Past exam of the geometry course of the University of Oxford / 2011 / 3 / c / i

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- Solution i

Solution

Differentiate to get

r_{u} = (a cos v, a sin v, 0), r_{v} = (- a u sin v, a u cos v, 1) .

Their squared lengths are

a^{2}

and

a^{2} u^{2} + 1

, and their inner product is zero. Hence the helicoid has first fundamental form

I = a^{2} d u^{2} + (a^{2} u^{2} + 1) d v^{2} .

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