Ken Shoemake
Abstract
Solid bodies roll and tumble through space. In computer animation, so do cameras. The rotations of these objects are best described using a four coordinate system, quaternions, as is shown in this paper. Of all quaternions, those on the unit sphere are most suitable for animation, but the question of how to construct curves on spheres has not been much explored. This paper gives one answer by presenting a new kind of spline curve, created on a sphere, suitable for smoothly in-betweening (i.e. interpolating) sequences of arbitrary rotations. Both theory and experiment show that the motion generated is smooth and natural, without quirks found in earlier methods.
- 1 BEZlER, P.E., Numerical Control Mathematics and Applications, John Wiley and Sons, London 0o72).Google Scholar
- 2 BOEHM, WOLFGANG, "Inserting new knots into }3- upline curves," Computer-Aided Design 12(4)pp. 199-201 (July 1980).Google ScholarCross Ref
- 3 BRADY, MICHAEL, ':Trajectory Planning," in Robot Motion: Planning and Control~ ed. Michael Brady, John M. Hollerb~ch, Timo~,hy L. Hohnson, Tomas Loz~no-Perez and Matthew T. Mason,The MIT Press (1982). Google ScholarDigital Library
- 4 BROU, PHILIPPE, "Using the Gaussian l:mage to Find the Orientation of Objects," The International Journal of Robotics Research 3(4) pp. 89-125 (Winter 1984).Google ScholarCross Ref
- 5 CAYLEY, ARTHUR, "On certain results relating to quaternions," Philosophical Magazine xxvi pp. 141-145 (February 1845).Google Scholar
- 6 COURANT, R. AND tIILBERT, D., Method8 o} Mathematical Physics, Volumv I, Interscience Publishers, Inc., New York (1953).Google Scholar
- 7 DAHLQUIST, GERMUND AND BJORCK, AKE, Numerical Methods, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1974). Translated by Ned Anderson. Google ScholarDigital Library
- 8 EULER, LEONHARD, "Decouverte d'un nouveau principe de m6canique (1752),'* pp. 81-108 in Opera crania, Ser. secunda, v. 5, Orell Fiisii Turici, Lausannae (1957).Google Scholar
- 9 EULER, LEONHARD, "Du mouvement de rotation des corps solides uutour d'un axe v~riable (1758)," in Opera crania, Set. secunda, v. 8, Orelt F iisli Turici, Lausannae ().Google Scholar
- 10 GABRIEL, STEVEN A. AND KAJIYA, JAMES T., "Spline Interpolation in Curved Manifolds," , (1985). SubmittedGoogle Scholar
- 11 GARDNER, MARTIN, New Mathematical Diversions from Scientific American, Fireside, St. Louis, Missouri (1971). Chapter 2Google Scholar
- 12 GOLDSTEIN, HERBERT, Classical Mechanics, second ~ditiou, Addison-Wesley Publishing Company, Inc., Reading, Mus~. (1980). Chapter 4 ~nd Appendix B.Google Scholar
- 13 GORDON, WILLIAM J. haND RIESENFELD, RICHARI~ F., "Bernstein-B~zier methods for the computeraided design of free-form curves and surfaces," J. ACM 21(2) pp. 293-310 (April 1974). Google ScholarDigital Library
- 14 GORDON, WILLIAM J. AND I~IESENFELD, RICtIARD F., "B-spline curves and surfaces," in Compute, Aided Geometric Design, ed. Robert E. Barnh}ll and Richard F. Riesenfeld,Academic Press, New York (1974).Google Scholar
- 15 HAMILTON, SIR WILLIAM ROWAN, "On quaternions; or on a new system of imaginaries in algebra," Philosophical Magazine xxv pp. 1%13 (July 1844).Google Scholar
- 16 HERSTEIN, I.N., Topics in Algebra, second edition, John Wiley and Sons, Inc., New York (1975).Google Scholar
- 17 KANE, THOMAS R., L1KINS, Pt~TER W. AND LEvi}N- SON, DAVID A., Spacecraft Dynamics, McGraw-Hill, ~n~. (10s3).Google Scholar
- 18 LANE, JEFFREY M., CARPENTER, LOREN C., WH|TTED, TURNER, AND BLINN~ JAMES F.~ "Scan line methods for displaying parametrically defined surfaces," Comm. ACM 23(1)pp. 23-34 (January 1980). Google ScholarDigital Library
- 19 MACLANE, SAUNDERS AND BIRKHOFF, GARRETT, Algebra, second edition, Mucmillaii Publishing Co., Inc., New York (1979).Google Scholar
- 20 MISNER, CHARLES W., THORNE, KIP S., AND WHEELER, JOHN ARCHIBALD, Gravitation, W.H. Freeman and Company, San Francisco (1973). Chapter 41 Spinors.Google Scholar
- 21 MITOHEI,L, E.E.L. AND ROGERS~ A.E., "Quaternion Parameters in the Simulation of a. Spinning Rigid Body," in Simulation The Dynamic Modeling of Idea~ and Systems with Computers, ed. John McLeod, P.E., (1988).Google Scholar
- 22 NEWMAN, WRLIAM M. AND SPROULL, ROBERT F., Principles of Interactive Computer Graphics, second edition, McGraw-Hill, Inc., New York (1079). Chapter 21 Curves ~nd surfaces. Google ScholarDigital Library
- 23 PICKERT, G. AND STEINER; H.-G., "Chapter 8 Complex numbers and quaternions," in Fundamentals of Mathematics, Volume I ~ Foundations of Mathematics: The Real Number System and Algebra, ed. H. Behnke, F. Bachrnann, K. Fladt, and W. S{iss, (1983). Translated by S.H. Gould.Google Scholar
- 24 SCIIMEIDLER, W. AND DREETZ, W., "Chapter 11 Functional analysis," in Fundamentals of Mathematics, Volume 11} ~ Analysis, ed. H. Behnke, F. Bachmann, K. Fladt, and W. Sfiss,MlT Press, Cambridge, Muss. (1983). Translated by S.H. Gould.Google Scholar
- 25 SCHOENBERG, I.J., "Contributions to the problem of approxima$ion of equidistant data by analytic functions,'~ Quart. Appf. Math. 4 pp. 45-99 and ,12-141 (1946).Google Scholar
- 26 SMITH, AI,~Y RAY, "Spline tutorial note~," Technical Memo No. 77, Computer Graphics Project, Lucasfilm Ltd. (May 198a).Google Scholar
- 27 S05$, W., GERICKE, H., AND BERGER, K.H., "Chapter 14 ~ Differential geomeery of curves and surfaces," in Fundamentals of Mathematics, Volume II Geometry, ed. l-l.'Behnke, F. Bachmann, K. Fladt, and W. ${iss,M{T Press (1983). Translated by S.H. Gould.Google Scholar
- 28 TAYLOR, RUSSELL H., "Planning and Execution of Straight Line Manipulator Trajectories," in Robot Motion: Planning and Control, ed. Michael Brady, John M. Hollerbach, Timothy L. Hohnson, Tomas Lozano-Perez and Matthew T. Mason,The h/liT Press (1982).Google Scholar
Index Terms
- Animating rotation with quaternion curves
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