 Solution

ID: past-exam-of-the-geometry-course-of-the-university-of-oxford/2010/a/2/iii/solution

Differentiating the arc-length identity gives

f^{'} f^{''} + h^{'} h^{''} = 0

. Multiply by

f^{'}

and use

(f^{'})^{2} + (h^{'})^{2} = 1

to obtain

(f^{'} h^{''} - f^{''} h^{'}) h^{'} = f^{'} h^{'} h^{''} - f^{''} (h^{'})^{2} = - f^{''} ((f^{'})^{2} + (h^{'})^{2}) = - f^{''} .

This derivation does not divide by

h^{'}

, so it remains valid where

h^{'} = 0

K = \frac{( f ^{'} h ^{''} - f ^{''} h ^{'} ) f h ^{'}}{f ^{2}} = - \frac{f ^{''}}{f} .

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