Ring torus

Definition

The ring torus is a form of embedding of the torus in three-dimensional Euclidean space. This surface type is not unique up to isometry or even up to similarity transformations, but rather, depends on two parameters for a description up to isometry and on one parameter for a description up to similarity transformations.

Implicit and parametric descriptions

Degree of generality	Implicit description	What the parameters mean	Parametric description	What the additional parameters mean	Comment
Arbitrary	Fill this in later
Up to rigid motions (rotations, translations, reflections)	$({\sqrt {x^{2}+y^{2}}}-c)^{2}+z^{2}=a^{2}$	$c$ is the radius of the central circle (spine) of the ring torus, and $a$ is the tube radius of the ring torus. This describes the ring torus where the axis of revolution is the $z$-axis.	$x=(c+a\cos v)\cos u,y=(c+a\cos v)\sin u,z=a\sin v$	$u$ is an angle giving local polar coordinates for the point any fixed location of the circle being rotated. $v$ is the angle giving polar coordinates for the center of the circle, on the spine circle.
Up to similarity transformations	We could rescale the above to normalize either one of $c$ and $a$ to 1, but we cannot normalize both simultaneously.