Computing in the Conformal Space Objects, Incidence Relations, and Geometric Constrains for Applications in AI, GIS, Graphics, Robotics, and Human-Machine Interaction

Abstract

This article presents a revisited proposal for the formulation of objects and geometric relations and constraints in the conformal space. For modeling, graphics engineering, kinematics, and dynamics, the solution of problems using only points and lines; or the formulation of rigid motion (SE(3) using vectors calculus, matrix algebra, or calculus is indeed very awkward. In contrast, we use incidence algebra and conformal geometric algebra to effectively represent geometric objects and compute relations and constraints between geometric entities. In conformal geometric algebra, one can compute efficiently the linear transformations SO(3) and SE(3) of these geometric entities using rotors, translators, and motors. Since these operators and geometric entities have no redundant coefficients, they can be computed very fast. The authors present a new and complete set of equations using incidence algebra and conformal geometric algebra. The use of the proposed equations depends upon the applications. You can enclose certain objects with geometric shapes in your setting using points, lines, planes, circles, spheres, hyperplanes, and hyperspheres. Then, quadratic programming for optimization can be applied to find the minimal directed distance or a minimal path to be followed among many geometric objects. These methods and equations can be used to tackle a variety of problems in graphics, augmented virtual reality, GIS, Robotics, and Human-Machine Interaction. For real-time applications, the procedures and equations presented in this work can be used to develop efficient algorithms, which can be sped up using FPGA or CUDA (Nvidia).