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GEOMETRIC COMPUTATION, DUALITY AND PROJECTIVE SPACE

This paper presents solution of some selected problems that can be easily solved in the projective space. Projective space and homogeneous coordinates are mostly used in computer graphics and used especially for geometric transformations. Nevertheless the projective formulation offers an elegant solution to several geometrical problems, too. If the principle of duality is used, quite surprising solutions can be found and new useful theorems can be generated as well, e.g. an equation of a line in E as an intersection of two planes, computation of barycentric coordinates etc. The projective extension of the Euclidean space also offers higher computational stability as it avoids division operation in general. In this paper we will show that a solution of a system of linear equations is equivalent to the extended cross (outer) product and how this can be used for solving geometrical problems. As the duality between points and lines in E or between points and planes in E is known, many geometrical algorithms can be easily converted to the dual representation, solved in the dual space and the result can be converted back to the projective or Euclidean space, e.g. point-in-polygon test is dual to a test if a line intersect a polygon. The presented approach offers higher robustness and it is convenient for matrix-vector supporting hardware namely for the GPU implementation.