4 Projective geometric transformations

Geometric transformations, projective space, and homogeneous coordinates are associated concepts that not only enable, but also explain one another. The first one is the most applicable to real-world problems, so it usually gets the most attention. To exploit it in the most effective way, however, you should also know a little about the other two.

In this chapter, you’ll learn how to do geometric transformations such as translation, rotation, scaling, and shear. You’ll learn how to generalize them into a matrix multiplication. But you don’t need a book to do all that. Any good framework has transformation routines; you can learn a few functions and get the job done.

4.1 Some special cases of geometric transformations

4.1.1 Translation

4.1.2 Scaling

4.1.3 Rotation

4.1.4 Section 4.1 summary

4.2 Generalizations

4.2.1 Linear transformations in Euclidean space

4.2.2 Bundling rotation, scaling, and translation in a single affine transformation

4.2.3 Generalizing affine transformations to projective transformations

4.2.4 An alternative to projective transformations

4.2.5 Section 4.2 summary

4.3 Projective space and homogeneous coordinates

4.3.1 Expanding the whole space with homogeneous coordinates

4.3.2 Making all the transformations a single matrix multiplication: Why?

4.3.3 Section 4.3 summary