{"id":1662,"date":"2019-10-26T16:50:26","date_gmt":"2019-10-26T14:50:26","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=1662"},"modified":"2021-10-26T16:53:47","modified_gmt":"2021-10-26T14:53:47","slug":"une-enveloppe-astroidale-obtenue-en-python-avec-turtle","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2019\/10\/26\/une-enveloppe-astroidale-obtenue-en-python-avec-turtle\/","title":{"rendered":"Enveloppe astro\u00efdale Python avec Turtle"},"content":{"rendered":"\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"320\" height=\"240\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/10\/Astroide.gif\" alt=\"\" class=\"wp-image-1663\"\/><figcaption>L&#8217;enveloppe de cette famille de cercles est une astro\u00efde<\/figcaption><\/figure><\/div>\n\n\n\n<p>L&#8217;objectif de cet article est de construire une <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Enveloppe_(g%C3%A9om%C3%A9trie)\" target=\"_blank\" rel=\"noreferrer noopener\">enveloppe<\/a> astro\u00efdale en Python avec cette suite de cercles rouges; on va utiliser pour cela le module <em>Turtle<\/em>.<\/p>\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\/10\/26\/une-enveloppe-astroidale-obtenue-en-python-avec-turtle\/#Enveloppe_astroidale_en_Python_approche_mathematique\" >Enveloppe astro\u00efdale en Python : approche math\u00e9matique<\/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\/10\/26\/une-enveloppe-astroidale-obtenue-en-python-avec-turtle\/#Avec_Turtle\" >Avec Turtle<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Enveloppe_astroidale_en_Python_approche_mathematique\"><\/span>Enveloppe astro\u00efdale en Python : approche math\u00e9matique<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Avant toute chose, il est n\u00e9cessaire de comprendre comment sont obtenus tous les cercles rouges.<\/p>\n\n\n\n<p>Si on regarde et analyse bien la figure, les trac\u00e9s sugg\u00e8rent que pour un angle \\(\\alpha\\) donn\u00e9, exprim\u00e9 en degr\u00e9, on trace un segment d&#8217;origine O (si on se place dans un rep\u00e8re, c&#8217;est l&#8217;origine) et d&#8217;angle \\(\\alpha\\), qui coupe l&#8217;un des c\u00f4t\u00e9 du carr\u00e9 inscrit dans le cercle principal.<\/p>\n\n\n\n<p>Prenons le c\u00f4t\u00e9 en haut \u00e0 droite (donc dans le quadrant x &gt; 0 et y &gt; 0 si on se ram\u00e8ne \u00e0 un rep\u00e8re). Il a pour \u00e9quation \\(y=-x+R\\) si on consid\u00e8re que le cercle principal a pour rayon \\(R\\). Notons I le point d&#8217;intersection de la droite d&#8217;\u00e9quation \\(y = x\\tan(\\alpha)\\), qui forme un angle de \\(\\alpha\\) avec l&#8217;horizontale, avec le segment d&#8217;\u00e9quation \\(y=-x+R\\). Alors, ses coordonn\u00e9es v\u00e9rifient:$$\\begin{cases}y_I=-x_I+R\\\\y_I=x_I\\tan(\\alpha)\\end{cases} $$Donc:$$x_I\\tan(\\alpha)=-x_I+R$$d&#8217;o\u00f9:$$x_I=\\frac{R}{\\tan(\\alpha)+R}.$$<\/p>\n\n\n\n<p>Une fois les coordonn\u00e9es de I connues, on calcule la longueur IM, o\u00f9 M est le point du cercle principal de coordonn\u00e9es \\(R\\cos\\alpha;R\\sin\\alpha)\\), \u00e0 l&#8217;aide de la formule vue en classe de Seconde:$$IM = \\sqrt{(x_I-x_M)^2 + (y_I-y_M)^2}.$$On peut alors tracer le cercle de centre I et de rayon IM : c&#8217;est un des cercles rouges.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Avec_Turtle\"><\/span>Avec Turtle<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Il faut faire appel \u00e0 quelques m\u00e9thodes du module Turtle; inutile donc d&#8217;\u00e9crire:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"false\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">from turtle import *<\/pre>\n\n\n\n<p>En effet, le mieux est de n&#8217;importer que les m\u00e9thodes qui nous int\u00e9ressent. Il en est de m\u00eame pour le module math, o\u00f9 seules les m\u00e9thodes sin, cos, tan et pi sont n\u00e9cessaires (pour la racine carr\u00e9e, on \u00e9l\u00e8ve \u00e0 la puissance 0.5).<\/p>\n\n\n\n<p>On commence donc par tracer un cercle (avec Turtle, c&#8217;est un peu&#8230; comment dire poliment ? &#8230; je trouve pas ! D\u00e9sol\u00e9 !) en se d\u00e9pla\u00e7ant d&#8217;abord en bas de la fen\u00eatre puis en tra\u00e7ant le cercle. Ensuite, on en profite pour tracer le carr\u00e9 inscrit dans le cercle (avec &#8220;goto&#8221;, comme le stylo est d\u00e9j\u00e0 baiss\u00e9, \u00e7a trace les segments).<\/p>\n\n\n\n<p>Maintenant, on fait une boucle it\u00e9rative sur l&#8217;angle variant de 0 \u00e0 359. Si vous observez bien, je ne me suis pas emb\u00eat\u00e9 avec les cas o\u00f9 l&#8217;angle est \u00e9gal \u00e0 90\u00b0, 180\u00b0 et 270\u00b0 car \u00e7a n&#8217;a que peu d&#8217;importance au final du point de vue visuel). En fait 180\u00b0 ne pose pas de probl\u00e8me pour la tangente, mais peu importe&#8230; Ouais, je suis une grosse feignasse !&#8230;<\/p>\n\n\n\n<p>Remarquez aussi que j&#8217;ai pris \\(R=300\\) car la fen\u00eatre par d\u00e9faut fait 800&#215;800. &#8220;300&#8221; me semblait un bon compromis. Voil\u00e0 donc le programme:<\/p>\n\n\n\n<figure class=\"wp-block-image\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/10\/astroide-avec-Python-et-Turtle.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"662\" height=\"708\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/10\/astroide-avec-Python-et-Turtle.png\" alt=\"\" class=\"wp-image-1667\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/10\/astroide-avec-Python-et-Turtle.png 662w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/10\/astroide-avec-Python-et-Turtle-300x321.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/10\/astroide-avec-Python-et-Turtle-600x642.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/10\/astroide-avec-Python-et-Turtle-281x300.png 281w\" sizes=\"auto, (max-width: 662px) 100vw, 662px\" \/><\/a><\/figure>\n\n\n\n<p>Alors l\u00e0, les plus observateurs.trices. d&#8217;entre vous me diront : &#8220;t&#8217;es qu&#8217;un charlatant ! Le GIF n&#8217;est pas exactement ce que fait ce programme&#8230;&#8221; et c&#8217;est vrai ! C&#8217;est en fait un ancien GIF qui tra\u00eenait sur mon disque dur&#8230; quand je vous disais que j&#8217;\u00e9tais une grosse feignasse ! En fait, l&#8217;image finale donne ceci:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"802\" height=\"707\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/enveloppe-astroidale-python-turtle.png\" alt=\"enveloppe astro\u00efdale obtenue avec Python et Turtle\" class=\"wp-image-3626\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/enveloppe-astroidale-python-turtle.png 802w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/enveloppe-astroidale-python-turtle-300x264.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/enveloppe-astroidale-python-turtle-600x529.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/enveloppe-astroidale-python-turtle-768x677.png 768w\" sizes=\"auto, (max-width: 802px) 100vw, 802px\" \/><\/figure><\/div>\n\n\n\n<p>En attendant, si vous souhaitez t\u00e9l\u00e9charger le programme directement plut\u00f4t que de vous emb\u00eater \u00e0 le r\u00e9\u00e9crire \u00e0 la main, c&#8217;est ci-dessous pour les abonn\u00e9\u00b7e\u00b7s:<\/p>\n\n\n\n<div class=\"wp-block-file aligncenter um_article\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/10\/astroide.zip\">astroide<\/a><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/10\/astroide.zip\" class=\"wp-block-file__button\" download>T\u00e9l\u00e9charger<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>L&#8217;objectif de cet article est de construire une enveloppe astro\u00efdale en Python avec cette suite de cercles rouges; on va utiliser pour cela le module Turtle. Enveloppe astro\u00efdale en Python : approche math\u00e9matique Avant toute chose, il est n\u00e9cessaire de comprendre comment sont obtenus tous les cercles rouges. Si on [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-1662","post","type-post","status-publish","format-standard","hentry","category-python"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Enveloppe astro\u00efdale Python avec Turtle - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"L&#039;objectif de cet article est de construire une enveloppe astro\u00efdale avec une suite de cercles \u00e0 l&#039;aide de Python et de son module Turtle.\" \/>\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\/10\/26\/une-enveloppe-astroidale-obtenue-en-python-avec-turtle\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Dessiner une enveloppe astro\u00efdale en Python avec Turtle\" \/>\n<meta property=\"og:description\" content=\"L&#039;objectif de cet article est de construire une enveloppe astro\u00efdale avec une suite de cercles \u00e0 l&#039;aide de Python et de son module Turtle.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.mathweb.fr\/euclide\/2019\/10\/26\/une-enveloppe-astroidale-obtenue-en-python-avec-turtle\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathweb.fr\" \/>\n<meta property=\"article:published_time\" content=\"2019-10-26T14:50:26+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-10-26T14:53:47+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/enveloppe-astroidale-python-turtle.png\" \/>\n\t<meta property=\"og:image:width\" content=\"802\" \/>\n\t<meta property=\"og:image:height\" content=\"707\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"author\" content=\"St\u00e9phane Pasquet\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Dessiner une enveloppe astro\u00efdale en Python avec Turtle\" \/>\n<meta name=\"twitter:description\" content=\"L&#039;objectif de cet article est de construire une enveloppe astro\u00efdale avec une suite de cercles \u00e0 l&#039;aide de Python et de son module Turtle.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/enveloppe-astroidale-python-turtle.png\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"St\u00e9phane Pasquet\" \/>\n\t<meta name=\"twitter:label2\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2019\\\/10\\\/26\\\/une-enveloppe-astroidale-obtenue-en-python-avec-turtle\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2019\\\/10\\\/26\\\/une-enveloppe-astroidale-obtenue-en-python-avec-turtle\\\/\"},\"author\":{\"name\":\"St\u00e9phane Pasquet\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"headline\":\"Enveloppe astro\u00efdale Python avec Turtle\",\"datePublished\":\"2019-10-26T14:50:26+00:00\",\"dateModified\":\"2021-10-26T14:53:47+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2019\\\/10\\\/26\\\/une-enveloppe-astroidale-obtenue-en-python-avec-turtle\\\/\"},\"wordCount\":627,\"commentCount\":2,\"publisher\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"image\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2019\\\/10\\\/26\\\/une-enveloppe-astroidale-obtenue-en-python-avec-turtle\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2019\\\/10\\\/Astroide.gif\",\"articleSection\":[\"Python\"],\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2019\\\/10\\\/26\\\/une-enveloppe-astroidale-obtenue-en-python-avec-turtle\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2019\\\/10\\\/26\\\/une-enveloppe-astroidale-obtenue-en-python-avec-turtle\\\/\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2019\\\/10\\\/26\\\/une-enveloppe-astroidale-obtenue-en-python-avec-turtle\\\/\",\"name\":\"Enveloppe astro\u00efdale Python avec Turtle - 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