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Surface with normal lines passing through a fixed point


Prove that if all the normal lines to a regular surface S pass through a fixed point, then the surface is a portion of the sphere.

Proof: Denote by x: U \subset \Bbb{R}^2 \to S a regular parametrization for a neighborhood of a point M \in S. Then there exists a point P such that x(u,v)+\alpha(u,v)n(u,v)=P \forall (u,v) \in U. Differentiating with respect to u we get
x_u+\alpha_u\cdot n+\alpha\cdot n_u=0. Taking the scalar product with n, and using x_u \perp n and n\perp n_u, we have \alpha_u=0. Similarly \alpha_v=0. Therefore \alpha is constant and x(u,v) is at constant distance \alpha of P. Doing this for all points of S, we see that S is contained in the sphere centered in P with radius \alpha.

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