Smooth interpolation on homogeneous matrix groups for computer animation

Homogeneous matrices are widely used to represent geometric transformations in computer graphics, with interpolation between those matrices being of high interest for computer animation. Many approaches have been proposed to address this problem, including computing matrix curves from curves in Euclidean space by registration, representing one-parameter curves on manifold by rational representations, changing subdivisional methods generating curves in Euclidean space to corresponding methods working for matrix curve generation, and variational methods. In this paper, we propose a scheme to generate rational one-parameter matrix curves based on exponential map for interpolation, and demonstrate how to obtain higher smoothness from existing curves. We also give an iterative technique for rapid computing of these curves. We take the computation as solving an ordinary differential equation on manifold numerically by a generalized Euler method. Furthermore, we give this algorithm's bound of the error and prove that the bound is proportional to the shift length when the shift length is sufficiently small. Compared to direct computation of the matrix functions, our Euler solution is faster.