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 Incoming links: Parallel of a surface of revolution

I = f (v)^{2} d u^{2} + q (v)^{2} d v^{2}, q (v)^{2} = f^{'} (v)^{2} + g^{'} (v)^{2} .

The geodesic Lagrangian is consequently

L = \frac{1}{2} (f (v)^{2} u^{'2} + q (v)^{2} v^{'2}) .

u

\frac{d}{d s} \frac{\partial L}{\partial u ^{'}} = \frac{d}{d s} (f (v)^{2} u^{'}) = 0.

Thus

f (v (s))^{2} u^{'} (s)

is constant.

The positively oriented unit tangent to the parallel of a surface of revolution through

γ (s)

e_{u} = ϕ_{u} / f (v)

. Since

γ

has unit speed and

θ

is its angle with that parallel,

cos θ (s) = ⟨ γ^{'} (s), e_{u} ⟩ = f (v (s)) u^{'} (s) .

f (v (s))

yields

f (v (s)) cos θ (s) = f (v (s))^{2} u^{'} (s) = constant,