Homogeneous Coordinates for Projective Geometry

Chapter 1 presented Euclidean geometry alongside analytic geometry in the Euclidean plane using Cartesian coordinates. Such a familiar mapping needed no introduction. By contrast, Chapter 11 illustrated how projective spaces differ from affine spaces but gave no hint to how one could perform computation in projective spaces. Such an accompanying analytic geometry requires assigning coordinates to points and other objects—the topic of this chapter. Homogeneous coordinates have been discovered four times independently— by Bobillier, Plücker, Feuerbach, and Möbius [19]. Chapter 13 discusses Möbius’s version, which remains interesting in its own right. The first known notes by Möbius on barycentric calculus date from 1818 [29, p. 49]. It is worth quoting Boyer’s impression.

The year 1827 is of considerable importance in the history of analytic geometry in Germany […]. It is sometimes said that Descartes arithmetized geometry, but this is not strictly correct. For almost two hundred years after his time coordinates were in essence geometric. […] The arithmetization of coordinates took place not in 1637 but in the crucial years 1827–1829. [19, p. 242]