Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 25I / c

Table of contents
- Solution c

Solution

For

R > r > 0

, the closed curve

γ (t) = ((R + r cos 2 t) cos t, (R + r cos 2 t) sin t, r sin 2 t)

is embedded on a torus and satisfies

γ (t + π) = Rot_{z} (π) γ (t) .

After arc-length parametrization, it meets all the requirements of part (b). Its bending and twisting vary around the two waves, so neither curvature nor torsion is constant.

Figure 1.
A symmetric closed curve and its reflections
. Details.

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