Polygonal Tori

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.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

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:
,
,
.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.