Theorems in Projective Geometry

Abstract

This chapter is more mathematical. The theorems of Pappus, Desargues, and Pascal are introduced to show that there is a non-metrical geometry such as Poncelet had described. However, they are proved in a modern way, by reducing them to simple special cases and then using a mixture of elementary vector methods and theorems from elementary Euclidean geometry. The reductions use simple projective transformations, which leads to discussion of the idea of augmenting the familiar plane with a line at infinity. The concept of the cross-ratio of four points on a line is introduced and shown to be invariant under a projective transformation. Poncelet’s porism is described.