Generalized Homogeneous Coordinates for Computational Geometry

Abstract

The standard algebraic model for Euclidean space Eⁿ is an n-dimensional real vector space ℝⁿ or, equivalently, a set of real coordinates. One trouble with this model is that, algebraically, the origin is a distinguished element, whereas all the points of Eⁿ are identical. This deficiency in the vector space model was corrected early in the 19th century by removing the origin from the plane and placing it one dimension higher. Formally, that was done by introducing homogeneous coordinates [110]. The vector space model also lacks adequate representation for Euclidean points or lines at infinity. We solve both problems here with a new model for Eⁿ employing the tools of geometric algebra. We call it the homogeneous model of Eⁿ.