{"id":3210,"date":"2020-08-21T10:30:21","date_gmt":"2020-08-21T08:30:21","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?page_id=3210"},"modified":"2023-04-16T16:18:15","modified_gmt":"2023-04-16T14:18:15","slug":"calcul-approche-de-longueur-dune-portion-de-courbe-en-python","status":"publish","type":"page","link":"https:\/\/www.mathweb.fr\/euclide\/calcul-approche-de-longueur-dune-portion-de-courbe-en-python\/","title":{"rendered":"Calcul approch\u00e9 de longueur d\u2019une portion de courbe en Python"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Sur cette page, nous allons voir une m\u00e9thode pour d\u00e9terminer une valeur approch\u00e9e de la longueur d&rsquo;une courbe repr\u00e9sentative d&rsquo;une fonction sur un intervalle donn\u00e9 en Python.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"640\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe-1024x640.png\" alt=\"Longueur d'une courbe en Python\" class=\"wp-image-3211\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe-1024x640.png 1024w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe-300x188.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe-600x375.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe-768x480.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe.png 1080w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Approche math\u00e9matique<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Une courbe peut \u00eatre vue comme une succession de segments minuscules. Prenons cette courbe:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"476\" height=\"385\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe-1.png\" alt=\"Longueur d'une courbe en Python\" class=\"wp-image-3212\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe-1.png 476w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe-1-300x243.png 300w\" sizes=\"auto, (max-width: 476px) 100vw, 476px\" \/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Divisons l&rsquo;intervalle [-0,5 ; 1] en 3 par exemple:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"488\" height=\"385\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe-2.png\" alt=\"Longueur d'une courbe en Python\" class=\"wp-image-3213\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe-2.png 488w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe-2-300x237.png 300w\" sizes=\"auto, (max-width: 488px) 100vw, 488px\" \/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">On a ici trac\u00e9 3 segments; certes, ils ne \u00ab\u00a0collent\u00a0\u00bb pas \u00e0 la courbe, mais on peut imaginer que si l&rsquo;on divise l&rsquo;intervalle [-0,5 ; 1] en plus de 3, les segments vont \u00eatre tr\u00e8s petits et donc vont coller \u00e0 la courbe. La longueur totale de ces segments va donc \u00eatre tr\u00e8s proche de la longueur de la courbe.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Longueur d&rsquo;une portion de courbe en Python<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Nous allons avant tout d\u00e9finir une fonction Python qui repr\u00e9sente la fonction math\u00e9matique \\(f(x)=x^3-x^2-0,2x+1\\) dont la courbe repr\u00e9sentative est donn\u00e9e pr\u00e9c\u00e9demment.<\/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=\"\">def f(x):\n    return x**3 - x**2 - 0.2*x +1<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Ensuite, nous cr\u00e9ons une fonction Python <em>longueur(a,b,n<\/em>) qui admet 3 arguments : <em>a<\/em> et <em>b<\/em> sont les bornes de l&rsquo;intervalle, et <em>n<\/em> le nombre de subdivisions souhait\u00e9es (le nombre de segments \u00e0 consid\u00e9rer).<\/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=\"4\" data-enlighter-title=\"\" data-enlighter-group=\"\">def longueur(a,b,n):\n    pas = (b - a) \/ n\n    x = a\n    longueur = 0\n    while x &lt;= b-pas:\n        longueur += ( pas**2 + ( f(x+pas) - f(x) )**2 )**0.5\n        x += pas\n        \n    return longueur<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">On commence par d\u00e9finir le <em>pas<\/em>, c&rsquo;est-\u00e0-dire la longueur des subdivisions de l&rsquo;intervalle [<em>a<\/em> ; <em>b<\/em>] : c&rsquo;est la longueur de l&rsquo;intervalle que l&rsquo;on a divis\u00e9 par <em>n<\/em> (\u2192 ligne 5). <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ensuite, on part de <em>x<\/em> = <em>a<\/em> (\u2192 ligne 6) et on initialise la longueur \u00e0 0 (\u2192 ligne 7). Nous allons parcourir l&rsquo;intervalle par petits sauts : <em>x<\/em>, puis <em>x<\/em> + <em>pas<\/em>, puis <em>x<\/em> + 2<em>pas<\/em>, etc.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"339\" height=\"444\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe-3.png\" alt=\"Longueur d'une courbe en Python\" class=\"wp-image-3214\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe-3.png 339w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe-3-300x393.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/08\/Python-longueur-courbe-3-229x300.png 229w\" sizes=\"auto, (max-width: 339px) 100vw, 339px\" \/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Pour un <em>x<\/em> donn\u00e9, on calcule la longueur du segment entre <em>x<\/em> et <em>x<\/em> + <em>pas<\/em> \u00e0 l&rsquo;aide de la formule:$$AB = \\sqrt{(x_B-x_A)^2 + (y_B-y_A)^2}$$qui donne dans notre cas:$$\\sqrt{(x+pas-x)^2+(f(x+pas)-f(x))^2}=\\sqrt{pas^2+(f(x+pas)-f(x))^2}.$$C&rsquo;est ainsi que l&rsquo;on obtient la ligne 9 : on ajoute \u00e0 la longueur le nombre calcul\u00e9.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ce calcul est fait jusqu&rsquo;\u00e0 ce que <em>x<\/em> soit \u00e9gal \u00e0 <em>b<\/em> &#8211; <em>pas<\/em>. Ensuite, la boucle <em>while<\/em> s&rsquo;arr\u00eate et la fonction Python retourne la derni\u00e8re valeur de <em>longueur<\/em> calcul\u00e9e, qui correspond donc \u00e0 la somme des longueurs des segments.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En ex\u00e9cutant ce programme, on a par exemple:<\/p>\n\n\n\n<pre class=\"wp-block-preformatted\">>>> longueur(-0.5,1,100)\n1.674759304224313\n\n>>> longueur(-0.5,1,1000)\n1.693754517038786\n\n>>> longueur(-0.5,1,10000)\n1.6937551333505372<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">N&rsquo;oubliez pas que si vous avez des difficult\u00e9s en math\u00e9matiques, <a href=\"https:\/\/courspasquet.fr\" target=\"_blank\" rel=\"noreferrer noopener\">je peux vous aider par webcam<\/a> !<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/ressources-python\/\">[Retour aux ressources Python]<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Sur cette page, nous allons voir une m\u00e9thode pour d\u00e9terminer une valeur approch\u00e9e de la longueur d&rsquo;une courbe repr\u00e9sentative d&rsquo;une fonction sur un intervalle donn\u00e9 en Python. Approche math\u00e9matique Une courbe peut \u00eatre vue comme une succession de segments minuscules. Prenons cette courbe: Divisons l&rsquo;intervalle [-0,5 ; 1] en 3 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-3210","page","type-page","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Longueur d&#039;une portion de courbe en Python - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Nous allons voir une m\u00e9thode pour d\u00e9terminer une valeur approch\u00e9e de la longueur d&#039;une courbe sur un intervalle donn\u00e9 en Python.\" \/>\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\/calcul-approche-de-longueur-dune-portion-de-courbe-en-python\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Longueur d&#039;une portion de courbe en Python - 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