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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 $\alpha$.