Umbilic Torus

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The umbilic torus appears in catastrophe theory in the context of classifying singularities. A clipping plan is included to demonstrate that the cross section is a deltoid curve.

Show cross section:

This umbilic torus is defined parametrically by

$\begin{align} x &= \left[ 7 + \cos \left( \frac{u}{3} - 2v \right) + 2 \cos \left( \frac{u}{3} + v \right) \right] \sin u \\ y &= \left[ 7 + \cos \left( \frac{u}{3} - 2v \right) + 2 \cos \left( \frac{u}{3} + v \right) \right] \cos u \\ z &= \sin \left( \frac{u}{3} - 2v \right) + 2 \sin \left( \frac{u}{3} + v \right) \end{align}$

with $-\pi \le u \le \pi$ and $-\pi \le v \le \pi$ .

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