Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 225 / 3 / a / ii

Table of contents
- Solution ii

Solution

For these increasing warps, the square-root velocity function is

Q (h) (t) = h^{'} (t)

. The chain rule gives

Q (h_{i} \circ γ) (t) = Q (h_{i}) (γ (t)) γ^{'} (t) .

Therefore

∥ Q (h_{i} \circ γ) - Q (h_{j} \circ γ) ∥_{2}^{2} = \int_{0}^{1} ∣ Q (h_{i}) (γ (t)) - Q (h_{j}) (γ (t)) ∣^{2} γ^{'} (t) d t = \int_{0}^{1} ∣ Q (h_{i}) (u) - Q (h_{j}) (u) ∣^{2} d u,

where the last equality again uses the change of variables formula. Taking square roots proves invariance under common right composition by

γ

.

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