 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-2/25i/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

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.

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