A Fast Computation of 3 × 3 Matrix Exponentials and its Application in CG

Abstract

It is often useful to compute a lot of matrix exponentials in computer graphics (CG). The exponential of a matrix is used for the smooth deformation of 2D or 3D meshed CG objects. Hence, we need to compute a large number of the exponentials of 3 × 3 rotational matrices and 3 × 3 real symmetric matrices. For rotational matrices, Rodrigues' formula is known to compute their exponentials. We investigated the polynomial methods introduced by Moler and Van Loan to compute an exponential of 3 × 3 real symmetric matrices, and we introduce an algorithm for eigenvalues of 3 × 3 real symmetric matrices. We introduce a simple formula for the matrix exponential of a 3 × 3 real symmetric matrix using a formula introduced by Kaji et al. in 2013 and Viète’s Formula. Since our matrix exponential algorithm do not use eigenvectors, we are able to reduce the computational cost using a fast eigenvalue computation algorithm. Then, we incorporated our implementation into a shape deforming tool developed by Kaji et al. As a result, we achieved a notable performance improvement. In fact we show our algorithms for matrix exponentials is about 76% faster than a standard algorithm for given 3 × 3 real symmetric matrices. For the deformation of a CG model, our algorithm was about 19% faster than a standard algorithm.