I would like to know if there is a method, and what is called the area of mathematics that studies this kind of things, for projecting surfaces on other surfaces.
Example: suppose to have a half-sphere (3d) centered in $(1,1,1)$. I want to find its projection on the plane $z=0$ as seen from the direction defined by the vector $(1,0,1)$ (the line $z=x$ on the $y=0$). How is then given the parametric formula of the surface "projected"? Basically this is how to find the shape of the shadow of an umbrella, and I would like to generalize it.
Best Answer
If your surface is represented as $S = S(u,v)$ given a direction $\vec n$
the projection is given as
$$ S_{\vec n}(u,v) = S(u,v)-\langle S(u,v), \vec n \rangle \frac{\vec n}{||\vec n||^2} $$
Here $<\cdot,\cdot>$ represents the scalar product of two vectors.
Here
$$ S(u,v) = \{\cos (v) (\cos (u)+3),\sin (v) (\cos (u)+3),\sin (u)\}\\ \vec n = (1,1,1) $$
NOTE
Regarding the slanted projection with direction $\vec n$ in a plane $\Pi\to \langle p-p_0,\vec a \rangle = 0$
this projection is generated by the intersection of each line associated to the point $S(u^*,v^*)$
$$ L\to p = S(u^*,v^*)+\lambda \vec n $$
with the plane $\Pi$ so the determination of
$$ \lambda^* = -\frac{\langle S_*-p_0,\vec a\rangle}{\langle \vec a, \vec n\rangle} $$
gives the projection image over $\Pi$ hence
$$ S_{\Pi} = S(u,v)-\left(\frac{\langle S(u,v)-p_0,\vec a\rangle}{\langle \vec a, \vec n\rangle}\right)\vec n $$
so choosing a projection plane as
$$ \Pi\to\langle p-p_0,\vec a \rangle = 0 $$
with
$$ p_0 = -(5,5,5)\\ \vec a = (0,0,1)\\ \vec n = (1,1,1) $$
we have