Polygonal Tori

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This Demonstration shows a torus with a polygonal cross section.


To get a better view of its inside, open up the meridional and longitudinal gaps.

The Serret–Frenet formulas (using Mathematica's built-in FrenetSerretSystem) are used to generate the parametric equation of the torus surface.


Contributed by: Erik Mahieu (July 2015)
Open content licensed under CC BY-NC-SA



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:




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