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.

 

[blue_raver22]

I can provide some insights into this topic. First, let me explain what a differentiable manifold is. A differentiable manifold is a mathematical space that is locally similar to Euclidean space, but may have a more complex global structure. In simpler terms, it is a space that looks like a curved surface, but each point on the surface behaves like a point on a flat plane.

Now, to address the question of projecting a differentiable manifold onto a plane. This is a common problem in mathematics and computer graphics, and there are various techniques that can be used to achieve this projection. One approach is to use a mapping function that transforms points on the manifold to points on the plane. This function can be based on the geometry of the manifold and the desired properties of the projection.

In your specific case, where you want to create a game similar to Portal, you can look into techniques such as stereographic projection or conformal mapping. These methods are commonly used in computer graphics and can help you achieve the desired projection from your 3-manifold to a 2-dimensional plane.

If you are not familiar with these techniques, I would recommend consulting with a mathematician or a computer graphics expert who can assist you in implementing them in your game. Additionally, there are many online resources and tutorials available that can help you understand and apply these techniques.

In conclusion, projecting a differentiable manifold onto a plane is a well-studied problem in mathematics and computer graphics, and there are various techniques that can be used to achieve this. With a little research and assistance from experts, you can successfully implement this projection in your game.