{"id":9735,"date":"2024-05-02T15:09:54","date_gmt":"2024-05-02T13:09:54","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=9735"},"modified":"2024-05-02T15:16:06","modified_gmt":"2024-05-02T13:16:06","slug":"diagrammes-de-voronoi-point-de-vue-mathematiques-et-python","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/02\/diagrammes-de-voronoi-point-de-vue-mathematiques-et-python\/","title":{"rendered":"Diagrammes de Vorono\u00ef, point de vue math\u00e9matiques et Python"},"content":{"rendered":"\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Un diagramme de Vorono\u00ef (math\u00e9matiques et Python), est une repr\u00e9sentation graphique. Nous allons voir en quoi elle consiste.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Pour celles et ceux qui d\u00e9sirent en savoir plus, vous pouvez consulter la page <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Diagramme_de_Vorono%C3%AF\" target=\"_blank\" rel=\"noreferrer noopener\">wikipedia<\/a>.<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_87_1 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\/2024\/05\/02\/diagrammes-de-voronoi-point-de-vue-mathematiques-et-python\/#Diagramme_de_Voronoi_point_de_vue_mathematiques_on_verra_ensuite_pour_Python\" >Diagramme de Vorono\u00ef: point de vue math\u00e9matiques (on verra ensuite pour Python)<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/02\/diagrammes-de-voronoi-point-de-vue-mathematiques-et-python\/#Definitions\" >D\u00e9finitions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/02\/diagrammes-de-voronoi-point-de-vue-mathematiques-et-python\/#En_mathematiques\" >En math\u00e9matiques<\/a><\/li><\/ul><\/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\/2024\/05\/02\/diagrammes-de-voronoi-point-de-vue-mathematiques-et-python\/#Diagramme_de_Voronoi_apres_les_mathematiques_une_implementation_en_Python\" >Diagramme de Vorono\u00ef: apr\u00e8s les math\u00e9matiques, une impl\u00e9mentation en Python<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/02\/diagrammes-de-voronoi-point-de-vue-mathematiques-et-python\/#Premier_exemple\" >Premier exemple<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/02\/diagrammes-de-voronoi-point-de-vue-mathematiques-et-python\/#Deuxieme_exemple\" >Deuxi\u00e8me exemple<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/02\/diagrammes-de-voronoi-point-de-vue-mathematiques-et-python\/#Troisieme_exemple_avec_de_la_couleur_cest_plus_fun\" >Troisi\u00e8me exemple: avec de la couleur (c&rsquo;est plus fun!)<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/02\/diagrammes-de-voronoi-point-de-vue-mathematiques-et-python\/#Applications_du_diagramme_de_Voronoi\" >Applications du diagramme de Vorono\u00ef<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Diagramme_de_Voronoi_point_de_vue_mathematiques_on_verra_ensuite_pour_Python\"><\/span>Diagramme de Vorono\u00ef: point de vue math\u00e9matiques (on verra ensuite pour Python)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Consid\u00e9rons dans un rep\u00e8re orthonorm\u00e9 un syst\u00e8me <em>S<\/em> de points \\(A_k(x_k;y_k)\\). Le diagramme de Vorono\u00ef associ\u00e9 \u00e0 ce syst\u00e8me <em>S<\/em> permet de voir quels sont les points du plan les plus proches d&rsquo;un des points de <em>S<\/em>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Definitions\"><\/span>D\u00e9finitions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">L&rsquo;ensemble:$$\\text{Vor}_S(A_k) = \\{ M(x;y)\\in\\mathbb{R}^2,\\  MA_k \\leqslant MA_i,\\ \\forall i \\neq k\\}$$ est l&rsquo;ensemble des points du plan qui sont plus proches de \\(A_k\\) que des autres points de <em>S<\/em>.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Cet ensemble est appel\u00e9 <em>polygone de Vorono\u00ef<\/em>, ou <em>polygone de Thiessen<\/em>, du point \\(A_k\\). On peut aussi l&rsquo;appeler une <em>cellule<\/em>.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Le point \\(A_k\\), quant \u00e0 lui, est appel\u00e9 <em>germe<\/em> (ou <em>centre de la cellule<\/em>).<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Un <em>pavage de Vorono\u00ef<\/em> est l&rsquo;ensemble des polygones de Vorono\u00ef.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"En_mathematiques\"><\/span>En math\u00e9matiques<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Un pavage de Vorono\u00ef avec deux points n&rsquo;est pas compliqu\u00e9 \u00e0 trouver. Il s&rsquo;agit de deux demi-plans d\u00e9limit\u00e9s par la m\u00e9diatrice du segment form\u00e9 par les deux points.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Avec trois points, il suffit de construire les m\u00e9diatrices des segments form\u00e9s par ces trois points:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-1.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"221\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-1-300x221.png\" alt=\"diagramme voronoi math\u00e9matiques python\" class=\"wp-image-9737\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-1-300x221.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-1.png 554w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On l&rsquo;aura compris, il suffit au final de tracer les m\u00e9diatrices; ce sont elles qui partitionnent le plan pour donner le pavage de Vorono\u00ef.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-2.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"225\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-2-300x225.png\" alt=\"diagramme voronoi math\u00e9matiques python\" class=\"wp-image-9738\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-2-300x225.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-2.png 553w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><figcaption class=\"wp-element-caption\">Pavage de Vorono\u00ef pour un syst\u00e8me de 7 points<\/figcaption><\/figure>\n<\/div>\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Diagramme_de_Voronoi_apres_les_mathematiques_une_implementation_en_Python\"><\/span>Diagramme de Vorono\u00ef: apr\u00e8s les math\u00e9matiques, une impl\u00e9mentation en Python<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Il existe d\u00e9j\u00e0 un module qui permet de tracer de tels diagrammes: dans scipy.spatial, il y a une fonction Voronoi.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Premier_exemple\"><\/span>Premier exemple<span class=\"ez-toc-section-end\"><\/span><\/h3>\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=\"\">import matplotlib.pyplot as plt\nimport numpy as np\nfrom scipy.spatial import Voronoi, voronoi_plot_2d\n\npoints = np.random.rand(10,2) # choisit au hasard 10 points dans le carr\u00e9 unit\u00e9\nvor = Voronoi(points)\nfig = voronoi_plot_2d(vor)\nplt.show()\n<\/pre>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Ce programme choisit 10 points au hasard. Leurs coordonn\u00e9es sont comprises entre 0 et 1, puis d\u00e9termine le pavage de Vorono\u00ef. On obtient par exemple:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-3.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"227\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-3-300x227.png\" alt=\"\" class=\"wp-image-9739\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-3-300x227.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-3.png 557w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Deuxieme_exemple\"><\/span>Deuxi\u00e8me exemple<span class=\"ez-toc-section-end\"><\/span><\/h3>\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=\"\">import matplotlib.pyplot as plt\nimport numpy as np\nfrom scipy.spatial import Voronoi, voronoi_plot_2d\n\npoints = np.random.rand(10,2)\nvor = Voronoi(points)\nfig = voronoi_plot_2d(vor, show_vertices=False, line_colors='red',line_width=2, line_alpha=0.6, point_size=2)\nplt.show()\n<\/pre>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-4-300x224.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"224\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-4-300x224.png\" alt=\"\" class=\"wp-image-9740\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-4-300x224.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-4.png 567w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Note : j&rsquo;ai oubli\u00e9 de mettre \u00e0 l&rsquo;\u00e9chelle les graphiques pr\u00e9c\u00e9dents, donc il est normal que les bords des polygones ne semblent pas perpendiculaires aux segments joignant deux points du syst\u00e8me. L&rsquo;orthogonalit\u00e9 se verra mieux sur les graphiques suivants.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Troisieme_exemple_avec_de_la_couleur_cest_plus_fun\"><\/span>Troisi\u00e8me exemple: avec de la couleur (c&rsquo;est plus fun!)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"generic\" data-enlighter-theme=\"\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">import numpy as np\nfrom scipy.spatial import Voronoi, voronoi_plot_2d\nimport matplotlib.pyplot as plt\n\npoints = np.random.rand(20, 2)\nvor = Voronoi(points)\ncolors = plt.cm.tab20(np.arange(len(vor.regions)))\n\nfig, ax = plt.subplots()\nvoronoi_plot_2d(vor, ax=ax, show_vertices=False, line_colors='k', line_width=2, line_alpha=0.6)\n\n\nfor region_index, region in enumerate(vor.regions):\n    if not -1 in region and len(region) > 0:  # V\u00e9rifier si la r\u00e9gion est finie\n        polygon = [vor.vertices[i] for i in region]\n        ax.fill(*zip(*polygon), color=colors[region_index], alpha=0.3)\n\n# Ajouter les points d'origine\nax.plot(points[:, 0], points[:, 1], 'o', markersize=1)\n\n# Optionnel : ajuster les limites de la figure pour mieux voir le pavage\nax.set_xlim(0, 1)\nax.set_ylim(0, 1)\n\nplt.show()<\/pre>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-6.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"280\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-6-300x280.png\" alt=\"\" class=\"wp-image-9742\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-6-300x280.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-6.png 560w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Applications_du_diagramme_de_Voronoi\"><\/span>Applications du diagramme de Vorono\u00ef<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">La polyvalence des diagramme de Vorono\u00ef en fait un outil puissant. Dans l&rsquo;analyse et la mod\u00e9lisation de structures spatiales dans de nombreux domaines scientifiques et appliqu\u00e9s, par exemple:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Sciences g\u00e9ospatiales<\/strong> : pour diviser une r\u00e9gion en zones contigu\u00ebs en fonction de la proximit\u00e9 des points de donn\u00e9es. Cela peut \u00eatre utile dans la cartographie, la t\u00e9l\u00e9d\u00e9tection, la g\u00e9ologie, etc.<\/li>\n\n\n\n<li><strong>Planification des r\u00e9seaux<\/strong> : dans les t\u00e9l\u00e9communications, pour planifier l&#8217;emplacement des tours de t\u00e9l\u00e9phonie mobile. Cela permet de maximiser la couverture du signal et minimiser les chevauchements.<\/li>\n\n\n\n<li><strong>Mod\u00e9lisation des mouvements de particules<\/strong> : en physique et en biologie, pour mod\u00e9liser le d\u00e9placement des particules. Cela se fait dans un espace en fonction des attracteurs ou des obstacles pr\u00e9sents.<\/li>\n\n\n\n<li><strong>Traitement d&rsquo;images<\/strong> : pour segmenter les images en r\u00e9gions homog\u00e8nes en fonction des niveaux de gris ou des couleurs.<\/li>\n\n\n\n<li><strong>Optimisation<\/strong> : dans des probl\u00e8mes d&rsquo;optimisation pour diviser l&rsquo;espace en r\u00e9gions qui minimisent la distance \u00e0 un ensemble de points donn\u00e9s.<\/li>\n\n\n\n<li><strong>Mod\u00e9lisation des \u00e9cosyst\u00e8mes<\/strong> : pour mod\u00e9liser les territoires de chasse ou de recherche des animaux en fonction de leur emplacement.<\/li>\n\n\n\n<li><strong>Analyse des donn\u00e9es spatiales<\/strong> : analyser la distribution spatiale des donn\u00e9es. Cela permet d&rsquo;identifier les clusters, les zones denses ou les r\u00e9gions isol\u00e9es.<\/li>\n\n\n\n<li><strong>Conception architecturale<\/strong> : pour g\u00e9n\u00e9rer des motifs de conception spatiale. Mais aussi pour d\u00e9terminer la r\u00e9partition des espaces en fonction des besoins des utilisateurs.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Un diagramme de Vorono\u00ef (math\u00e9matiques et Python), est une repr\u00e9sentation graphique. Nous allons voir en quoi elle consiste. Pour celles et ceux qui d\u00e9sirent en savoir plus, vous pouvez consulter la page wikipedia.<\/p>\n","protected":false},"author":1,"featured_media":9745,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,5],"tags":[418,135],"class_list":["post-9735","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","category-python","tag-diagrammes","tag-matplotlib"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Diagrammes de Vorono\u00ef, point de vue math\u00e9matiques et Python - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Un diagramme de Vorono\u00ef, du point de vue math\u00e9matiques et Python, est une repr\u00e9sentation graphique. 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