The torus and the elliptic cyclide

The most used parameterization of the ordinary torus (the donut) is:

\[ \textrm{torus}_{R,r}(u, v) = \begin{pmatrix} (R + r \cos v) \cos u

\\ (R + r \cos v) \sin u \\ r \sin v \end{pmatrix}. \]

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