{"id":648,"date":"2019-05-09T17:48:06","date_gmt":"2019-05-09T15:48:06","guid":{"rendered":"https:\/\/blogs.futura-sciences.com\/lehning\/?p=648"},"modified":"2019-05-16T13:22:54","modified_gmt":"2019-05-16T11:22:54","slug":"une-astroide-dans-un-autobus","status":"publish","type":"post","link":"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/05\/09\/une-astroide-dans-un-autobus\/","title":{"rendered":"Une astro\u00efde dans un autobus"},"content":{"rendered":"<p>La notion d\u2019enveloppe de droites recouvre deux notions en g\u00e9n\u00e9ral \u00e9quivalentes. D\u2019un c\u00f4t\u00e9, il s\u2019agit d\u2019une courbe s\u00e9parant deux domaines entre eux, de l\u2019autre une courbe tangente \u00e0 toutes les droites. La seconde se pr\u00eate mieux au calcul.<\/p>\n<h2>Exemple d&#8217;une porte d&#8217;autobus<\/h2>\n<p>Prenons un exemple concret, celui d\u2019une porte d\u2019autobus coulissante \u00e0 deux battants s\u2019ouvrant selon le sch\u00e9ma\u00a0:<\/p>\n<figure id=\"attachment_651\" aria-describedby=\"caption-attachment-651\" style=\"width: 228px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-651 size-medium\" src=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/02\/Enveloppes-Droites-01-228x300.png\" alt=\"\" width=\"228\" height=\"300\" srcset=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/02\/Enveloppes-Droites-01-228x300.png 228w, https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/02\/Enveloppes-Droites-01.png 711w\" sizes=\"auto, (max-width: 228px) 100vw, 228px\" \/><figcaption id=\"caption-attachment-651\" class=\"wp-caption-text\">Porte d\u2019autobus \u00e0 deux battants se repliant. Au sol, on obtient deux segments de droite. Celui de gauche pivote pour d\u00e9crire un quart de disque. Celui de droite est plus int\u00e9ressant \u00e0 \u00e9tudier.<\/figcaption><\/figure>\n<p>La projection sur le sol de la porte de droite (sur la figure) est un segment qui d\u00e9finit une droite coupant deux droites perpendiculaires selon un segment AB de longueur constante (celle de la porte enti\u00e8re).<\/p>\n<figure id=\"attachment_653\" aria-describedby=\"caption-attachment-653\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-653 size-medium\" src=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/02\/Enveloppes-Droites-02-300x187.png\" alt=\"\" width=\"300\" height=\"187\" srcset=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/02\/Enveloppes-Droites-02-300x187.png 300w, https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/02\/Enveloppes-Droites-02-768x480.png 768w, https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/02\/Enveloppes-Droites-02.png 1017w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-653\" class=\"wp-caption-text\">La projection de la porte sur le sol de l&#8217;autobus est constitu\u00e9e des deux segments \u00e9gaux AI et IO. Comme IO = IB, le segment AB est de longueur constante.<\/figcaption><\/figure>\n<h2>Apparition d&#8217;une enveloppe<\/h2>\n<p>Il est facile de tracer un grand nombre de segments AB en faisant varier l&#8217;angle <em>t de 0 \u00e0 90\u00b0<\/em> on voit alors appara\u00eetre une courbe en n\u00e9gatif : leur enveloppe.<\/p>\n<figure id=\"attachment_654\" aria-describedby=\"caption-attachment-654\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-654 size-medium\" src=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/02\/Enveloppes-Droites-03-300x300.png\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/02\/Enveloppes-Droites-03-300x300.png 300w, https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/02\/Enveloppes-Droites-03-150x150.png 150w, https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/02\/Enveloppes-Droites-03-768x768.png 768w, https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/02\/Enveloppes-Droites-03.png 770w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-654\" class=\"wp-caption-text\">Les segments AB restent tangents \u00e0 une m\u00eame courbe. Cette courbe est leur enveloppe.<\/figcaption><\/figure>\n<p>Si on fait varier l&#8217;angle\u00a0<em>t<\/em> de 0 \u00e0 360\u00b0, on obtient une courbe en forme d&#8217;astre, appel\u00e9e pour cela astro\u00efde.<\/p>\n<figure id=\"attachment_655\" aria-describedby=\"caption-attachment-655\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-655 size-medium\" src=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/02\/Enveloppes-Droites-04-300x295.png\" alt=\"\" width=\"300\" height=\"295\" srcset=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/02\/Enveloppes-Droites-04-300x295.png 300w, https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/02\/Enveloppes-Droites-04.png 567w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-655\" class=\"wp-caption-text\">L&#8217;astro\u00efde en entier.<\/figcaption><\/figure>\n<h2>Point de Monge<\/h2>\n<p>Gaspard Monge (1746 &#8211; 1818), l&#8217;un des cr\u00e9ateurs de l&#8217;\u00e9cole polytechnique et de l&#8217;\u00e9cole normale sup\u00e9rieure o\u00f9 il a ensuite enseign\u00e9 a trouv\u00e9 un moyen de d\u00e9crire l&#8217;enveloppe d&#8217;une famille de droites d\u00e9pendant d&#8217;un param\u00e8tre D (<em>t<\/em>) comme le lieu d&#8217;un point mobile, appel\u00e9 depuis point de Monge en son hommage, ou simplement point caract\u00e9ristique de D (<em>t<\/em>). Il se d\u00e9finit comme la limite du point d&#8217;intersection de \u00a0D (<em>t<\/em>) et\u00a0D (<i>t<\/i> + d<em>t<\/em>) quand d<em>t<\/em> tend vers z\u00e9ro, ce qui permet son calcul \u00e0 travers la notion de d\u00e9riv\u00e9e : le point de Monge est \u00e0 l&#8217;intersection de\u00a0\u00a0D (<em>t<\/em>) et\u00a0\u00a0D&#8217; (<em>t<\/em>) qui s&#8217;obtient en d\u00e9rivant l&#8217;\u00e9quation de\u00a0\u00a0D (<em>t<\/em>) par rapport \u00e0\u00a0 <em>t<\/em>.<\/p>\n<p>Pour les f\u00e9rus de calculs, si\u00a0<em>a<\/em> est le longueur de la porte, l&#8217;\u00e9quation de D (<em>t<\/em>) est \u00a0<em>x<\/em>\u00a0sin\u00a0<em>t<\/em>\u00a0+\u00a0<em>y<\/em>\u00a0cos\u00a0<em>t<\/em> = <em>a <\/em>cos\u00a0<em>t<\/em>\u00a0sin\u00a0<em>t <\/em>donc les coordonn\u00e9es du point de Monge est solution du syst\u00e8me :<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-659 aligncenter\" src=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/03\/equation-01.jpg\" alt=\"\" width=\"215\" height=\"46\" \/><\/p>\n<p>On en d\u00e9duit ses coordonn\u00e9es :<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-660 aligncenter\" src=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/03\/equation-02.jpg\" alt=\"\" width=\"58\" height=\"38\" \/><\/p>\n<p>ce qui permet de tracer l&#8217;enveloppe de la famille de droites D.<\/p>\n<!-- AddThis Advanced Settings generic via filter on the_content --><!-- AddThis Share Buttons generic via filter on the_content -->","protected":false},"excerpt":{"rendered":"<p>La notion d\u2019enveloppe de droites recouvre deux notions en g\u00e9n\u00e9ral \u00e9quivalentes. D\u2019un c\u00f4t\u00e9, il s\u2019agit d\u2019une courbe s\u00e9parant deux domaines entre eux, de l\u2019autre une courbe tangente \u00e0 toutes les droites. La seconde se pr\u00eate mieux au calcul. Exemple d&#8217;une porte d&#8217;autobus Prenons un exemple concret, celui d\u2019une porte d\u2019autobus coulissante \u00e0 deux battants s\u2019ouvrant &hellip; <a href=\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/05\/09\/une-astroide-dans-un-autobus\/\" class=\"more-link\">Continuer la lecture de <span class=\"screen-reader-text\">Une astro\u00efde dans un autobus<\/span>  <span class=\"meta-nav\">&rarr;<\/span><\/a><!-- AddThis Advanced Settings generic via filter on get_the_excerpt --><!-- AddThis Share Buttons generic via filter on get_the_excerpt --><\/p>\n","protected":false},"author":12,"featured_media":650,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[29,4,98],"tags":[244,245,243,247,246],"class_list":["post-648","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths-2","category-regard-sur-le-monde","category-viequotidienne","tag-astroide","tag-autobus","tag-enveloppe-de-droites","tag-lieu","tag-monge"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v24.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Une astro\u00efde dans un autobus, par Herv\u00e9 Lehning<\/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:\/\/blogs.futura-sciences.com\/lehning\/2019\/05\/09\/une-astroide-dans-un-autobus\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Une astro\u00efde dans un autobus, par Herv\u00e9 Lehning\" \/>\n<meta property=\"og:description\" content=\"La notion d\u2019enveloppe de droites recouvre deux notions en g\u00e9n\u00e9ral \u00e9quivalentes. 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