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October 8th, 2012, 06:16 AM   #1
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Geometry: perspective projection

Hello, I have a simple problem - here it is:

1) We have a camera in R3 with point of view C[0,0,-1].
2) The projection plane of the camera is the plane z=0 (this is Ax+By+Cz+D=0 where A=B=D=0 and C=1).

3) if we want to project a point (for example G[g1,g2,g3]) to the projection plane, we construct a line through the point and C (GC). The intersection between the line (GC) and the projection plane (z=0) is the projection of our point (G').

4) We have 3 points in space (L[l1,l2,l3], M[m1,m2,m3] and N[n1,n2,n3]), which form a triangle with width 128, height 96 and right angle at M.

5) We know the coordinates of the projections of the points L, M and N - they are L'[l1',l2',l3'], M'[m1',m2',m3'] and N'[n1',n2',n3'].

We are looking for the coordinates of L, M and N.
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October 9th, 2012, 09:36 AM   #2
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Re: Geometry: perspective projection

I would use vectors. For a point L' in the projection plane, the actual point L = L' + a*L'' for some value of a, where L'' is any vector which is parallel to the line between C and L.

Also M = M' + b*M'' and N = N' + c*N''.

Plug in your constraints and solve the resulting simultaneous equations for a, b and c.

You actually have a massive simplification in your case because you know that L, M and N are in the same z plane, and also the L', M' and N' are in the same z plane. This means that if you choose L'', M'' and N'' to each have a z component of 1, then a, b and c will have the same value.
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