Suppose we have a smooth curve $C$ lying in $\mathbb{R}^2$, and let us consider the orthogonal projection function $P_C(x)$ onto the curve, described by $$P_C(x) = \arg\min_{y \in C} \Vert x - y \Vert$$ where $\Vert \cdot \Vert$ is a norm, it can be $\Vert \cdot \Vert_2^2$, or $\Vert \cdot \Vert_1$.
My question is: is there a general relationship between the second derivative of $P_C(x)$ and the curvature of the curve $C$? For example, relationship between the norm, whether it is "positive definite", etc. If no, under what restrictions on the curve $C$ and/or location of $x$ can we say something about their relationships? Does there exist work that discusses this problem or some problems related to it?
To visualize the problem somewhat, we consider the picture below:
Denoting the blue curve as $C_1$ and black curve as $C_2$, $C_1$ clearly has greater curvature than $C_2$, but what about $\Vert D^2P_{C_1}(x) \Vert$ vs. $\Vert D^2P_{C_2}(x) \Vert$?
Yes, there is. You can find it in Gilbarg and Trudinger in the appendix: Boundary curvatures and the Distance Function. The authors deal with the case in which you have an open set $\Omega\subset \mathbb{R}^n$ and the boundary $\partial\Omega$ is a manifold of class $C^2$. They characterize the second derivative of the distance function $d(x)=\text{dist}(x,\partial\Omega)$ at a point $x_0$ close to the boundary in terms of the principal curvatures of $\partial\Omega$ at $y_0\in\partial \Omega$ where $d(x_0)=\vert x_0-y_0\vert$. They prove that up to a change of coordinate axes you can write $$D^2 d(x_0)=\text{diag}[\frac{-\kappa_1}{1-\kappa_1d(x_0)},\cdots,\frac{-\kappa_{n-1}}{1-\kappa_{n-1}d(x_0)},0].$$ If you write the boundary $\partial\Omega$ near $y_0$ as the hypersurface $x_n=\varphi(x')$, where $x'=(x_1,\ldots,x_{n-1})$, then the principal curvatures $\kappa_i$ at $y_0$ are defined as the eigenvalues of $D^2_{x'} \varphi(y_0')$, where $y_0=(y_0',y_{0n})$. The change of coordinates is such that the $x_n$ coordinates axis lies in the direction of the normal to $\partial\Omega$ at $y_0$ and the other $x_i$ coordinates axes lie in the direction of the eigenvalues of $\kappa_i$.
If $n=2$ then $D^2_{x'} \varphi(y_0')=\varphi''(y_0')=\kappa_1$, which is proportional to the signed curvature to the curve $\frac{\varphi''(y_0')}{(1+(\varphi''(y_0'))^2)^{3/2}}$.