Optimized line and line segment clipping in E2 and Geometric Algebra

Abstract

Algorithms for line and line segment clipping are well known algorithms especially in the field of computer graphics. They are formulated for the Euclidean space representation. However, computer graphics uses the projective extension of the Euclidean space and homogeneous coordinates for representation geometric transformations with points in the E or E space. The projection operation from the E to the E space leads to the necessity to convert coordinates to the Euclidean space if the clipping operation is to be used. In this contribution, an optimized simple algorithm for line and line segment clipping in the E space, which works directly with homogeneous representation and not requiring the conversion to the Euclidean space, is described. It is based on Geometric Algebra (GA) formulation for projective representation. The proposed algorithm is simple, efficient and easy to implement. The algorithm can be efficiently modified for the SSE4 instruction use or the GPU application, too.