On Rational Geometry of Conic Sections

Simple geometric objects and transformations appear in representations and algorithms of geometric facilities in computer applications such as modelling, robotics, or graphics. Usually, these applications only support objects and transformations fully describable by rational parameters, and a computer display of points of the objects at least implicitly requires points with rational coordinates. In this setting we investigate some basic questions of the geometry of rational conic sections, when the geometry is defined by the group of rational projective transformations, the group of rational affine transformations, or the group of rational rigid transformations. Some results follow classical results, while others turn out to be quite different. In particular, we obtain a complete classification scheme for nondegenerate rational conics for rational affine geometry and a constructive method for production of a minimal set of representatives of all equivalence classes.