Fast Study Quadric Interpolation in the Conformal Geometric Algebra Framework

Abstract

Interpolating trajectories of points and geometric entities is an important problem for kinematics. To describe these trajectories, several algorithms have been proposed using matrices, quaternions, dual-quaternions, and the Study quadric; the last one allows the embedding of motors as 8D vectors into projective space P7, where the interpolation of rotations and translations becomes a linear problem. Furthermore, conformal geometric algebra (CGA) is an effective and intuitive framework for representing and manipulating geometric entities in Euclidean spaces, and it allows the use of quaternions and dual-quaternions formulated as Motors. In this paper, a new methodology for accelerating the Study quadric Interpolation based on Conformal Geometric Algebra is presented. This methodology uses General Purpose Graphics Processing Units (GPUs) and it is applied for medical robotics, but it can also be extended to other areas such as aeronautics, robotics, and graphics processing.