Analysis Of Applications With Conformal Geometric Algebra

Since the emergence of computer graphics in the mid-1970s, we are basically using linear algebra for the mathematical framework. Now,we will want to use another system is geometric algebra, in particular the five-dimensional conformal geometric algebra, it unifies mathematical systems used computer graphics in a simple and intuitive way. This paper discusses the conformal geometric algebra in the application of computer application.The main contents and contributions of this thesis are summarized as follows:(1) We analysis the characteristics of the structure of geometric algebra, the description of transform,the calculation done by means of system. Geometric algebra is a more general mathematical language established on the basis of the Clifford algebra. Based on the analysis of traditional matrix algebra, Herman Grassman vector algebra and WR Hamilton quaternion algebra and geometry algebra, we derived transformation of the linear expression in three-dimensional by the nature computing of geometric algebra. Experiments show that the expression of some transformation is more simple, efficient using geometric algebra than Goldman quaternion algebra , and the mathematical description of equivalent.(2) 2D-3D pose estimation includes a few spaces,so CGA is good at denoting the scence in only one frame.The rotation operation is a linear operation in Euler space, but translation operation not. Because the nonlinear characteristics of translational displacement operation, rigid displacement is no longer a linear operation. We apply screw and twist in the 2d-3d pose estimation,and get linear description, experimental verification that the method is more simple in three-dimensional geometrical interpretation of movement than based on matrix algebra(3) Applying CGA to model omnidirectional system of united model generally, combineing with obtained projective invariant, provide easier bases for further complicated applications. CGA is good at handle these questions that is because, that it not only express points, lines and planes, also it could indicate point pairs, circles and spheres (geometric objects needed in the UM). All these geometric objects are based on sphere(such as points can be seen as sphere with zero radius, and plane can be seen as sphere with unlimited radius and so on). Further more, it allows different operations and translations using the same and simpler means. In the end, we can define the model free of coordinate, just use geometric relations of objects.