Details

The parametric equation of a circular torus with a polygonal cross section is obtained as follows.

The parametric of the ring circle with radius is:

.

The polar equation of a regular -gon with circumradius is:

.

Using the Mathematica function FrenetSerretSystem, we define the normal and binormal vectors and :

.

This gives the parametric equation of the elliptical torus:

,

with vertices and the circumradius of the cross-sectional polygon.

Expanded, with , this becomes:

,

,

.