Euclidean Conics

Abstract

From 15.D we know there are three possible cases of Euclidean affine conics: the ellipse, the hyperbola and the parabola. Only the parabola does not have a center; it has a single axis of (orthogonal) symmetry and a vertex. The ellipse and the hyperbola have a center and two axes of orthogonal symmetry; the ellipse has four vertices (with the exception of the circle), and the hyperbola has two. The equations, in an appropriate orthonormal basis, are

$$\begin{array}{l} \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - 1 = 0(ellipse),\;\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} - 1 = 0(hyperbola),\\ {y^2} - 2px = 0(parabola). \end{array} $$

In this chapter X denotes a Euclidean affine plane, $\tilde X$ its projective completion, $\tilde X$ ^C the complexification of $\tilde X$ (see 9.D) and {I,J} the cyclical points of X. In general, C will be the non-empty image of a proper conic α in X.