Similar articles

Abstract: 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.

[1] Alexa, M., 2002. Linear Combination of Transformations. Proceedings of SIGGRAPH’02, p.380-387.

[2] Barr, A.H., Currin, B., Gabriel, S., Hughes, J.F., 1992. Smooth interpolation of orientations with angular velocity constraints using quaternions. Computer Graphics (SIGGRAPH’92), 26(2):313-320.

[3] Bloom, C., Blow, J., Muratori, C., 2004. Errors and Omissions in Marc Alexa’s “Linear Combination of Transformations”. http://www.cbloom.com/3d/techdocs/lcot_errors.pdf.

[4] Do Carmo, M.P., 1992. Riemannian Geometry. Birkhauser, Springer, Boston, MA.

[5] Godinho, L., Natário, J., 2004. An Introduction to Riemannian Geometry with Applications. Http://www.math.ist.utl.pt/~lgodin/.

[6] Hofer, M., Pottmann, H., 2004. Energyminimizing splines in manifolds. Transactions on Graphics, 23(3):284-293.

[7] Hofer, M., Pottmann, H., Ravani, B., 2004. From curve design algorithms to the design of rigid body motions. The Visual Computer, 20(5):279-297.

[8] Hunt, K.H., 1978. Kinematic Geometry of Mechanisms. Oxford University Press.

[9] Kim, M.J., Myung-Soo, K., Shin, S.Y., 1995. A General Construction Scheme for Unit Quaternion Curves with Simple High Order Derivatives. Proc. of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH’95, p.369-376.

[10] McCarthy, J.M., 1990. Introduction to Theoretical Kinematics. MIT Press, Cambridge MA.

[11] Noakes, L., 2004. Spherical Splines.

[12] Park, F.C., Ravani, B., 1997. Smooth invariant interpolation of rotations. ACM Transactions on Graphics (TOG), 16(3):277-295.

[13] Pottmann, H., Hofer, M., 2005. A variational approach to spline curves on surfaces. Computer Aided Geometric Design, 22(7):693-709.

[14] Shoemake, K., 1985. Animating rotation with quaternion curves. Computer Graphics (SIGGRAPH’85), 19(3):245-254.

[15] Wallner, J., Dyn, N., 2005. Convergence and C¹ analysis of subdivision schemes on manifolds by proximity. Computer Aided Geometric Design, 22(7):593-622.

[16] Zefran, M., Kuman, V., Croke, C.B., 1998. On the generation of smooth three-dimensional rigid body motions. IEEE Transactions on Robotics and Automation, 14(4):576-589.

[17] Zefran, M., Kumar, V., Croke, C.B., 1999. Metrics and connections for rigid-body kinematics. The International Journal of Robotics Research, 18(242):1-16.