Given a surface $S$ in $\mathbb{R}^3$, we can calculate the area of its projection $S'$ onto a given plane $P$ using the formula $[S'] = \iint_{S} \cos\beta \space dA$, where $\beta$ is the angle between the surface and $P$ at a given point on $S$.
Is there a corresponding formula for the perimeter of a surface projected onto a plane? It seems like it would be messier, since not all lengths are scaled equally under projection.