Elliptical Torus

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This Demonstration shows a torus for which both the tube and the cross section of the tube are elliptical.


To get a better view of its structure, you can open the torus with gaps in either the meridional or longitudinal directions.

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


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



The parametric equation of an elliptical ring torus with an elliptical cross section is obtained as follows.

The ring ellipse has parametric equation in the - plane, where and are the semimajor and semiminor axes.

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


This gives the parametric equation of the elliptical torus:


where and are the semimajor and semiminor axes of the cross-sectional ellipse.

Expanded, this becomes:




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