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# Euclidean Conics

Chapter
Part of the Problem Books in Mathematics book series (PBM)

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

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## Copyright information

© Marcel Berger 1984

## Authors and Affiliations

1. 1.U.E.R. de Mathematique et InformatiqueUniversité Paris VIIParis, Cedex 05France
2. 2.Centre National de la Recherche ScientifiqueParisFrance
3. 3.P.U.K.GrenobleFrance