Making a cross section with an implicit equation is easy to do, simply set any one or more variables to zero. But, how would I do this using a parametric equation? Say we have a torus:
$ x = (\cos(u) +2)\cos(v) $
$ y = (\cos(u) +2)\sin(v) $
$ z = \sin(u) $
Or, more specifically, a rotating torus:
$ x = (\cos(u) +2)\cos(v)\cos(a) - \sin(u)\sin(a) $
$ y = (\cos(u) +2)\sin(v) $
$ z = (\cos(u) +2)\cos(v)\sin(a) + \sin(u)\cos(a) $
$$ 0 < u,v < 2\pi $$ vary 'a' from 0 to 2pi to rotate on plane xz
Is there a way to algebraically modify this (without converting to implicit), to make a 2D slice on plane xy? A projection is easy to do, simply set z=0. But how about a cross section? It seems like if we're defining a 1D surface (as the 2D slice) , the parametric equation would be just a function of $u$ . I've already naively tried to set $v=0$ , or $u=v$ , both of which didn't work. So, it's some other process I'm not familiar with.