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

(1)

In terms of the old coordinates, the new coordinates are

(2)

(3)

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

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 132-133, 1969.Kabai,
S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica.
Püspökladány, Hungary: Uniconstant, p. 125, 2002.Melzak,
Z. A. Invitation
to Geometry. New York: Wiley, pp. 63-72, 1983.Schmidt,
H. Die
Inversion und ihre Anwendungen. Munich, Germany: Oldenbourg, 1950.Villarceau,
M. "Théorème sur le tore." Nouv. Ann. Math.7,
345-347, 1848.