Oriented Projective Geometry

Abstract

Oriented projective geometry is a model for geometric computation that combines the elegance of classical projective geometry with the ability to talk about oriented lines and planes, signed angles, line segments, convex figures, and many other concepts that cannot be defined within the classical version. Classical projective geometry is the implicit framework of many geometric computations, since it underlies the well-known homogeneous coordinate representation. It is argued here that oriented projective geometry — and its analytic model, based on signed homogeneous coordinates — provide a better foundation for computational geometry than their classical counterparts.The differences between the classical and oriented versions are largely confined to the mathematical formalism and its interpretation. Computationally, the changes are minimal and do not increase the cost and complexity of geometric algorithms. Geometric algorithms that use homogeneous coordinates can be easily converted to the oriented framework at little cost. The necessary changes are largely a matter of paying attention to the order of operands and to the signs of coordinates, which are frequently ignored or left unspecified in the classical framework.