Let the curve r(s) be parametrized by arc length and have kappa(s) > 0 and tau(s

Question

Let the curve r(s) be parametrized by arc length and have kappa(s) > 0 and tau(s) 0. Suppose that the curve lies on the sphere with centre c and radius R. Prove that r(s) - c = -rho(s) (s) - rho?(s) sigma (s) Bhat(s) where rho(s) = 1 / kappa(s) and sigma (s) = 1 / tau(s). In particular R2 = rho(s)2 + p?(s)2 sigma 2. Prove that, conversely, if rho(s)2 + rho?(s)2 sigma (s)2 is a constant and p?(s) 0, then r(s) lies on a sphere. Hint: Prove that r(s) + rho(s) (s) + p?(s) sigma (s) Bhat(s) is a constant.

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