{"id":378825,"date":"2023-10-01T18:00:00","date_gmt":"2023-10-02T00:00:00","guid":{"rendered":"https:\/\/laustep.github.io\/stlahblog\/posts\/torusAndCyclide.html"},"modified":"2023-10-01T18:00:00","modified_gmt":"2023-10-02T00:00:00","slug":"the-torus-and-the-elliptic-cyclide","status":"publish","type":"post","link":"https:\/\/www.r-bloggers.com\/2023\/10\/the-torus-and-the-elliptic-cyclide\/","title":{"rendered":"The torus and the elliptic cyclide"},"content":{"rendered":"\r\n\r\n
\r\n[This article was first published on Saturn Elephant<\/a><\/strong>, and kindly contributed to R-bloggers<\/a>]. (You can report issue about the content on this page here<\/a>)\r\n
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\n The most used parameterization of the ordinary torus (the donut) is:\n \\[ \\textrm{torus}_{R,r}(u, v) = \\begin{pmatrix} (R + r \\cos v) \\cos u\n \\\\ (R + r \\cos v) \\sin u \\\\ r \\sin v \\end{pmatrix}. \\]<\/span\n >\n <\/p>\n

\n The\n elliptic Dupin cyclide<\/a\n >\n is a generalization of the torus. It has three nonnegative parameters\n \\(c < \\mu < a\\)<\/span>, and its usual\n parameterization is, letting\n \\(b = \\sqrt{a^2 – c^2}\\)<\/span>:\n \\[ \\textrm{cyclide}_{a, c, \\mu}(u, v) = \\begin{pmatrix} \\dfrac{\\mu (c\n – a \\cos u \\cos v) + b^2 \\cos v}{a – c \\cos u \\cos v} \\\\ \\dfrac{b (a –\n \\mu \\cos u) \\sin v}{a – c \\cos u \\cos v} \\\\ \\dfrac{b (c \\cos v – \\mu)\n \\sin u}{a – c \\cos u \\cos v} \\end{pmatrix}. \\]<\/span\n >\n The picture below shows such a cyclide in its symmetry plane\n \\(\\{z = 0\\}\\)<\/span>:\n <\/p>\n

\n \n <\/p>\n

For \\(c=0\\)<\/span>, this is the torus.<\/p>\n

\n Here is a cyclide in 3D (image taken from\n this post<\/a\n >):\n <\/p>\n

\n \n <\/p>\n

\n I think almost everything you can do with a torus, you can do it with a\n cyclide. For example, a parameterization of the\n \\((p,q)\\)<\/span>-torus knot is\n \\[ \\textrm{torus}_{R, r}(pt, qt), \\qquad 0 \\leqslant t < 2\\pi.\n \\]<\/span\n >\n Then, the\n \\((p,q)\\)<\/span>-cyclide knot<\/em> is\n parameterized by\n \\[ \\textrm{cyclide}_{a, c, \\mu}(pt, qt), \\qquad 0 \\leqslant t <\n 2\\pi. \\]<\/span\n >\n \n <\/p>\n

Here is a cyclidoidal helix<\/em>:<\/p>\n

\n \n <\/p>\n

And here is a rotoid dancing around a cyclide:<\/p>\n

\n \n <\/p>\n

\n I found the way to do this animation for the torus on\n this website<\/a>, and then I adapted\n it to the cyclide.\n <\/p>\n

\n The R code used to generate these animations is available in\n this gist<\/a\n >.\n <\/p>\n <\/div>\n <\/div>\n<\/div>\n\n

\r\n
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The most used parameterization of the ordinary torus (the donut) is:
\n \\[ \\textrm{torus}_{R,r}(u, v) = \\begin{pmatrix} (R + r \\cos v) \\cos u
\n \\\\ (R + r \\cos v) \\sin u \\\\ r \\sin v \\end{pmatrix}. \\]<\/p>\n

…<\/p>\n","protected":false},"author":2532,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[4],"tags":[],"aioseo_notices":[],"jetpack-related-posts":[],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/posts\/378825"}],"collection":[{"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/users\/2532"}],"replies":[{"embeddable":true,"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/comments?post=378825"}],"version-history":[{"count":1,"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/posts\/378825\/revisions"}],"predecessor-version":[{"id":378826,"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/posts\/378825\/revisions\/378826"}],"wp:attachment":[{"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/media?parent=378825"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/categories?post=378825"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/tags?post=378825"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}