{"id":1426,"date":"2019-07-14T16:40:16","date_gmt":"2019-07-14T14:40:16","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=1426"},"modified":"2025-02-26T11:09:35","modified_gmt":"2025-02-26T10:09:35","slug":"introduction-aux-matrices-de-rotation","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2019\/07\/14\/introduction-aux-matrices-de-rotation\/","title":{"rendered":"Introduction aux matrices de rotation"},"content":{"rendered":"\n<p>Consid\u00e9rons la configuration suivante :<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/matrice_rotation.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"295\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/matrice_rotation-300x295.png\" alt=\"\" class=\"wp-image-1427\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/matrice_rotation-300x295.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/matrice_rotation.png 452w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p>Dans le rep\u00e8re orthonorm\u00e9 d&#8217;origine O, A(x;y) est un point quelconque et A'(x&#8217;;y&#8217;) est son image par la rotation de centre O et d&#8217;angle \\(\\theta\\). On cherche \u00e0 exprimer x&#8217; et y&#8217; en fonction de x, y et \\(\\theta\\)&#8230;<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/07\/14\/introduction-aux-matrices-de-rotation\/#Un_peu_de_trigonometrie\" >Un peu de trigonom\u00e9trie<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/07\/14\/introduction-aux-matrices-de-rotation\/#Notation_matricielle\" >Notation matricielle<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/07\/14\/introduction-aux-matrices-de-rotation\/#Obtenir_une_equation_de_la_droite_image_par_rotation\" >Obtenir une \u00e9quation de la droite image par rotation<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/07\/14\/introduction-aux-matrices-de-rotation\/#Un_peu_de_Python\" >Un peu de Python<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Un_peu_de_trigonometrie\"><\/span>Un peu de trigonom\u00e9trie<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Compl\u00e9tons la figure pr\u00e9c\u00e9dente en introduisant un angle :<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/matrice_rotation02.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/matrice_rotation02-300x300.png\" alt=\"\" class=\"wp-image-1428\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/matrice_rotation02-300x300.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/matrice_rotation02-100x100.png 100w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/matrice_rotation02-150x150.png 150w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/matrice_rotation02.png 443w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p>Dans le triangle rectangle rouge (OAx), on a : $$(1)\\qquad\\begin{cases}x&amp;=OA\\cos\\varphi\\\\y&amp;=OA\\sin\\varphi\\end{cases}$$<\/p>\n\n\n\n<p>Dans le triangle rectangle orange (OA&#8217;x&#8217;), on a : $$\\begin{cases}x&#8217;&amp;=OA&#8217;\\cos(\\theta+\\varphi)=OA&#8217;\\big[\\cos\\theta\\cos\\varphi-\\sin\\theta\\sin\\varphi \\big]\\\\y&#8217;&amp;=OA&#8217;\\sin(\\theta+\\varphi)=OA&#8217;\\big[\\sin\\theta\\cos\\varphi+\\sin\\varphi\\cos\\theta \\big]\\end{cases}$$Or, OA = OA&#8217; donc ceci revient \u00e0 \u00e9crire, en tenant compte des \u00e9galit\u00e9s (1) : $$\\begin{cases}x&#8217; &amp; = x\\cos\\theta\\ &#8211; y\\sin\\theta\\\\y&#8217; &amp; = x\\sin\\theta + y\\cos\\theta\\end{cases}$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Notation_matricielle\"><\/span>Notation matricielle<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Du syst\u00e8me :  $$\\begin{cases}x&#8217; &amp; =  x\\boxed{\\cos\\theta}+ y\\boxed{-\\sin\\theta}\\\\y&#8217; &amp; = x\\boxed{\\sin\\theta} + y\\boxed{\\cos\\theta}\\end{cases}$$ on peut &#8220;extraire&#8221; les coefficients encadr\u00e9s et les mettre dans un tableau: $$\\begin{array}{|c|c|}\\hline\\cos\\theta &amp; -\\sin\\theta\\\\\\hline\\sin\\theta &amp; \\cos\\theta\\\\\\hline\\end{array}$$que l&#8217;on va tout de suite repr\u00e9senter diff\u00e9remment, pour simplifier, sous la forme suivante : $$\\begin{pmatrix}\\cos\\theta &amp; -\\sin\\theta\\\\\\sin\\theta &amp; \\cos\\theta\\end{pmatrix}.$$C&#8217;est cette repr\u00e9sentation que l&#8217;on va appeler &#8220;matrice de rotation&#8221;. Le syst\u00e8me :  $$\\begin{cases}x&#8217; &amp; =  x\\cos\\theta\\ &#8211; y\\sin\\theta\\\\y&#8217; &amp; = x\\sin\\theta + y\\cos\\theta\\end{cases}$$ peut alors s&#8217;\u00e9crire de mani\u00e8re matricielle ainsi : $$\\begin{pmatrix}x&#8217;\\\\y&#8217;\\end{pmatrix}= \\begin{pmatrix}\\cos\\theta &amp; -\\sin\\theta\\\\\\sin\\theta &amp; \\cos\\theta\\end{pmatrix}  \\begin{pmatrix}x \\\\ y\\end{pmatrix}.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Obtenir_une_equation_de_la_droite_image_par_rotation\"><\/span>Obtenir une \u00e9quation de la droite image par rotation<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Consid\u00e9rons une droite (d) d&#8217;\u00e9quation r\u00e9duite \\( y = mx + p\\). Cherchons une \u00e9quation cart\u00e9sienne de l&#8217;image (d&#8217;) de (d) par la rotation de centre O et d&#8217;angle \\(\\theta\\).<\/p>\n\n\n\n<p>Notons \\(\\vec{u}\\binom{1}{m}\\) un vecteur directeur de (d) et \\(\\vec{u&#8217;}\\binom{a}{b}\\) un vecteur directeur de (d&#8217;) de sorte que \\(\\|\\vec{u}\\|^2=\\|\\vec{u&#8217;}\\|^2=1+m^2\\). Alors,$$\\vec{u}\\cdot\\vec{u&#8217;}=\\|\\vec{u}\\|\\times\\|\\vec{u&#8217;}\\|\\times\\cos\\theta=(1+m^2)\\cos\\theta$$et par suite:$$\\begin{cases}a^2+b^2&amp;=1+m^2\\\\\\vec{u}\\cdot\\vec{u&#8217;}&amp;=a+bm=(1+m^2)\\cos\\theta\\end{cases}$$<\/p>\n\n\n\n<p>On peut alors \u00e9crire, dans un premier temps:$$a=(1+m^2)\\cos\\theta-bm$$ et en substituant dans l &#8216;autre \u00e9quation, on obtient ensuite:$$(1+m^2)^2\\cos^2\\theta-2bm(1+m^2)\\cos\\theta+(1+m^2)b^2=1+m^2$$soit, en simplifiant par \\(1+m^2\\):$$b^2-(2m\\cos\\theta) b+(1+m^2)\\cos^2\\theta-1=0.$$C&#8217;est une \u00e9quation du second degr\u00e9 en \\(b\\) de discriminant:$$\\Delta=4m^2\\cos^2\\theta-4\\big[ (1+m^2)\\cos^2\\theta-1\\big] = 4\\sin^2\\theta$$dont les solutions sont:$$b=m\\cos\\theta\\pm|\\sin\\theta|.$$On en d\u00e9duit alors:$$a=\\cos\\theta\\pm m|\\sin\\theta|.$$ Prenons, pour simplifier:$$b=m\\cos\\theta+|\\sin\\theta|\\quad,\\quad a=\\cos\\theta &#8211; m|\\sin\\theta|.$$<\/p>\n\n\n\n<p>Une \u00e9quation cart\u00e9sienne de (d&#8217;) est de la forme:$$bx-ay+c=0.$$Or, le point de coordonn\u00e9es \\( \\begin{pmatrix}\\cos\\theta &amp; -\\sin\\theta\\\\\\sin\\theta &amp; \\cos\\theta\\end{pmatrix} \\begin{pmatrix}0 \\\\ p\\end{pmatrix} = \\begin{pmatrix}-p\\sin\\theta \\\\ p\\cos\\theta\\end{pmatrix} \\) appartient \u00e0 (d&#8217;), comme image du point d&#8217;intersection de (d) avec l&#8217;axe des ordonn\u00e9es par la rotation <em>r<\/em>. On en d\u00e9duit alors que:$$c=pa\\cos\\theta+pb\\sin\\theta$$soit:$$c=p\\cos^2\\theta+p\\sin\\theta|\\sin\\theta|-m\\cos\\theta|\\sin\\theta|+mp\\cos\\theta\\sin\\theta$$Une \u00e9quation cart\u00e9sienne de (d&#8217;) est donc:$$\\begin{array}{l}\\big(m\\cos\\theta\\pm|\\sin\\theta|\\big)x-\\big(\\cos\\theta \\pm m|\\sin\\theta|\\big)y\\\\+p\\sin\\theta\\big(m\\cos\\theta+|\\sin\\theta|\\big)+p\\cos^2\\theta+p\\sin\\theta|\\sin\\theta|-m\\cos\\theta|\\sin\\theta|+mp\\cos\\theta\\sin\\theta\\big)=0\\end{array}$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Un_peu_de_Python\"><\/span>Un peu de Python<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Si nous avons fait cela, c&#8217;est peut-\u00eatre pour nous simplifier la vie un peu plus tard&#8230; On peut, avec cette derni\u00e8re formule, \u00e9crire un programme Python qui nous donne directement une \u00e9quation cart\u00e9sienne de (d&#8217;) en fonction des param\u00e8tres m et p que l&#8217;on entre au clavier.<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">from math import cos,sin,pi\nimport matplotlib.pyplot as plt\n\nm = float(input(\"Entrer le coefficient directeur 'm' : \"))\np = float(input(\"Entrer l'ordonn\u00e9e \u00e0 l'origine 'p' : \"))\nt = float(input(\"Entrer l'angle de la rotation de centre O (en degr\u00e9) : \"))\nt = t*pi\/180\n\nb = m*cos(t) + abs( sin(t) )\na = -m*abs( sin(t) ) + cos(t)\nc = p*cos(t)*a + p*sin(t)*b\n\nprint(f'{b}x+{-a}y+{c}=0')\n\n# trac\u00e9\n\nfig = plt.figure()\n\n\naxes = fig.add_subplot(111)\naxes.set_frame_on(True)\n\naxes.yaxis.tick_left()\naxes.xaxis.set_visible(True)\n(xmin, xmax) = (-5,5)\n(ymin, ymax) = (-20,20)\naxes.add_artist(plt.Line2D((xmin, xmax), (0, 0),\n                              color = 'magenta', linewidth = 1))\naxes.add_artist(plt.Line2D((0, 0), (ymin, ymax),\n                              color = 'magenta', linewidth = 1))\n\n\nx = [-5 , 5]\ny = [-5*m+p , 5*m+p]\n\nplt.plot(x,y,label='(d)')\n\ny = [b*(-5)\/a+c\/a , b*5\/a+c\/a]\n\nplt.plot(x,y,color='r',label=\"(d')\")\n\nlegend = fig.legend(loc='upper center', shadow=True, fontsize='x-large')\n\nplt.show()<\/pre>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/rotation_droite.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"640\" height=\"478\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/rotation_droite.png\" alt=\"\" class=\"wp-image-1445\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/rotation_droite.png 640w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/rotation_droite-300x224.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/rotation_droite-600x448.png 600w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/a><figcaption class=\"wp-element-caption\">Graphique obtenu avec la droite d&#8217;\u00e9quation y = 3x+2 et avec un angle de 60\u00b0<\/figcaption><\/figure>\n<\/div>\n\n\n<p>Par d\u00e9faut, le rep\u00e8re n&#8217;\u00e9tant pas orthonorm\u00e9, l&#8217;angle que l&#8217;on voit n&#8217;est pas repr\u00e9sentatif. Voyons donc un autre code :<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">from math import cos,sin,pi\nimport matplotlib.pyplot as plt\n\nm = float(input(\"Entrer le coefficient directeur 'm' : \"))\np = float(input(\"Entrer l'ordonn\u00e9e \u00e0 l'origine 'p' : \"))\nt = float(input(\"Entrer l'angle de la rotation de centre O (en degr\u00e9) : \"))\nt = t*pi\/180\n\nb = m*cos(t) + abs( sin(t) )\na = -m*abs( sin(t) ) + cos(t)\nc = p*cos(t)*a + p*sin(t)*b\n\nprint(f'{b}x+{-a}y+{c}=0')\n\n# trac\u00e9\n\nfig = plt.figure()\n\nplt.axis('equal')\nplt.grid(axis='both',color='lightgray', linestyle='--')\n\n\nx = [-8 , 8]\ny = [0 , 0]\nplt.plot(x,y,linewidth=1,color='black')\n\nx = [0 , 0]\ny = [-6 , 6]\nplt.plot(x,y,linewidth=1,color='black')\n\n\nx = [-5 , 5]\ny = [-5*m+p , 5*m+p]\n\nplt.plot(x,y,label='(d)')\n\ny = [b*(-5)\/a+c\/a , b*5\/a+c\/a]\n\nplt.plot(x,y,color='r',label=\"(d')\")\n\nlegend = fig.legend(loc='upper center', shadow=True, fontsize='x-large')\n\nplt.show()<\/pre>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/rotation_droite2.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"640\" height=\"478\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/rotation_droite2.png\" alt=\"\" class=\"wp-image-1446\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/rotation_droite2.png 640w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/rotation_droite2-600x448.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/07\/rotation_droite2-300x224.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/a><figcaption class=\"wp-element-caption\">Graphique obtenu avec (d) : y = x + 1 et avec un angle de 90\u00b0<\/figcaption><\/figure>\n<\/div>\n\n\n<p>Sur ce dernier exemple, on voit parfaitement l&#8217;angle de la rotation (ici, 90\u00b0).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consid\u00e9rons la configuration suivante : Dans le rep\u00e8re orthonorm\u00e9 d&#8217;origine O, A(x;y) est un point quelconque et A'(x&#8217;;y&#8217;) est son image par la rotation de centre O et d&#8217;angle \\(\\theta\\). On cherche \u00e0 exprimer x&#8217; et y&#8217; en fonction de x, y et \\(\\theta\\)&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,6,5],"tags":[135,134,101,133],"class_list":["post-1426","post","type-post","status-publish","format-standard","hentry","category-enseignement","category-mathematiques","category-python","tag-matplotlib","tag-matrice","tag-python","tag-rotation"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Introduction aux matrices de rotation - Mathweb.fr<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/07\/14\/introduction-aux-matrices-de-rotation\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Introduction aux matrices de rotation - Mathweb.fr\" \/>\n<meta property=\"og:description\" content=\"Consid\u00e9rons la configuration suivante : Dans le rep\u00e8re orthonorm\u00e9 d&#8217;origine O, A(x;y) est un point quelconque et A&#039;(x&#8217;;y&#8217;) est son image par la rotation de centre O et d&#8217;angle (theta). 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