# Ring torus

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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.