you slut.
I'm not even sure what a 4-dimensional torus would be, but wikipedia tells me that the topological equivalent of a torus in 4D is: (cos(a),sin(a),cos(b),sin(b)) (=V) (parametrically).
So projecting from the point (0,0,0,1) (=A), to a point on the torus described by a,b, to the hyperplane (x,y,z,0), we have the equation (V-A)*t+A=(x,y,z,0). Only the fourth coordinate helps us find t, so we get:
(sin(b)-1)t+1=0, t=-1/(sin(b)-1).
-1/(sin(b)-1)=-(sin(b)+1)/(sin^2(b)-1)=(sin(b)+1)/cos^2(b)
So then plugging t back in (and dropping the fourth component because it's zero with that t):
x=cos(a)*(sin(b)+1)/cos(b)^2
y=sin(a)*(sin(b)+1)/cos(b)^2
z=(sin(b)+1)/cos(b)
I was going to try to do this with mathematica, but... "Solve::nsmet: This system cannot be solved with the methods available to Solve."
whoop de doo...
Anyways, long story short it turns out that you have to have cos(b)^2=(z/x)^2+(z/y)^2 which can obviously only be satisfied for specific x,y,z with that sum less than or equal to one, that region looks like this:
So you could only plot shapes within that region :/
