 Curvatures of a parametrized surface of revolution

ID: curvatures-of-a-parametrized-surface-of-revolution

Put

q = (f^{'2} + g^{'2})^{1/2}

for the profile parametrization

X (u, v) = (f (v) cos u, f (v) sin u, g (v)) .

N = (- g^{'} cos u, - g^{'} sin u, f^{'}) / q

k_{1} = \frac{g ^{'}}{f q}, k_{2} = \frac{f ^{'} g ^{''} - g ^{'} f ^{''}}{q ^{3}} .

Consequently

H = (k_{1} + k_{2}) /2

and

K = k_{1} k_{2}

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