Spiric section

[1] A spiric section is sometimes defined as the curve of intersection of a torus and a plane parallel to its rotational symmetry axis.

The name spiric is due to the ancient notation spira of a torus.,[2][3] Start with the usual equation for the torus: Interchanging y and z so that the axis of revolution is now on the xy-plane, and setting z=c to find the curve of intersection gives In this formula, the torus is formed by rotating a circle of radius a with its center following another circle of radius b (not necessarily larger than a, self-intersection is permitted).

Expanding the equation gives the form seen in the definition where In polar coordinates this becomes or Spiric sections on a spindle torus, whose planes intersect the spindle (inner part), consist of an outer and an inner curve (s. picture).

Examples include the hippopede and the Cassini oval and their relatives, such as the lemniscate of Bernoulli.

The Cassini oval has the remarkable property that the product of distances to two foci are constant.