the classification of conics in affine geometry

We begin by showing that each affine conic is congruent to one of three standard forms.

Theorem AC2

  1. Each parabola is affine congruent to P0 : y2 = x.
  2. Each ellipse or circle is affine congruent to E0 : x2 + y2 = 0.
  3. Each hyperbola is affine congruent to H0 : xy  = 1.

proof of theorem AC2

Since two curves which are affine congruent to the same curve are affine congruent, we see
that the set of affine conics consists of at most three congruence classes. As yet, we cannot
be sure that there are actually three (distinct) classes - for example, there could be an affine
transformation mapping a parabola to an ellipse, circle or hyperbola. There are several ways
to show that the classes are distinct, as we shall see. Each method involves ideas which are
interesting in their own right.

affine symmetry groups of conics

tangents in affine geometry

main affine page