The Geometry of Vision

The basic principle of perspective drawing is that parallel lines converge. Looking at examples, it is easy to see that pictures drawn "in perspective" correctly model our vision, while pictures that are not drawn "in perspective" fail to do so. But why is this the case? Namely, why is it that when we look at parallel lines in space, we actually observe them converging to a point? And why do different families of parallel lines seem to converge to different points? Rather than study points and lines in space, we will study the images of points and lines in our vision, in order to answer these questions. In fact, using "homogeneous coordinates" we can make mathematically precise statements about the images we see. The mathematical laws these images obey are different from the usual "Euclidean" laws that we are used to studying, but nevertheless give rise to a rich geometric structure.