The Projective Geometry of Conics

J.C. Álvarez Paiva

This chapter is dedicated to the projective study of conics. We start by studying the action of the linear group on the space of quadratic forms. After proving that a nondegenerate conic on the projective plane is equivalent to a circle, we show that the cross-ratio of four points on a conic is well-defined, prove the theorems of Chasles and Steiner characterizing conics in terms of projective transformations, and prove the theorems of Pascal and Brianchon.

In the last section, we study the action of the subgroup of projective transformations preserving a given conic ${\cal C}$ on the exterior and the interior of ${\cal C}$. The action on the interior is transitive and preserves a metric. We then show that in this metric -- the Cayley-Klein model of hyperbolic geometry -- geodesics are straight lines.

Notes

• Anything preceded by a * may be left for a second reading.

• Next to the exercises there is a two-digit number in parentheses which describes its degree of difficulty. The simplest exercises are identified by a (00), while the hardest -- those that could take you a week of intensive brain work -- are identified by a (50).