Homogeneous Coordinates

We present a geometric definition of conic sections in the oriented projective plane and describe some of their nice properties. This definition leads to a very simple and unambiguous representation for affine conics and conic arcs. A conic (of any type) is represented by the homogeneous coordinates of its foci and one point on it, hence, the metric plays a major role in this case as opposed to the traditional algebraic characterization of conics as second degree polynomial curves. This representation is particularly suitable for the implementation of geometric solutions of problems that involve the concept of distance. Furthermore, we discuss point location with respect to conic curves which constitutes an important elementary operation for the solution of many such problems

Abstract--The objective of most stereo vision work has been to extract the 3D shape of a world scene from two of its images which have been captured from two different standpoints. One of the major challenges in computer vision has been the elaboration of methods, ...

This paper presents solution of selected problems using principle of duality and projective space representation. It will be shown that alternative formulation in the projective space offers quite surprisingly simple solutions that lead to more robust and faster algorithms which are convenient for use within parallel architectures as GPU (Graphical Processor Units-NVIDIA-TESLA/Fermi) or SCC (Intel-Single-chip Cloud Computing), which can speed up solutions of numerical problems in magnitude of 10-100. There are many geometric algorithms based on computation of intersection of lines, planes etc. Sometimes, very complex mathematical notations are used to express simple mathematical solutions, even if their formulation in the projective space offers much more simple solution. It is shown that a solution of a system of linear equations is equivalent to generalized cross product, which leads with the duality principle to new algorithms. This is presented on a new formulation of a line in ...