{"id":728,"date":"2018-09-14T13:59:37","date_gmt":"2018-09-14T11:59:37","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=728"},"modified":"2021-10-26T17:11:00","modified_gmt":"2021-10-26T15:11:00","slug":"des-equations-de-coeurs","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/14\/des-equations-de-coeurs\/","title":{"rendered":"Des \u00e9quations de c\u0153urs"},"content":{"rendered":"\n<p>On peut \u00eatre matheux et romantique. La preuve : toutes ces \u00e9quations de c\u0153urs&#8230; Tiens ! C&#8217;est un bon pr\u00e9texte pour parler de courbes param\u00e9tr\u00e9es !<\/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-1'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/14\/des-equations-de-coeurs\/#Pre-requis_courbes_parametrees\" >Pr\u00e9-requis : courbes param\u00e9tr\u00e9es<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/14\/des-equations-de-coeurs\/#Le_coeur_dEugene_Beutel_1909\" >Le c\u0153ur d&#8217;Eug\u00e8ne Beutel (1909)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/14\/des-equations-de-coeurs\/#Le_coeur_de_Raphael_Laporte_1993\" >Le c\u0153ur de Rapha\u00ebl Laporte (1993)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/14\/des-equations-de-coeurs\/#Le_coeur_de_Dwight_Boddorf_2008\" >Le c\u0153ur de Dwight Boddorf (2008)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/14\/des-equations-de-coeurs\/#Le_coeur_de_Pierre_Daniel_2013\" >Le c\u0153ur&nbsp; de Pierre Daniel (2013)<\/a><\/li><\/ul><\/nav><\/div>\n<h1 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Pre-requis_courbes_parametrees\"><\/span>Pr\u00e9-requis : courbes param\u00e9tr\u00e9es<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<p>Dans le cursus scolaire fran\u00e7ais, nous voyons assez t\u00f4t, et longtemps, que certains ph\u00e9nom\u00e8nes peuvent se traduire par des courbes, engendr\u00e9es par des \u00e9quations cart\u00e9siennes, c&#8217;est-\u00e0-dire des \u00e9quations de la forme \\(y=f(x)\\) pour les plus simples.<\/p>\n\n\n\n<p>Mais il existe d&#8217;autres types de courbes comme par exemple les courbes param\u00e9tr\u00e9es (engendr\u00e9es par des \u00e9quations param\u00e9triques) ou les courbes polaires.<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Les \u00e9quations param\u00e9triques de courbes planes sont de la forme : \\[ \\left\\{\\begin{array}{l}x=f(t)\\\\y=g(t)\\end{array}\\right.\\] o\u00f9&nbsp;<em>f<\/em> et&nbsp;<em>g<\/em> sont deux fonctions cart\u00e9siennes, et o\u00f9 (<em>x<\/em> ;&nbsp;<em>y<\/em>) repr\u00e9sentent les coordonn\u00e9es des points de la courbe param\u00e9tr\u00e9e.<\/li><li>Les \u00e9quations polaires de courbes planes sont de la forme : \\[\\rho = f(\\theta)\\] o\u00f9 \\(\\rho\\) repr\u00e9sente la distance de l&#8217;origine O du rep\u00e8re \\(O;\\vec{i},\\vec{j})\\) au point M de la courbe, et \\(\\theta\\) l&#8217;angle \\(\\Big(\\widehat{\\vec{i};\\vec{OM}}\\Big)\\).<\/li><\/ul>\n\n\n\n<h1 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_coeur_dEugene_Beutel_1909\"><\/span>Le c\u0153ur d&#8217;Eug\u00e8ne Beutel (1909)<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<p>Ce c\u0153ur a pour \u00e9quation cart\u00e9sienne :&nbsp;\\[(x^2+y^2-1)^3=x^2y^3.\\]<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-eugene-beutel.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"188\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-eugene-beutel-300x188.png\" alt=\"\" class=\"wp-image-731\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-eugene-beutel-300x188.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-eugene-beutel-600x377.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-eugene-beutel.png 631w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure><\/div>\n\n\n\n<p>Pour construire cette courbe, nous partons de l&#8217;\u00e9quation d&#8217;origine et nous prenons la racine cubique de chaque membre :&nbsp;\\[x^2+y^2-1=yx^{2\/3}\\] puis nous mettons tout dans le membre de gauche : \\[&nbsp;y^2-x^{2\/3}y+x^2-1=0.\\qquad (E) \\]<\/p>\n\n\n\n<p>Nous reconnaissons alors une \u00e9quation polynomiale d&#8217;inconnue&nbsp;<em>y<\/em> de degr\u00e9 2 et de discriminant :&nbsp;\\[ \\Delta=x^{4\/3}+4-4x^2.\\]<\/p>\n\n\n\n<p>\\(\\Delta\\geq0\\) pour \\(x\\in[0;1,13902816469]\\) (pour obtenir cela, on trace la courbe de la fonction&nbsp;<em>f<\/em> d\u00e9finie par \\(f(x)=x^{4\/3}+4-4x^2\\), on remarque qu&#8217;elle est au-dessus de l&#8217;axe des abscisses de 0 \u00e0 une certaine valeur de <em>x<\/em>, puis en-dessous. On r\u00e9sout alors \u00e0 l&#8217;aide d&#8217;un logiciel de calculs formels l&#8217;\u00e9quation \\(x^{4\/3}+4-4x^2=0\\), et nous obtenons la valeur 1,13902816469).<\/p>\n\n\n\n<p>Ainsi, nous obtenons deux solutions pour l&#8217;\u00e9quation (E) :&nbsp;\\[ y=\\frac{x^{2\/3}\\pm\\sqrt{x^{4\/3}+4-4x^2}}{2}.\\]<\/p>\n\n\n\n<p>En tra\u00e7ant les courbes d&#8217;\u00e9quations respectives :&nbsp;\\[ y=\\frac{x^{2\/3}-\\sqrt{x^{4\/3}+4-4x^2}}{2}\\] et \\[ y=\\frac{x^{2\/3}+\\sqrt{x^{4\/3}+4-4x^2}}{2}\\]&nbsp;sur [0 ; 1,139], nous obtenons la partie droite du c\u0153ur de Beutel.<\/p>\n\n\n\n<p>Par sym\u00e9trie par rapport \u00e0 l&#8217;axe des ordonn\u00e9es, nous obtenons l&#8217;autre partie, et ainsi le c\u0153ur en entier.<\/p>\n\n\n\n<h1 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_coeur_de_Raphael_Laporte_1993\"><\/span>Le c\u0153ur de Rapha\u00ebl Laporte (1993)<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<p>L&#8217;\u00e9quation de cette courbe fut trouv\u00e9e par Rapha\u00ebl Laporte alors qu&#8217;il avait 16 ans, et ce pour sa petite amie&#8230; L\u00e0, si c&#8217;est pas romantique, je ne sais pas ce qui l&#8217;est !<\/p>\n\n\n\n<p>Cette \u00e9quation param\u00e9trique est :&nbsp;\\[\\left\\{\\begin{array}{l}x=\\sin^3t\\\\y=\\cos t-\\cos^4t\\end{array}\\right.\\]<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-raphael-laporte.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"279\" height=\"300\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-raphael-laporte-279x300.png\" alt=\"\" class=\"wp-image-732\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-raphael-laporte-279x300.png 279w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-raphael-laporte-300x322.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-raphael-laporte.png 486w\" sizes=\"auto, (max-width: 279px) 100vw, 279px\" \/><\/a><\/figure><\/div>\n\n\n\n<p>Il faut tout de m\u00eame rappeler qu&#8217;en 1993, les courbes param\u00e9tr\u00e9es \u00e9taient encore au programme de Terminale scientifique&#8230; ce qui n&#8217;est plus le cas en France depuis tr\u00e8s (trop ?) longtemps.<\/p>\n\n\n\n<h1 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_coeur_de_Dwight_Boddorf_2008\"><\/span>Le c\u0153ur de Dwight Boddorf (2008)<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<p>Ce c\u0153ur est donn\u00e9 par l&#8217;\u00e9quation polaire suivante :&nbsp;\\[\\rho = \\vert \\tan\\theta \\vert^{\\vert\\cot\\theta\\vert}\\quad,\\quad0\\leq\\theta\\leq\\pi\\]<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-dwight-boddorf.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"256\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-dwight-boddorf-300x256.png\" alt=\"\" class=\"wp-image-733\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-dwight-boddorf-300x256.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-dwight-boddorf.png 522w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure><\/div>\n\n\n\n<h1 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_coeur_de_Pierre_Daniel_2013\"><\/span>Le c\u0153ur&nbsp; de Pierre Daniel (2013)<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<p>L&#8217;\u00e9quation param\u00e9trique de ce&nbsp;c\u0153ur est :&nbsp;\\[\\left\\{\\begin{array}{l}x^2=\\frac{(1-t^2)^3}{1+t^2}\\\\y=\\frac{4t}{1+t^2}-t^2\\end{array}\\right.\\]<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-pierre-daniel.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"247\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-pierre-daniel-300x247.png\" alt=\"\" class=\"wp-image-734\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-pierre-daniel-300x247.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/coeur-pierre-daniel.png 584w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure><\/div>\n\n\n\n<p>Pour en voir davantage :<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><a href=\"http:\/\/www.mathcurve.com\/courbes2d\/ornementales\/ornementales.shtml\" target=\"_blank\" rel=\"noopener\">http:\/\/www.mathcurve.com\/courbes2d\/ornementales\/ornementales.shtml<\/a><\/li><li><a href=\"http:\/\/www.mathematische-basteleien.de\/heart.htm\" target=\"_blank\" rel=\"noopener\">http:\/\/www.mathematische-basteleien.de\/heart.htm<\/a><\/li><li><a href=\"http:\/\/mathworld.wolfram.com\/HeartCurve.html\" target=\"_blank\" rel=\"noopener\">http:\/\/mathworld.wolfram.com\/HeartCurve.html<\/a><\/li><\/ul>\n\n\n\n<div class=\"wp-block-file aligncenter um_article\"><a  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Des-\u00e9quations-de-coeurs-1.pdf\" data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">Des \u00e9quations de coeurs<\/a><a  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Des-\u00e9quations-de-coeurs-1.pdf\" class=\"wp-block-file__button\" download data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">T\u00e9l\u00e9charger<\/a><\/div>\n\n\n\n<p>Pour obtenir les sources \\(\\LaTeX\\) du document PDF:<\/p>\n\n\n\n<div class=\"wp-block-file aligncenter um_article\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Des-\u00e9quations-de-coeurs.zip\">Des \u00e9quations de coeurs<\/a><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/Des-\u00e9quations-de-coeurs.zip\" class=\"wp-block-file__button\" download>T\u00e9l\u00e9charger<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>On peut \u00eatre matheux et romantique. La preuve : toutes ces \u00e9quations de c\u0153urs&#8230; Tiens ! C&#8217;est un bon pr\u00e9texte pour parler de courbes param\u00e9tr\u00e9es !<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[72,73,71],"class_list":["post-728","post","type-post","status-publish","format-standard","hentry","category-mathematiques","tag-courbes-parametrees","tag-courbes-polaires","tag-coeurs"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Des \u00e9quations de c\u0153urs - 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\/2018\/09\/14\/des-equations-de-coeurs\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Des \u00e9quations de c\u0153urs - Mathweb.fr\" \/>\n<meta property=\"og:description\" content=\"On peut \u00eatre matheux et romantique. 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