Geometric Computing in Computer Graphics and Robotics using Conformal Geometric Algebra

Abstract

In computer graphics and robotics a lot of different mathematical systems like vector algebra, homogenous coordinates, quaternions or dual quaternions are used for different applications. Now it seems that a change of paradigm is lying ahead of us based on Conformal Geometric Algebra unifying all of these different approaches in one mathematical system. Conformal Geometric Algebra is a very powerful mathematical framework. Due to its geometric intuitiveness, compactness and simplicity it is easy to develop new algorithms. Algorithms based on Conformal Geometric Algebra lead to enhanced quality, a reduction of development time, better understandable and better maintainable solutions. Often a clear structure and greater elegance results in lower runtime performance. However, it will be shown that algorithms based on Conformal Geometric Algebra can even be faster than conventional algorithms. The main contribution of this thesis is the geometrically intuitive and - nevertheless - efficient algorithm for a computer animation application, namely an inverse kinematics algorithm for a virtual character. This algorithm is based on an embedding of quaternions in Conformal Geometric Algebra. For performance reasons two optimization approaches are used in a way to make the application now three times faster than the conventional solution. With these results we are convinced that Geometric Computing using Conformal Geometric Algebra will become more and more fruitful in a great variety of applications in computer graphics and robotics.