Chapter 6 - An Introduction to Projective Geometry

Summary

Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre.… There are 12 passengers aboard. The wind is blowing East-North-East. The clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?

Gustave Flaubert

All the transformations so far considered have taken points into points. The most characteristic feature of the “projective” plane is the principle of duality, which enables us to transform points into lines and lines into points. One such transformation, somewhat resembling inversion, is “reciprocation” with respect to a fixed circle. Every point except the center O is reciprocated into a line, every line not through O is reciprocated into a point, and every circle is reciprocated into a “conic” having O for a “focus”. After some discussion of the various kinds of conic, we shall close the chapter with a careful comparison of inversive geometry and projective geometry.

Reciprocation

For this variant of inversion, we use (as in Section 5.3, page 108) a circle ω with center O and radius K Each point P (different from O) determines a corresponding line p, called the polar of P; it is the line perpendicular to OP through the inverse of P (see Figure 6.1A).