{"id":2008,"date":"2020-02-08T15:52:23","date_gmt":"2020-02-08T14:52:23","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=2008"},"modified":"2022-06-30T09:32:29","modified_gmt":"2022-06-30T07:32:29","slug":"distance-moyenne-entre-deux-points-aleatoires-dun-carre","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/02\/08\/distance-moyenne-entre-deux-points-aleatoires-dun-carre\/","title":{"rendered":"Distance moyenne entre deux points al\u00e9atoires d&rsquo;un carr\u00e9"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Distance moyenne entre deux points al\u00e9atoires d&rsquo;un carr\u00e9: consid\u00e9rons un carr\u00e9 de c\u00f4t\u00e9 10 et pla\u00e7ons en son int\u00e9rieur deux points de mani\u00e8re al\u00e9atoire, et int\u00e9ressons-nous \u00e0 la distance moyenne entre les deux points.<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 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\/2020\/02\/08\/distance-moyenne-entre-deux-points-aleatoires-dun-carre\/#Dabord_une_figure%E2%80%A6_en_LaTeX_pour_visualiser_la_distance_moyenne_entre_deux_points_aleatoires_dun_carre\" >D&rsquo;abord, une figure&#8230; en \\(\\LaTeX\\) pour visualiser la distance moyenne entre deux points al\u00e9atoires d&rsquo;un carr\u00e9<\/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\/2020\/02\/08\/distance-moyenne-entre-deux-points-aleatoires-dun-carre\/#Simulation_en_Python_pour_la_distance_moyenne_entre_deux_points_aleatoires_dun_carre\" >Simulation en Python pour la distance moyenne entre deux points al\u00e9atoires d&rsquo;un carr\u00e9<\/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\/2020\/02\/08\/distance-moyenne-entre-deux-points-aleatoires-dun-carre\/#Approche_mathematique\" >Approche math\u00e9matique<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Dabord_une_figure%E2%80%A6_en_LaTeX_pour_visualiser_la_distance_moyenne_entre_deux_points_aleatoires_dun_carre\"><\/span>D&rsquo;abord, une figure&#8230; en \\(\\LaTeX\\) pour visualiser la distance moyenne entre deux points al\u00e9atoires d&rsquo;un carr\u00e9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"latex\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">\\documentclass{standalone}\n\\usepackage{tikz}\n\\begin{document}\n\\begin{tikzpicture}\n\\clip (-.2,-1) rectangle (11,10.1);\n\\coordinate (A) at (0,0);\n\\coordinate (B) at (10,0);\n\\coordinate (C) at (10,10);\n\\coordinate (D) at (0,10);\n\\draw (A) -- (B) node[below,midway,scale=2] {10} -- (C) node[right,midway,scale=2] {10} -- (D) -- cycle;\n\\pgfmathparse{10*random()}\\let\\x\\pgfmathresult\n\\pgfmathparse{10*random()}\\let\\y\\pgfmathresult\n\\fill (\\x,\\y) circle (1mm);\n%\n\\pgfmathparse{10*random()}\\let\\xx\\pgfmathresult\n\\pgfmathparse{10*random()}\\let\\yy\\pgfmathresult\n\\fill (\\xx,\\yy) circle (1mm);\n%\n\\pgfmathparse{sqrt((\\x-\\xx)*(\\x-\\xx) + (\\y-\\yy)*(\\y-\\yy))}\\let\\d\\pgfmathresult\n\\draw[line width=1mm] (\\x,\\y) -- (\\xx,\\yy) node[midway,fill=white,scale=1.33] {\\d};\n\\end{tikzpicture}\n\\end{document}<\/pre>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Simulation_en_Python_pour_la_distance_moyenne_entre_deux_points_aleatoires_dun_carre\"><\/span>Simulation en Python pour la distance moyenne entre deux points al\u00e9atoires d&rsquo;un carr\u00e9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/06\/image-8.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"296\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/06\/image-8-300x296.png\" alt=\"Distance moyenne entre deux points al\u00e9atoires\" class=\"wp-image-7711\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/06\/image-8-300x296.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/06\/image-8-100x100.png 100w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/06\/image-8-600x593.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/06\/image-8.png 743w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Afin d&rsquo;avoir une estimation de la distance moyenne que l&rsquo;on peut esp\u00e9rer obtenir, je vais maintenant simplifier les choses et prendre <strong>un carr\u00e9 de c\u00f4t\u00e9 1<\/strong>. Ensuite, j&rsquo;utilise le programme suivant:<\/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 random import random\n\ndef dist(): \n    # choisit 2 points au hasard et retourne la distance entre eux\n    x1 = random()\n    x2 = random()\n    y1 = random()\n    y2 = random()\n\n    return ( (x1 - x2) ** 2 + (y1 - y2) ** 2 ) ** 0.5\n\ndef simul():\n    f = 0\n    N = 10000\n    for k in range(1000):\n        m = 0\n        for k in range(N):\n            m += dist()\n        f += m\/N\n\n    return f \/ 1000<\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>&gt;&gt;&gt; for _ in range(5):\n       print( simul() )\n\n0.5212892093208212\n0.5214659571727084\n0.52142262283233\n0.5214123153216067\n0.5214725134207495<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">On peut raisonnablement conjecturer que la distance moyenne est d&rsquo;environ 0,521.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Approche_mathematique\"><\/span>Approche math\u00e9matique<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Attention, \u00e7a va d\u00e9coller&#8230; !<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Notons \\(A(x_1;y_1)\\) et \\(B(x_2;y_2)\\) les deux points pris de mani\u00e8re al\u00e9atoire. La distance AB est donc \u00e9gale, d&rsquo;apr\u00e8s l&rsquo;ami Pyth. \u00e0 :$$\\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$et en cons\u00e9quence, la distance moyenne entre A et B est:$$\\int_0^1\\int_0^1\\int_0^1\\int_0^1 \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \\text{d}x_1\\text{d}x_2\\text{d}y_1\\text{d}y_2.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Bon, c&rsquo;est pas trop fun d&rsquo;avoir \u00e0 calculer cette int\u00e9grale, alors on va vite trouver un moyen de la simplifier&#8230; On va alors noter \\(x = |x_1-x_2|\\) et \\(y=|y_1-y_2|\\). L&rsquo;ami Pyth. nous dit alors que \\(AB = \\sqrt{x^2+y^2}\\). De plus, \\(x_1,\\ x_2,\\ y_1,\\ y_2\\) suivent la loi uniforme sur [0;1] (la fonction de distribution est 1) donc <em>x<\/em> et <em>y<\/em> suivent une loi dite <em>triangulaire<\/em> (cf. <a rel=\"noreferrer noopener\" aria-label=\"ici (s\u2019ouvre dans un nouvel onglet)\" href=\"https:\/\/fr.wikipedia.org\/wiki\/Loi_triangulaire\" target=\"_blank\">ici<\/a>) [si <em>w<\/em>=|<em>u-v<\/em>| et si <em>u<\/em> et <em>v<\/em> suivent une loi uniforme alors <em>w<\/em> suit une loi triangulaire et sa fonction de r\u00e9partition est 2(1-<em>w<\/em>)]. On peut donc exprimer la distance moyenne par:$$4\\int_0^1\\int_0^1\\sqrt{x^2+y^2}(1-x)(1-y)\\text{d}x\\text{d}y.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Maintenant, posons:$$\\begin{cases}x = r\\cos\\theta\\\\y=r\\sin\\theta\\end{cases}$$avec \\(0 \\leq\\theta\\leq\\frac{\\pi}{4}\\) et \\(0 \\leq r \\leq \\frac{1}{\\cos\\theta}\\). La distance moyenne devient alors:$$4\\int_0^{\\pi\/4}2\\int_0^{1\/\\cos\\theta}\\sqrt{r^2\\cos^2\\theta+r^2\\sin^2\\theta}(1-r\\cos\\theta)(1-r\\sin\\theta)r\\text{d}r\\text{d}\\theta$$soit:$$8 \\int_0^{\\pi\/4}\\int_0^{1\/\\cos\\theta} r^2-r^3\\cos\\theta-r^3\\sin\\theta+r^4\\cos\\theta\\sin\\theta\\text{d}r\\text{d}\\theta.$$L&rsquo;int\u00e9grale int\u00e9rieure (de variable d&rsquo;int\u00e9gration <em>r<\/em>) peut \u00eatre facilement calcul\u00e9e:$$8\\int_0^{\\pi\/4}\\frac{1}{12\\cos^3\\theta}-\\frac{\\tan\\theta}{20\\cos^3\\theta}\\text{d}\\theta.$$Une primitive de \\(\\displaystyle \\frac{\\tan\\theta}{\\cos^3\\theta}\\) est \\(\\displaystyle\\frac{1}{3\\cos^3\\theta}\\) mais une primitive de \\(\\displaystyle \\frac{1}{\\cos^3\\theta}\\) est plus cors\u00e9e \u00e0 trouver. On arrive tout de m\u00eame \u00e0 en trouver une: \\(\\displaystyle\\frac{1}{2}\\left( \\frac{\\tan\\theta}{\\cos\\theta} + \\ln\\left| \\frac{1}{\\cos\\theta}+\\tan\\theta \\right| \\right)\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u00c0 l&rsquo;aide de ces primitives, on trouve alors que la distance moyenne est exactement \u00e9gale \u00e0:$$\\frac{2+\\sqrt2+5\\ln\\big(\\sqrt2+1\\big)}{15}\\approx0,521.$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Distance moyenne entre deux points al\u00e9atoires d&rsquo;un carr\u00e9: consid\u00e9rons un carr\u00e9 de c\u00f4t\u00e9 10 et pla\u00e7ons en son int\u00e9rieur deux points de mani\u00e8re al\u00e9atoire, et int\u00e9ressons-nous \u00e0 la distance moyenne entre les deux points.<\/p>\n","protected":false},"author":1,"featured_media":7714,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,6,5],"tags":[104,88,352,169,101,351],"class_list":["post-2008","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-latex","category-mathematiques","category-python","tag-integrales","tag-latex","tag-maths","tag-probabilites","tag-python","tag-simulations"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Distance moyenne entre deux points al\u00e9atoires d&#039;un carr\u00e9 - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Distance moyenne entre deux points al\u00e9atoires d&#039;un carr\u00e9 : calcul math\u00e9matique et simulation en Python. 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