{"id":2914,"date":"2023-12-29T23:26:42","date_gmt":"2023-12-29T22:26:42","guid":{"rendered":"https:\/\/eva.louis-le-grand.net\/maths\/?p=2914"},"modified":"2024-01-16T18:37:01","modified_gmt":"2024-01-16T17:37:01","slug":"gateau-de-villarceau","status":"publish","type":"post","link":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/2023\/12\/29\/gateau-de-villarceau\/","title":{"rendered":"G\u00e2teau de Villarceau"},"content":{"rendered":"\n

Par 1 point du tore, il passe 4 cercles.<\/h2>\n\n\n\n

Comme l’illustrent ces gratins de spaghettis toriques, par un point du tore, il passe 2 cercles, un ‘horizontal’ en rose et un ‘vertical’ en bleu, et deux cercles de Villarceau<\/a> en vert et son sym\u00e9trique.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\n\n\n\n

On peut aussi obtenir une courbe en forme de 8<\/a><\/p>\n\n\n\n

<\/p>\n\n\n\n

 
\"Rendered<\/p>\n<\/code><\/pre>\n\n\n\n
Asymptote:<\/p>\n\n\n\n
\"\begin{asymptote}<\/code><\/pre>\n\n\n\n
Autre exemple:<\/p>\n\n\n\n
\"Rendered<\/p><\/code><\/pre>\n\n\n\n
Avec Asymptote:<\/p>\n\n\n\n
\\begin{asy}\n\nsettings.render = 8;\nsettings.prc = false;\n\nimport graph3;\nimport contour;\nsize3(8cm);\n\ncurrentprojection = orthographic(10,1,4);\ndefaultrender = render(merge = true);\n\n\/\/ create torus as surface of rotation\nint umax = 40;\nint vmax = 40;\nsurface torus = surface(Circle(c=2Y, r=0.6, normal=X, n=vmax), c=O, axis=Z, n=umax);\ntorus.ucyclic(true);\ntorus.vcyclic(true);\n\npen meshpen = 0.3pt + gray;\n\ndraw(torus, surfacepen=material(diffusepen=white+opacity(0.6), emissivepen=white));\nfor (int u = 0; u < umax; ++u)\n  draw(torus.uequals(u), p=meshpen);\nfor (int v = 0; v < vmax; ++v)\n  draw(graph(new triple(real u) {return torus.point(u,v); }, 0, umax, operator ..),\n       p=meshpen);\n\npair a = (floor(umax\/2) + 2, 3);\ndot(torus.point(a.x, a.y), L=\"\"a\"\", align=W);\npair b = (5, floor(vmax\/2));\ndot(torus.point(b.x, b.y), L=\"\"b\"\", align=2Z + X);\n\npath3 abpath(int ucycles, int vcycles) {\n  pair bshift = (ucycles*umax, vcycles*vmax);\n  triple f(real t) {\n    pair uv = (1-t)*a + t*(b+bshift);\n    return torus.point(uv.x, uv.y);\n  }\n  return graph(f, 0, 1, operator ..);\n}\n\nreal linewidth = 0.8pt;\n\ndraw(abpath(0,0), p=linewidth + orange);\ndraw(abpath(1,0), p=linewidth + red);\ndraw(abpath(1,-1), p=linewidth + darkgreen);\n\\end{asy}<\/code><\/pre>\n\n\n\n
Dessiner tore et geodesic en latex:<\/p>\n\n\n\n
https:\/\/tex.stackexchange.com\/questions\/156498\/drawing-torus-points-and-arches-with-coordinates-read-from-a-file<\/a><\/p>\n\n\n\n
https:\/\/tex.stackexchange.com\/questions\/149946\/how-can-i-visualize-a-torus-with-three-paths\/151031#151031<\/a><\/p>\n\n\n\n
https:\/\/tex.stackexchange.com\/questions\/70090\/3d-helix-torus-with-hidden-lines?rq=1<\/a><\/p>\n\n\n\n
<\/p>\n\n\n\n
\n\n\n\n
Cercles de Villarceau<\/h2>\n\n\n\n
\"\begin{texgraph}[name=villarceau, file]<\/code><\/pre>\n\n\n\n
Les cercles en bleu et rouge sur un petit mod\u00e8le en pl\u00e2tre.<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
D\u00e9monstration<\/h2>\n\n\n\n
Il y a deux moyens de montrer l’existence des cercles de Villarceau une fois bien appr\u00e9hend\u00e9s par le g\u00e2teau.<\/p>\n\n\n\n
Soit avec des \u00e9quations param\u00e9triques.<\/p>\n\n\n\n
Soit avec la g\u00e9om\u00e9trie projective en 4 dimensions.<\/p>\n\n\n\n
Equations param\u00e9triques<\/h2>\n\n\n\n
…<\/p>\n\n\n\n
G\u00e9om\u00e9trie projective en 4 dimensions.<\/h2>\n\n\n\n
…<\/p>\n","protected":false},"excerpt":{"rendered":"
Par 1 point du tore, il passe combien de cercles?<\/p>\n","protected":false},"author":1,"featured_media":2922,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-2914","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-non-classe"],"_links":{"self":[{"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/posts\/2914","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/comments?post=2914"}],"version-history":[{"count":7,"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/posts\/2914\/revisions"}],"predecessor-version":[{"id":3371,"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/posts\/2914\/revisions\/3371"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/media\/2922"}],"wp:attachment":[{"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/media?parent=2914"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/categories?post=2914"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/tags?post=2914"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}