“Steady Affine Motions and Morphs” by Rossignac and Vinacua

  • ©Jarek Rossignac and Alvar Vinacua




    Steady Affine Motions and Morphs



    We propose to measure the quality of an affine motion by its steadiness, which we formulate as the inverse of its Average Relative Acceleration (ARA). Steady affine motions, for which ARA=0, include translations, rotations, screws, and the golden spiral. To facilitate the design of pleasing in-betweening motions that interpolate between an initial and a final pose (affine transformation), B and C, we propose the Steady Affine Morph (SAM), defined as At∘ B with A = C ∘ B-1. A SAM is affine-invariant and reversible. It preserves isometries (i.e., rigidity), similarities, and volume. Its velocity field is stationary both in the global and the local (moving) frames. Given a copy count, n, the series of uniformly sampled poses, Ai/n∘ B, of a SAM form a regular pattern which may be easily controlled by changing B, C, or n, and where consecutive poses are related by the same affinity A1/n. Although a real matrix At does not always exist, we show that it does for a convex and large subset of orientation-preserving affinities A. Our fast and accurate Extraction of Affinity Roots (EAR) algorithm computes At, when it exists, using closed-form expressions in two or in three dimensions. We discuss SAM applications to pattern design and animation and to key-frame interpolation.


    Abdel-Malek, K., Blackmore, D., and Joy, K. 2006. Swept volumes: Foundations, perspectives, and applications. Int. J. Shape Model. 12, 1, 87–127.Google ScholarCross Ref
    Agoston, M. K. 2005. Computer Graphics and Geometric Modeling: Mathematics. Springer. Google ScholarDigital Library
    Alexa, M. 2002. Linear combination of transformations. In Proceedings of the SIGGRAPH Conference. J. Hughes, Ed., ACM Press/ACM SIGGRAPH, 380–387. Google ScholarDigital Library
    Alexa, M., Cohen-Or, D., and Levin, D. 2000. As-Rigid-As-Possible shape interpolation. In Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’00). ACM Press, New York, 157–164. Google ScholarDigital Library
    Alexander, C., Ishikawa, S., and Silverstein, M. 1977. A Pattern Language: Towns, Buildings, Construction. Oxford University Press, Oxford, UK.Google Scholar
    Arnold, V. I. 1981. Ordinary Differential Equations. The MIT Press.Google Scholar
    Barr, A. H., Currin, B., Gabriel, S., and Hughes, J. F. 1992. Smooth interpolation of orientations with angular velocity constraints using quaternions. In Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’92). ACM Press, 313–320. Google ScholarDigital Library
    Bregler, C., Loeb, L., Chuang, E., and Deshpande, H. 2002. Turning to the masters: Motion capturing cartoons. In Proceedings of the 29th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’02). ACM, New York, 399–407. Google ScholarDigital Library
    Buss and Fillmore. 2001a. Spherical averages and applications to spherical splines and interpolation. ACM Trans. Graph. 20. Google ScholarDigital Library
    Buss, S. R. and Fillmore, J. P. 2001b. Spherical averages and applications to spherical splines and interpolation. ACM Trans. Graph. 20, 2, 95–126. Google ScholarDigital Library
    Cheng, S. H., Higham, N. J., Kenney, C. S., and Laub, A. J. 2000. Approximating the logarithm of a matrix to specified accuracy. SIAM J. Matrix Anal. Appl. 22, 4, 1112–1125. Google ScholarDigital Library
    Choi, J. and Szymczak, A. 2003. On coherent rotation angles for as-rigid-as-possible shape interpolation. Tech. rep., Georgia Institute of Technology.Google Scholar
    Clifford, W. 1882. Mathematical Papers. Macmillan, London.Google Scholar
    Culver, W. C. 1966. On the existence and uniqueness of the real logarithm of a matrix. Proc. Amer. Math. Soc. 17, 5, 1146–1151.Google ScholarCross Ref
    Davies, P. I. and Higham, N. J. 2003. A schur-parlett algorithm for computing matrix functions. SIAM J. Matrix Anal. Appl. Google ScholarDigital Library
    Duff, T. 1986. Splines in animation and modeling. In SIGGRAPH Course Notes on State of the Art Image Synthesis. ACM Press.Google Scholar
    Eaton, J. W. J. W. 2000. GNU Octave: A High-Level Interactive Language for Numerical Computations Ed. 3, Octave version 2.0.13. Network Theory Ltd.Google Scholar
    Fu, H., Tai, C. L., and Au, O. K. 2005. Morphing with laplacian coordinates and spatial-temporal texture. In Proceedings of the Pacific Graphics’05 Conference. 100–102.Google Scholar
    Golub, G. H. and Van Loan, C. F. 1996. Matrix Computations 3rd Ed. Johns Hopkins University Press, Baltimore, MD. Google ScholarDigital Library
    Grassia, F. S. 1998. Practical parameterization of rotations using the exponential map. J. Graph. Tools 3, 3, 29–48. Google ScholarDigital Library
    Higham, D. J. and Higham, N. J. 2005. MATLAB Guide. SIAM. Google ScholarDigital Library
    Higham, N. J. 1986. Newton’s method for the matrix square root. Math. Comp.. Google ScholarDigital Library
    Hyun, D.-E., Jüttler, B., and Kim, M.-S. 2002. Minimizing the distortion of affine spline motions. In Graph. Models. Google ScholarDigital Library
    Jang, J. and Rossignac, J. 2009. Octor: Subset selection in recursive pattern hierarchies. In Graphical Models. To appear. Google ScholarDigital Library
    Johnston, O. and Thomas, F. 1995. The Illusion of Life: Disney Animation. Disney Press.Google Scholar
    Kavan, L., Collins, S., O’Sullivan, C., and Žára, J. 2006. Dual quaternions for rigid transformation blending. Tech. rep. TCD-CS-2006-46, Trinity College, Dublin.Google Scholar
    Kavan, L., Collins, S., Žára, J., and O’Sullivan, C. 2008. Geometric skinning with approximate dual quaternion blending. ACM Trans. Graph. 27, 4, 105:1–105:23. Google ScholarDigital Library
    Kavan, L. and Žára, J. 2005. Spherical blend skinning: A real-time deformation of articulated models. In Proceedings of the Symposium on Interactive 3D Graphics and Games (I3D’05). ACM, New York, 9–16. Google ScholarDigital Library
    Kim, B. and Rossignac, J. 2003. Collision prediction. J. Comput. Inf. Sci. Engin. 3, 4, 295–301.Google ScholarCross Ref
    Kim, M.-J., Kim, M.-S., and Shin, S. Y. 1995. A general construction scheme for unit quaternion curves with simple high order derivatives. In Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’95). ACM Press, 369–376. Google ScholarDigital Library
    Kim, M. S. and Nam, K. W. 1995. Interpolating solid orientations with circular blending quaternion curves. Comput. Aid. Des. 27, 385–398.Google ScholarCross Ref
    Larive, M. and Gaildrat, V. 2006. Wall grammar for building generation. In Proceedings of the 4th International Conference on Computer Graphics and Interactive Techniques in Australasia and Southeast Asia (GRAPHITE’06). ACM, New York, 429–437. Google ScholarDigital Library
    Lee, J. and Shin, S. Y. 2002. General construction of time-domain filters for orientation data. IEEE Trans. Visualiz. Comput. Graph., 8, 2, 119–128. Google ScholarDigital Library
    Llamas, I., Kim, B., Gargus, J., Rossignac, J., and Shaw, C. D. 2003. Twister: A space-warp operator for the two-handed editing of 3D shapes. In Proceedings of ACM SIGGRAPH Conference, J. Hodgins and J. C. Hart, Eds. 663–668. Google ScholarDigital Library
    Ma, L., Chan, T. K. Y., and Jiang, Z. 2000. Interpolating and approximating moving frames using B-splines. In Proceedings of the Pacific Conference on Computer Graphics and Applications. IEEE Computer Society, 154–164. Google ScholarDigital Library
    Meini, B. 2005. The matrix square root from a new functional perspective: Theoretical results and computational issues. SIAM J. Matrix Anal. Appl. 26, 2, 362–376. Google ScholarDigital Library
    Mortenson, M. 2007. Geometric Transformations for 3D Modeling. Industrial Press, New York. Google ScholarDigital Library
    Pauly, M., Mitra, N. J., Wallner, J., Pottmann, H., and Guibas, L. 2008. Discovering structural regularity in 3D geometry. ACM Trans. Graph. 27, 3, 1–11. Google ScholarDigital Library
    Powell, A. and Rossignac, J. 2008. ScrewBender: Smoothing piecewise helical motions. IEEE Comput. Graph. Appl. 28, 1, 56–63. Google ScholarDigital Library
    Power, J. L., Brush, A. J. B., Prusinkiewicz, P., and Salesin, D. H. 1999. Interactive arrangement of botanical l-system models. In Proceedings of the Symposium on Interactive 3D Graphics (I3D’99). ACM, New York, 175–182. Google ScholarDigital Library
    Roberts, K. S., Bishop, G., and Ganapathy, S. K. 1988. Smooth interpolation of rotational matrices. In Proceedings of the Computer Vision and Pattern Recognition (CVPR’88). IEEE Computer Science Press, 724–729.Google Scholar
    Rossignac, J. and Kim, J. J. 2001. Computing and visualizing pose-interpolating 3D motions. Comput.-Aid. Des. 33, 4, 279–291.Google ScholarCross Ref
    Rossignac, J., Kim, J. J., Song, S. C., Suh, K. C., and Joung, C. B. 2007. Boundary of the volume swept by a free-form solid in screw motion. Comput.-Aid. Des. 39, 9, 745–755. Google ScholarDigital Library
    Rossignac, J. and Schaefer, S. 2008. J-splines. Comput.-Aid. Des. 40, 10-11, 1024–1032. Google ScholarDigital Library
    Shoemake, K. 1985. Animating rotation with quaternion curves. In Proceedings of the 12th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’85). ACM Press, 245–254. Google ScholarDigital Library
    Shoemake, K. 1987. Quaternion calculus and fast animation. In SIGGRAPH Course Notes on State of the Art Image Synthesis. ACM Press, 101–121.Google Scholar
    Shoemake, K. and Duff, T. 1992. Matrix animation and polar decomposition. In Proceedings of Graphics Interface Conference. Morgan Kaufmann Publishers, 258–264. Google ScholarDigital Library
    van Emmerik, M., Rappoport, A., and Rossignac, J. 1993. Simplifying interactive design of solid models: A hypertext approach. Vis. Comput. 9, 5, 239–254.Google ScholarCross Ref
    Wang, W. and Joe, B. 1993. Orientation interpolation in quaternion space using spherical biarcs. In Proceedings of the Graphics Interface Conference. 24–32.Google Scholar
    Whited, B., Noris, G., Simmons, M., Sumner, R. W., Gross, M., and Rossignac, J. 2010. Betweenit: An interactive tool for tight inbetweening. Comput. Graph. Forum 29, 2, 605–614.Google ScholarCross Ref
    Wonka, P., Wimmer, M., Sillion, F., and Ribarsky, W. 2003. Instant architecture. In ACM SIGGRAPH Papers. ACM, New York, 669–677. Google ScholarDigital Library

ACM Digital Library Publication: