The use of dual quaternions in rigid body motion interpolation and visualization

Abstract

Dual quaternions are an excellent alternative tool to transformation matrices in the study of rigid body motion. They are known to represent the position and orientation of a rigid body in a concise and efficient manner, with the real part of the dual quaternion representing orientation without introducing singularities associated with other representations such as Euler angles. In this thesis, we have used dual quaternions to study the problem of motion interpolation of rigid bodies and to visualize the motion of the rigid body in three-dimensional space. In this work, we first studied and derived several significant properties of unit dual quaternion curves treated as space curves in four dimensions. For these space curves, we derived the expressions for the dual curvature and the dual torsion, and analyzed their significance in terms of rigid body motion with detailed case studies. An algorithm for the estimation of rigid body motion for a given curvature and torsion of the dual quaternion curve is presented. In addition, we propose a method for obtaining a closed?form solution for the transformation of the rigid body from the graph of the dual quaternion curvature. The algorithm appears to be much faster than conventional interpolation techniques and provides an exact solution. Towards the goal of motion interpolation of a rigid body and visualization of the motion, we use cubic basis functions in the quaternion space. This approach achieves a smooth interpolation of animation keyframes of both orientation and position simultaneously. The method was implemented in various animations to demonstrate its performance in real time. The first example is the motion of a cricket ball with user?defined initial conditions. The second example is a simulation of the very complicated orbits of the Moon and the Earth around the Sun. The program was able to perform an extremely fast and reasonably accurate prediction of solar eclipses for a user?defined time span.