Villarceau circles

They can be written in parametric form as and The slicing plane is chosen to be tangent to the torus at two points while passing through its center.

The angle of slicing is uniquely determined by the dimensions of the chosen torus.

A proof of the circles’ existence can be constructed from the fact that the slicing plane is tangent to the torus at two points.

Begin with a circle of radius r in the yz plane, centered at (0,R, 0): Sweeping this circle around the z-axis replaces y by (x2 + y2)1/2, and clearing the square root produces a quartic equation for the torus: The cross-section of the swept surface in the yz plane now includes a second circle, with equation This pair of circles has two common internal tangent lines, with slope at the origin found from the right triangle with hypotenuse R and opposite side r (which has its right angle at the point of tangency).

Thus, on these tangent lines, z/y equals ±r / (R2 − r2)1/2, and choosing the plus sign produces the equation of a plane bitangent to the torus: We can calculate the intersection of this plane with the torus analytically, and thus show that the result is a symmetric pair of circles of radius R centered at

[1] A more abstract — and more flexible — approach was described by Hirsch (2002),[2] using algebraic geometry in a projective setting.

In the homogeneous quartic equation for the torus, setting w to zero gives the intersection with the “plane at infinity”, and reduces the equation to This intersection is a double point, in fact a double point counted twice.

Hirsch extends this argument to any surface of revolution generated by a conic, and shows that intersection with a bitangent plane must produce two conics of the same type as the generator when the intersection curve is real.

The torus plays a central role in the Hopf fibration of the 3-sphere, S3, over the ordinary sphere, S2, which has circles, S1, as fibers.

[5] Mannheim (1903) showed that the Villarceau circles meet all of the parallel circular cross-sections of the torus at the same angle, a result that he said a Colonel Schoelcher had presented at a congress in 1891.