# Math behind the perspective projection matrix

I am looking for some history and the actual development of the Math behind a perspective projection Matrix. I googled for it. I mostly find the final matrix everywhere, not exactly a derivation of it or the history behind it. Could anyone please provide some links/name of books where i can find a detailed account of this?

Thanks! Mukund

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I learned the projection matrix along the line of mathematical modeling class and I'm willing to share the things I went through.

### Concepts

Basic idea (*Coo stands for coordinates, Para for parameter)

Camera Matrix        Persp Projection       Intrinsic Para Matrix
World Coo   <--->   Camera Coo   <--->   Image Plane Coo  <--->   Pixel Coo
↑                                                                ↑
└━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┘
Projecton Matrix


Pinhole camera model which is the easiest linear camodel we can conceive of.

Transformation/Rotation matrices means translation and rotation of objects based on our view.

Perspective projection has some explanation, and a link to camera matrix.

### Notations

Homogenous coordinates which is rather useful. You may see that we can scale the homogenous coordinates without changing the point's coordinate in the projection plane.

### Further Materials

Fundamentals of Computer Vision page has slides and notes to computer vision and digital camera related topics. See slide/notes #1.

Geometric Framework for Vision I: Single View and Two-View Geometry Andrew Zisserman has some fine material on projection matrices.

3D Math Primer for Graphics and Game Development, 2ndE, a good book on 3D gaming math basics. There are prepared slides and full code in C++.

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There's a lot other concepts involved in my post (application of it) other than the Perspective Projection Matrix. The idea is the first three concepts in the diagram I drew in code block. The final step is necessary when it comes to rasterized receivers. –  FrenzY DT. Aug 20 '12 at 9:03
Thanks a lot FrenzY DT! –  Mukund Aug 20 '12 at 14:16

I recommend the book Multiple View Geometry in Computer Vision on that topic. It contains an accessible introduction to projection matrices and projective geometry which first treats the two-dimensional case and then moving on to the three dimensional case.

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Thank you Dirk! –  Mukund Aug 20 '12 at 14:17