What curves have a closed-form formula for projecting a point onto them in multiple dimensions?

Question

What curves have a closed-form formula for projecting a point onto them in multiple dimensions? For example, give a simple, straight line $$ c(t) = v t $$ where $v\in\mathbb{R}^m$ and $c:\mathbb{R}\rightarrow \mathbb{R}^m$ and a point $p\in\mathbb{R}^m$, the orthogonal projection of $p$ onto the curve $c$ is $$ \frac{v^Tp}{v^Tv}v $$ or $c(t)$ where $t=\frac{v^Tp}{v^Tv}$. Basically, I'd like a more complicated curve, but I'd like the projection to be something closed-form. Better than that, I'd like it if the formula for the projection is not just closed-form, but differentiable.

Answer 1

This means that you would like the following equation

$F(t)=(P-C(t)) \cdot C^{'}(t)=0$

to have a close-form solution for its roots.

When $C(t)$ is a polynomial of degree 1 and 2, $F(t)$ is a polynomial of degree 1 and 3, which will have a close-form solution for its root(s).

When $C(t)$ is a circle (partial or not), I think there is also a close-form solutoion for the projected point.