Let $S\subseteq\mathbb{R}^3$ be a smooth surface, no boundary (not necessarily compact). For any $r\in\mathbb{R}^3\backslash S$, define a smooth function $h_r:S\to\mathbb{R},q\mapsto |q-r|$
Let $p\in S$ be a critical point of $h_r$, then $\vec{pr}\perp T_p S$. So $p$ is non-degenerated if $\frac1{|p-r|}\neq k_1,k_2$. $k_1,k_2$ are two principal curvatures ($\vec{pr}$ be the normal direction at $p$, $k_{1,2}$ need not to be different).
Call $h_r$ a Morse function, if every critical point $p\in S$ of $h_r$ is non-degenerated.
Show that $\{r\in\mathbb{R}^3\backslash S|h_r$ is Morse funtion$\}$ is a open and dense set in $\mathbb{R}^3$.
I find little useful information of the set, possibly its complement is not a discrete set.