Villarceau circles

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Villarceau circles are two special circles that appear when you cut a donut-shaped surface (a torus) with a plane that passes through the torus’s center and just touches it at two opposite points. Along with the usual latitude-like circles (in planes parallel to the torus’s equatorial plane) and longitude-like circles (in planes perpendicular to that), the Villarceau circles complete the set of circles you can get through a given point on the torus. In fact, through any point on the torus there are four circles: a latitude circle, a longitude circle, and the two Villarceau circles from the oblique cut.

These circles come in two families, each consisting of disjoint circles that together cover every point of the torus exactly once. If you rotate the bitangent cutting plane around the torus’s axis, you generate all the Villarceau circles. They are named after Yvon Villarceau, a French mathematician who described them in 1848.

Example: consider a horizontal torus with major radius 5 and minor radius 3. The torus is made by rotating a circle of radius 3 whose center moves on a circle of radius 5 in the xy-plane. In the plane z = 0 you see two concentric circles: the outer one with x^2 + y^2 = 64 and the inner one with x^2 + y^2 = 4. In the plane x = 0 you see two side-by-side circles: (y − 5)^2 + z^2 = 9 and (y + 5)^2 + z^2 = 9.

Two Villarceau circles can be produced by slicing with the plane 3y = 4z. They are centered at (3, 0, 0) and (−3, 0, 0), and each has radius 5. A convenient parametric form for these two circles is:
(x, y, z) = (±3 + 5 cos t, 4 sin t, 3 sin t), for t from 0 to 2π.
The cutting plane is tangent to the torus at two opposite points, (0, 16/5, 12/5) and (0, −16/5, −12/5). Rotating this plane around the z-axis yields all Villarceau circles for this torus.

In short, Villarceau circles are a pair of equal circles that arise from a special oblique cut through a torus, forming two complementary families that together cover the surface.