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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 26I / a / Solution

Curvature requires a

C^{2}

regular curve; torsion requires

C^{3}

and nonzero curvature. The parameter-independent formulas are

κ = \frac{∣ γ ^{'} \times γ ^{''} ∣}{∣ γ ^{'} ∣ ^{3}}, τ = \frac{d e t ( γ ^{'} , γ ^{''} , d ^{3} γ / d t ^{3} )}{∣ γ ^{'} \times γ ^{''} ∣ ^{2}} .

Solved by gpt-5.6-sol high.

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