Oliver Gross
Skip Abstract Section
Skip Supplemental Material SectionAbstract
We consider motion effected by shape change. Such motions are ubiquitous in nature and the human made environment, ranging from single cells to platform divers and jellyfish. The shapes may be immersed in various media ranging from the very viscous to air and nearly inviscid fluids. In the absence of external forces these settings are characterized by constant momentum. We exploit this in an algorithm which takes a sequence of changing shapes, say, as modeled by an animator, as input and produces corresponding motion in world coordinates. Our method is based on the geometry of shape change and an appropriate variational principle. The corresponding Euler-Lagrange equations are first order ODEs in the unknown rotations and translations and the resulting time stepping algorithm applies to all these settings without modification as we demonstrate with a broad set of examples.
Supplemental Material
Available for Download
zip
papers_572-supplemental.zip (86.1 MB)
supplemental material
- Alexander M. Bronstein, Michael M. Bronstein, and Ron Kimmel. 2008. Numerical Geometry of Non-rigid Shapes. Springer.Google Scholar
Digital Library
- Fang Da, David Hahn, Christopher Batty, Chris Wojtan, and Eitan Grinspun. 2016. Surface-only Liquids. ACM Trans. Graph. 35, 4 (2016), 78:1--12.Google ScholarDigital Library
- Daniel Daye. 2019. Rigged and Animated Scan of Timber Rattlesnake. https://sketchfab.com/. CC Attribution-NonCommercial.Google Scholar
- J. Elgeti, R. G. Winkler, and G. Gompper. 2015. Physics of Microswimmers---Single Particle Motion and Collective Behavior: a Review. Rep. Prog. Phys. 78 (2015).Google Scholar
- Leonhard Euler. 1744. Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes. Bousquet, Lausanne & Geneva. English translation at WikiSource.Google Scholar
- Cliff Frohlich. 1979. Do Springboard Divers Violate Angular Momentum Conservation? Amer. J. Phys. 47, 7 (1979), 583--592.Google ScholarCross Ref
- Cliff Frohlich. 1980. The Physics of Somersaulting and Twisting. Sci. Am. 242, 3 (1980), 154--165.Google ScholarCross Ref
- Graphics & Extended Reality Lab. 2022. Large Whip Snake. https://sketchfab.com/. CC Attribution.Google Scholar
- J. Gray and G. J. Hancock. 1955. The Propulsion of Sea-Urchin Spermatozoa. J. Exp. Biol. 32, 4 (1955), 802--814.Google ScholarCross Ref
- Jeffrey S. Guasto, Jonathan B. Estrada, Filippo Menolascina, Lisa J. Burton, Mohak Patel, Christian Franck, A. E. Hosoi, Richard K. Zimmer, and Roman Stocker. 2020. Flagellar Kinematics Reveals the Role of Environment in Shaping Sperm Motility. J. Roy. Soc. Interface 17, 20200525 (2020), 10 pages.Google ScholarCross Ref
- Ernst Hairer and Gerhard Wanner. 2015. Euler Methods, Explicit, Implicit, Symplectic. In Encyclopedia of Applied and Computational Mathematics, Björn Engquist (Ed.). Springer, 451--455.Google Scholar
- David L. Hu, Jasmine Nirody, Terri Scott, and Michael J. Shelley. 2009. The Mechanics of Slithering Locomotion. Proc. Nat. Acad. Sci. 106, 25 (2009), 10081--10085.Google ScholarCross Ref
- E. Ju, J. Won, J. Lee, B. Choi, J. Noh, and M. Gyu Choi. 2013. Data-Driven Control of Flapping Flight. ACM Trans. Graph. 32, 5 (2013), 151:1--12.Google ScholarDigital Library
- T. R. Kane and M. P. Scher. 1969. A Dynamical Explanation of the Falling Cat Phenomenon. Int. J. Solids Structures 5 (1969), 663--670.Google ScholarCross Ref
- Eva Kanso, Jerrold E. Marsden, Clancy W. Rowley, and J. B. Melli-Huber. 2005. Locomotion of Articulated Bodies in a Perfect Fluid. J. Non-L. Sci. 15 (2005), 255--289.Google ScholarCross Ref
- Gustav Kirchhoff. 1870. Ueber die Bewegung eines Rotationskörpers in einer Flüssigkeit. J. Reine Angew. Math. 1870, 71 (1870), 237--262.Google ScholarCross Ref
- Gustav Kirchhoff. 1876. Vorlesungen über mathematische Physik. Teubner, 233--250.Google Scholar
- V. V. Kozlov and S. M. Ramodanov. 2001. The Motion of a Variable Body in an Ideal Fluid. J. Appl. Math. Mech. 65, 4 (2001), 579--587.Google ScholarCross Ref
- Philip V. Kulwicki and Edward J. Schlei. 1962. Weightless Man: Self-Rotation Techniques. Technical Report AMRL-TDR-62-129. Beh. Sci. Lab., Wright-Patterson AFB.Google Scholar
- V. M. Kuznetsov, B. A. Lugovtsov, and Y. N. Sher. 1967. On the Motive Mechanism of Snakes and Fish. Arch. Rat. Mech. 25 (1967), 367--387.Google ScholarCross Ref
- L. D. Landau and E. M. Lifshitz. 1976. Mechanics (third ed.). Course of Theoretical Physics, Vol. 1. Butterworth Heinemann.Google Scholar
- Eric Lauga and Thomas R. Powers. 2009. The Hydrodynamics of Swimming Microorganisms. Rep. Prog. Phys. 72, 096601 (2009), 36pp.Google Scholar
- Michael Lentine, Jon Tomas Gretarsson, Craig Schroeder, Avi Robinson-Mosher, and Ronald Fedkiw. 2011. Creature Control in a Fluid Environment. IEEE Trans. Vis. Comp. Graph. 17, 5 (2011), 682--693.Google ScholarDigital Library
- P. Liljebäck, Ky. Y. Pettersen, Ø. Stavdahl, and J. T. Gravdahl. 2012. A Review on Modeling, Implementation, and Control of Snake Robots. Rob. Aut. Syst. 60, 1 (2012), 29--40.Google ScholarDigital Library
- M. Marey. 1894. Photographs of a Tumbling Cat. Nature 51 (1894), 80--81.Google ScholarCross Ref
- Jerrold E. Marsden and Matthew West. 2001. Discrete Mechanics and Variational Integrators. Act. Num. 10 (May 2001), 357--514.Google Scholar
- Gavin S. Miller. 1988. The Motion Dynamics of Snakes and Worms. In Proc. ACM/SIGGRAPH Conf. ACM, 169--173.Google ScholarDigital Library
- Sehee Min, Jungdam Won, Seunghwan Lee, Jungnam Park, and Jehee Lee. 2019. SoftCon: Simulation and Control of Soft-Bodied Animals with Biomimetic Actuators. ACM Trans. Graph. 38, 6 (2019), 208:1--12.Google ScholarDigital Library
- R. Montgomery. 1993. Dynamics and Control of Mechanical Systems; The Falling Cat and Related Problems. Number 1 in Fields Inst. Commun. Fields Institute, Chapter Gauge Theory of the Falling Cat, 193--218.Google Scholar
- Emmy Noether. 1918. Invariante Variationsprobleme. Nachr. König. Ges. Wiss. Math. Phys. Klasse (1918), 235--257. Engl. translation https://arxiv.org/abs/physics/0503066..Google Scholar
- Jim Ostrowski and Joel Burdick. 1998. The Geometric Mechanics of Undulatory Robotic Locomotion. I. J. Rob. Res. 17, 7 (1998), 683--701.Google ScholarCross Ref
- Marcel Padilla, Albert Chern, Felix Knöppel, Ulrich Pinkall, and Peter Schröder. 2019. On Bubble Rings and Ink Chandeliers. ACM Trans. Graph. 38, 4 (2019), 129:1--14.Google ScholarDigital Library
- Louis Poinsot. 1851. Théorie Nouvelle de la Rotation des Corps. Bachelier.Google Scholar
- E. M. Purcell. 1977. Life at Low Reynolds Number. Amer. J. Phys. 45, 3 (1977), 3--11.Google ScholarCross Ref
- Michel Rieutord. 2015. Fluid Dynamics: An Introduction. Springer.Google ScholarCross Ref
- Perrin E. Schiebel, Jennnifer M. Reiser, Alex M. Hubbard, Lillian Chen, D. Zeb Rocklin, and Daniel I. Goldman. 2019. Mechanical Diffraction Reveals the Role of Passive Dynamics in a Slithering Snake. Proc. Nat. Acad. Sci. 116, 11 (2019), 4798--4803.Google ScholarCross Ref
- Alfred Shapere and Frank Wilczek. 1989a. Geometry of Self-Propulsion at Low Reynolds Number. J. Fluid Mech. 198 (1989), 557--585.Google ScholarCross Ref
- Alfred Shapere and Frank Wilczek. 1989b. Gauge Kinematics of Deformable Bodies. Amer. J. Phys. 57, 6 (1989), 514--518.Google ScholarCross Ref
- Hon. J. W. Strutt, M. A. (Lord Rayleigh). 1871. Some General Theorems Relating to Vibrations. Proc. Lond. Math. Soc. s1-4, 1 (1871), 357--368.Google ScholarCross Ref
- Jie Tan, Yuting Gu, Greg Turk, and C. Karen Liu. 2011. Articulated Swimming Creatures. ACM Trans. Graph. 30, 4 (2011), 58:1--12.Google ScholarDigital Library
- Xiaoyuan Tu and Demetri Terzopoulos. 1994. Artificial Fishes: Physics, Locomotion, Perception, Behavior. In Proc. ACM/SIGGRAPH Conf. 43--50.Google ScholarDigital Library
- Steven Vogel. 1983. Life in Moving Fluids (2nd ed.). Princeton University Press.Google Scholar
- Hermann von Helmholtz. 1882. Zur Theorie der stationären Ströme in reibenden Flüssigkeiten. In Wissenschaftliche Abhandlungen. Vol. I. J. A. Barth, 223--230.Google Scholar
- Barlomiej Waszak. 2018. Limbless Movement Simulation with a Particle-Based System. Comp. Anim. Virt. Worlds 29, 2 (2018), 1--21.Google Scholar
- Steffen Weißmann and Ulrich Pinkall. 2012. Underwater Rigid Body Dynamics. ACM Trans. Graph. 31, 4 (2012), 104:1--7.Google ScholarDigital Library
- Wayne L. Wooten and Jessica K. Hodgins. 1996. Animation of Human Diving. Comp. Graph. Forum 15, 1 (1996), 3--13.Google ScholarCross Ref
- Jia-Chi Wu and Zoran Popović. 2003. Realistic Modeling of Bird Flight Animations. ACM Trans. Graph. 22, 3 (2003), 888--895.Google ScholarDigital Library
Index Terms
- Motion from Shape Change
Recommendations
Geometric modeling in shape space
SIGGRAPH '07: ACM SIGGRAPH 2007 papersWe present a novel framework to treat shapes in the setting of Riemannian geometry. Shapes -- triangular meshes or more generally straight line graphs in Euclidean space -- are treated as points in a shape space. We introduce useful Riemannian metrics ...
Geometric modeling in shape space
We present a novel framework to treat shapes in the setting of Riemannian geometry. Shapes -- triangular meshes or more generally straight line graphs in Euclidean space -- are treated as points in a shape space. We introduce useful Riemannian metrics ...
Visualization of Shape Motions in Shape Space
Analysis of dynamic object deformations such as cardiac motion is of great importance, especially when there is a necessity to visualize and compare the deformation behavior across subjects. However, there is a lack of effective techniques for ...
Comments