Projection of a differentiable manifold onto a plane

[Etherian]

For a game I am thinking about making I would need to know how to project points from a differentiable bounded 3-manifold to a Euclidean plane (the computer screen). The manifold would be made from a 3-dimensional space with two balls cut out of it and a hypercylinder glued onto it at the holes created. The result, hopefully, would be similar to the game Portal.

I only have a cursory knowledge of topology, so I do not know how to do such a projection or where to look to find out. An explanation of how or where to look for one would be a great help to me.

 

[jgm340]

It sounds like you'll be embedding this 3-manifold into R⁴. If this is the case, it's more a matter of linear algebra than anything else.

You can approximate this manifold as a union 3-simplexes (pyramids with four triangular sides) in R⁴. Perhaps you'll have 1000 of these simplexes to form a decent approximation. You can represent each simplex as a matrix S where the columns are the vertices of the simplex:

s₁₁ s₁₂ s₁₃ s₁₄
s₂₁ s₂₂ s₂₃ s₂₄
s₃₁ s₃₂ s₃₃ s₃₄
s₄₁ s₄₂ s₄₃ s₄₄

You want to project this onto some 2-dimensional plane in R⁴. Suppose this plane has normal u = (u₁, u₂, u₃, u₄). Then we want each of the vertices (columns of S) to be sent to a point on this plane (which we are assuming passes through the origin).

You can create a "shift" matrix M which, when subtracted from S, gives the projected location of each point. The i-th column of M will be a vector m_i = (s_i DOT u)/(u DOT u) u, where s_i is the i-th column of S.

Then you apply a rotation to (S - M) so that the plane lies in the x-y plane. Then you just use the first two coordinates of each column vector.

 

[Etherian]

Your response definitely helped me, but I believe the 2D plane P would have to be in the 3-manifold M to achieve the effect I am looking for. Like I said, though, I am still learning about this stuff. Also, I was hoping that by gluing and ungluing the two submanifolds I could get away with not having to define M using simplexes which would be expensive in memory and computation.

 

[Etherian]

Never mind about the previous post. I have everything I need to know now. Thanks for your help.