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Elliptical Torus

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.

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DETAILS

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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