Villarceau Circles

Villarceau circles are the oblique circles lying on a ring torus (Villarceau 1848, Schmidt 1950, Coxeter 1969, Melzak 1983). Exactly four distinct circles lying entirely on a ring torus pass through each point . Two are familiar: the circle in a plane parallel to the equatorial plane and the circle in a plane through the axis of the torus. The other two are Villarceau circles.

The assertion that there are exactly four, rather than merely at least four, was proved by Blum (1980). Takeuchi (2000, Proposition 4.2(i), p. 128) states this result explicitly and attributes it to Blum. In the notation below, the ring torus condition is .

Sym (2009, p. 1) discusses the historical connection with Darboux and states that a complete sufficiently smooth surface with exactly four circles through each point must be a standard torus, citing Takeuchi (2000). This characterization is incorrect as stated. Inversion of a ring torus about an inversion center off its surface gives a compact smooth ring cyclide and preserves circles and their incidence (De Comité 2015). Applying the inverse inversion shows that there are still exactly four circles through each point. Such a ring cyclide need not be a surface of revolution, so the four-circle property does not characterize a standard torus.

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, the equation of the torus becomes

Expanding the left side gives

But

so

In the plane, plugging in (1) 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