CODEWITHSUNDEEP
openglmatrix3d
Mohammed Li
Mohammed Li

Reputation: 305

How does a projection Matrix work?

I have to write a paper for my A-Levels about 3D-Programming. But I got a serious problem understanding the perspective projection Matrix and I need to fully explain the Matrix in detail. I've searched a lot of websites and youtube videos on this topic but very little even try to answer the question why the Matrix has these values at that place. Based on this http://www.songho.ca/opengl/gl_projectionmatrix.html I was able to find out how the w-row works, but I don't understand the other three.

I decided to use the "simpler" version for symmetric viewports only (right-handed Coord.):

![<code>r+l=0,</code>r-l=2r<code>(width);</code>t+b=0<code>,</code>t-b=2t<code>(height);</code>[n/r 0 0 0; 0 n/t 0 0; 0 0 -(f+n)/(f-n) -(2fn)/(f-n); 0 0 -1 0]`[1]

I am very thankful for every attempt to explain the first three rows to me!

Upvotes: 13

Views: 5110

Answers (1)

ratchet freak
ratchet freak

Reputation: 48176

The core reason for the matrix is to map the 3D coordinates to a 2D plane and have more distant objects be smaller.

For just this a much simpler matrix suffices (assuming your camera is at origin and looking at the Z axis):

1 0 0 0
0 1 0 0
0 0 0 0
0 0 1 0

After multiplying with this matrix and then renormalizing the w coordinate you have exactly that. Each x,y,z,1 point becomes x/z,y/z,0,1.

However there is no depth information (Z is 0 for all points) so a depth buffer/filter won't work. For that we can a a parameter to the matrix so the depth information remains available:

1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0

Now the resulting point contains the inverse depth in the Z coordinate. Each x,y,z,1 point becomes x/z,y/z,1/z,1.

The extra parameters are the result of mapping the coordinates into the (-1,-1,-1) - (1,1,1) device box (the bounding box where if you are outside of it the point won't get drawn) using a scale and a translate.

Upvotes: 10

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