Math behind the perspective projection matrix

Question

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

Answer 1

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++.

Answer 2

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.