Intersection computation in projective space using homogeneous coordinates

Abstract

There are many algorithms based on computation of intersection of lines, planes etc. Those algorithms are based on representation in the Euclidean space. Sometimes, very complex mathematical notations are used to express simple mathematical solutions. This paper presents solutions of some selected problems that can be easily solved by the projective space representation. Sometimes, if the principle of duality is used, quite surprising solutions can be found and new useful theorems can be generated as well. It will be shown that it is not necessary to solve linear system of equations to find the intersection of two lines in the case of E2 or the intersection of three planes in the case of E3. Plücker coordinates and principle of duality are used to derive an equation of a parametric line in E3 as an intersection of two planes. This new formulation avoids division operations and increases the robustness of computation. The presented approach for intersection computation is well suited especially for applications where robustness is required, e.g. large GIS/CAD/CAM systems etc.