&0\\0&-i\end{pmatrix}}\\&=a\,I+ic\,\sigma _{x}+ib\,\sigma _{y}+id\,\sigma _{z},\end{aligned}}}" loading="lazy"></dd></dl>

où les σi sont les <a href="Matrices_de_Pauli" title="Matrices de Pauli">matrices de spin de Pauli</a>. Ainsi, les paramètres d'Euler-Rodrigues sont les coefficients de la représentation d'une rotation tridimensionnelle dans SU(2).

<div class="mw-heading mw-heading2"><h2 id="Voir_également">Voir également</h2></div>

<ul><li><a href="Quaternions_et_rotation_dans_l'espace" title="Quaternions et rotation dans l'espace">Quaternions et rotation dans l'espace</a></li>

<li><a href="Angles_d'Euler" title="Angles d'Euler">Angles d'Euler</a></li>

<li><a href="Transformation_de_Cayley" title="Transformation de Cayley">Transformation de Cayley</a></li>

<li><a href="Rotation_en_quatre_dimensions" title="Rotation en quatre dimensions">Rotations en dimension 4</a></li>

<li><a href="Quadruplet_pythagoricien" title="Quadruplet pythagoricien">Quadruplet pythagoricien</a></li></ul>

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

<ul><li><a href="%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a>, <cite class="italique">The Theory of Spinors</cite>, Dover, <time>1981</time> (<a href="International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> 0-486-64070-1)<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Spinors&rft.pub=Dover&rft.aulast=Cartan&rft.aufirst=%C3%89lie&rft.date=1981&rft.isbn=0-486-64070-1&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler%E2%80%93Rodrigues"></li>

<li><a href="William_Rowan_Hamilton" title="William Rowan Hamilton">W. R. Hamilton</a>, <cite class="italique">Elements of Quaternions</cite>, Cambridge University Press, <time>1899</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Quaternions&rft.pub=Cambridge+University+Press&rft.aulast=Hamilton&rft.aufirst=W.+R.&rft.date=1899&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler%E2%80%93Rodrigues"></li>

<li>E.J. Haug, <cite class="italique">Computer-Aided Analysis and Optimization of Mechanical Systems Dynamics.</cite>, Springer-Verlag, <time>1984</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computer-Aided+Analysis+and+Optimization+of+Mechanical+Systems+Dynamics.&rft.pub=Springer-Verlag&rft.aulast=Haug&rft.aufirst=E.J.&rft.date=1984&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler%E2%80%93Rodrigues"></li>

<li><abbr class="abbr indicateur-langue" title="Langue : espagnol">(es)</abbr> Garza et Pacheco Quintanilla, « <cite style="font-style:normal" lang="es">Benjamin Olinde Rodrigues, matemático y filántropo, y su influencia en la Física Mexicana</cite> », Revista Mexicana de Física,‎ <time class="nowrap" datetime="2011-06" data-sort-value="2011-06">juin 2011</time>, <abbr class="abbr" title="pages">p.</abbr> 109–113 (<a rel="nofollow" class="external text" href="http://rmf.smf.mx/pdf/rmf-e/57/1/57_1_0109.pdf">lire en ligne</a> [<a rel="nofollow" class="external text" href="https://web.archive.org/web/20120423215549/http://rmf.smf.mx/pdf/rmf-e/57/1/57_1_0109.pdf">archive du <time class="nowrap" datetime="2012-04-23" data-sort-value="2012-04-23">23 avril 2012</time></a>] <abbr class="abbr indicateur-format format-pdf" title="Document au format Portable Document Format (PDF) d'Adobe">[PDF]</abbr>)<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Benjamin+Olinde+Rodrigues%2C+matem%C3%A1tico+y+fil%C3%A1ntropo%2C+y+su+influencia+en+la+F%C3%ADsica+Mexicana&rft.jtitle=Revista+Mexicana+de+F%C3%ADsica&rft.au=Garza&rft.au=Pacheco+Quintanilla&rft.date=2011-06&rft.pages=109%E2%80%93113&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler%E2%80%93Rodrigues"></li>

<li><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Shuster, « <cite style="font-style:normal" lang="en">A Survey of Attitude Representations</cite> », Journal of the Astronautical Sciences, <abbr class="abbr" title="volume">vol.</abbr> 41, <abbr class="abbr" title="numéro">no</abbr> 4,‎ <time>1993</time>, <abbr class="abbr" title="pages">p.</abbr> 439–517 (<a rel="nofollow" class="external text" href="http://malcolmdshuster.com/Pub_1993h_J_Repsurv_scan.pdf">lire en ligne</a> <abbr class="abbr indicateur-format format-pdf" title="Document au format Portable Document Format (PDF) d'Adobe">[PDF]</abbr>)<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+Survey+of+Attitude+Representations&rft.jtitle=Journal+of+the+Astronautical+Sciences&rft.issue=4&rft.au=Shuster&rft.date=1993&rft.volume=41&rft.pages=439%E2%80%93517&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler%E2%80%93Rodrigues"></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes_et_références">Notes et références</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> <abbr class="abbr indicateur-langue" title="Langue : latin">(la)</abbr> Leonhard Euler, « <cite style="font-style:normal" lang="la">Problema algebraicum ob affectiones prorsus singulares memorabile</cite> », Commentatio 407 Indicis Enestoemiani, Novi Comm. Acad. Sci. Petropolitanae 15,‎ <time>1770</time>, <abbr class="abbr" title="pages">p.</abbr> 75–106<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Problema+algebraicum+ob+affectiones+prorsus+singulares+memorabile&rft.jtitle=Commentatio+407+Indicis+Enestoemiani%2C+Novi+Comm.+Acad.+Sci.+Petropolitanae+15&rft.aulast=Euler&rft.aufirst=Leonhard&rft.date=1770&rft.pages=75%E2%80%93106&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler%E2%80%93Rodrigues">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Olinde Rodrigues, « <cite style="font-style:normal">Des lois géométriques qui régissent les déplacements d'un système solide dans l'espace, et de la variation des coordonnées provenant de ces déplacements considérés indépendamment des causes qui peuvent les produire</cite> », Journal de mathématiques pures et appliquées,‎ <time>1840</time>, <abbr class="abbr" title="pages">p.</abbr> 380-440 (<a rel="nofollow" class="external text" href="http://sites.mathdoc.fr/JMPA/PDF/JMPA_1840_1_5_A39_0.pdf">lire en ligne</a>)<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Des+lois+g%C3%A9om%C3%A9triques+qui+r%C3%A9gissent+les+d%C3%A9placements+d%27un+syst%C3%A8me+solide+dans+l%27espace%2C+et+de+la+variation+des+coordonn%C3%A9es+provenant+de+ces+d%C3%A9placements+consid%C3%A9r%C3%A9s+ind%C3%A9pendamment+des+causes+qui+peuvent+les+produire&rft.jtitle=Journal+de+math%C3%A9matiques+pures+et+appliqu%C3%A9es&rft.aulast=Rodrigues&rft.aufirst=Olinde&rft.date=1840&rft.pages=380-440&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler%E2%80%93Rodrigues">

</li>

</ol></div>

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