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June 2nd, 2012, 12:21 AM   #1
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Projection Matrix from 8 points

Hello there!

I'm currently looking into different solutions for a problem which has appeared in one of my coding projects. Honestly it's been a long while since I did any math more complex than 3d vectors, so I hope the mathematicians of this forum can provide me with some assistance.

What I need is a rather specific 4x4 projection matrix. I have eight points in 3d space, they represent the eight corners of the resulting frustum I'm after. To be more exact the points define two rectangles, the near and the far rectangles encompassing the frustum.

[attachment=0:1febjhww]Example.jpg[/attachment:1febjhww]

The eight corner points in example 1 are the points I have.
Example 2 is a top down view of how they're calculated. The black dot represents the camera. The red points are taken from the corners of rectangular object. The green points are then simply extended by drawing a vector from the camera towards each red point and extending past them.

That's all the relevant information I can think of right now. Any thoughts?

All help is greatly appreciated.
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File Type: jpg Example.jpg (21.9 KB, 455 views)
FredrikAkerblom is offline  
 
August 10th, 2012, 03:37 AM   #2
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Re: Projection Matrix from 8 points

Hey FredrikAkerblom.

The typical way to do a projection to a plane (perspective not an orthogonal) is to do x' = (x-a)/cz, y' = (y-b)/cz, z' = c. This will project a point onto the plane z' = c with a centre of x = a, y = b.

Projecting to a point between the plane that the point lies on (parallel to x-y plane) and the plane you are projecting on is a linear interpolation from the original point to the final point.

So if you have say your original point A and your projected point B, you can use the formula x = At + (1-t)B where A = (x,y,z) for original point and B = (x,y,z) for projected point.

You can generate for various values of t any point between the two where t = 0 to t = 1.
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