Practical Geometric Modeling Using Geometric Algebra Motors

The usage of Geometric Algebra motors instead of Euclidean vectors for describing the position and orientation of points on a surface has promising applications in Computer Science and Engineering. Common geometric transformations, such as rotations and translations of Euclidean points, are also applicable to motors. However, encoding vertex positions and orientations as motors adds the capability of computing motor interpolation on surfaces. Thanks to that, general curves and surfaces can be generated by a motor interpolation process using different basis functions and parameterizations. In applications, the generated surfaces can be visually manipulated and deformed in a predictable way by changing the motors. In this paper we look inside the theory behind those applications as well as practical details on how Geometric Algebra algorithms can be computed efficiently. We show that geometric deformations can be computed at interactive rates on surface models with millions of vertices using the GPU.