Beyond the Celestial Sphere: Oriented Projective Geometry and Computer Graphics

Abstract

Software for computer graphics represents three-dimensional space a little differently than one might expect. Euclidean geometry is not quite right. The usual approach uses what is called projective geometry, certainly one of the most beautiful systems in mathematics. Yet even with this approach, when the mathematics actually meets the computer code there are some awkward inconveniences. Perhaps the best solution may be what is called oriented projective geometry. This geometry was worked out in detail by Jorge Stolfi [10, 11] in 1987; it has also found more recent application in computer vision [6]. This paper is an elementary introduction to this still unfamiliar geometry from a coordinate-based point of view, restricted to three dimensions. It assumes only a background in linear algebra. For the reader who knows of classical projective geometry and homogeneous coordinates, it is best to set this knowledge aside and examine whether the oriented approach works on its own terms. At the end of the paper we compare and contrast it with the classical approach. Also, readers conversant with wedge products, Plucker coordinates, and the Hodge operator may be pleased to see them appear in such a concrete setting.