The torus and the elliptic cyclide
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The most used parameterization of the ordinary torus (the donut) is:
torusR,r(u,v)=((R+rcosv)cosu(R+rcosv)sinursinv).
The
elliptic Dupin cyclide
is a generalization of the torus. It has three nonnegative parameters
c<μ<a, and its usual
parameterization is, letting
b=√a2–c2:
cyclidea,c,μ(u,v)=(μ(c–acosucosv)+b2cosva–ccosucosvb(a–μcosu)sinva–ccosucosvb(ccosv–μ)sinua–ccosucosv).
For c=0, this is the torus.
Here is a cyclide in 3D (image taken from this post):
I think almost everything you can do with a torus, you can do it with a
cyclide. For example, a parameterization of the
(p,q)-torus knot is
torusR,r(pt,qt),0⩽t<2π.
Here is a cyclidoidal helix:
And here is a rotoid dancing around a cyclide:
I found the way to do this animation for the torus on this website, and then I adapted it to the cyclide.
The R code used to generate these animations is available in this gist.
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