Villarceau Circles

Four circles may be drawn through an arbitrary point on a torus. The first two circles are obvious: one is in the plane of the torus and the second perpendicular to it. The third and fourth circles (which are inclined with respect to the torus) are much more unexpected and are known as the Villarceau circles (Villarceau 1848, Schmidt 1950, Coxeter 1969, Melzak 1983).

To see that two additional circles exist, consider a coordinate system with origin at the center of torus, with pointing up. Specify the position of by its angle measured around the tube of the torus. Define for the circle of points farthest away from the center of the torus (i.e., the points with ), and draw the x-axis as the intersection of a plane through the z-axis and passing through with the -plane. Rotate about the y-axis by an angle , where

In terms of the old coordinates, the new coordinates are

So in coordinates, equation (◇) of the torus becomes

Expanding the left side gives

But

so

In the plane, plugging in (◇) and factoring gives

This gives the circles

and

in the plane. Written in matrix form with parameter , these are

In the original coordinates,

The point must satisfy

so

Plugging this in for and gives the angle by which the circle must be rotated about the z-axis in order to make it pass through ,

The four circles passing through are therefore