|Location:||MSRI: Simons Auditorium|
Geodesics on the group of rotations SO(3) manifest in disparate fields ranging from the kinematics of the human eye to rigid body dynamics to computer graphics. In kinematics of the human eye, where the motion of the eye is assumed to be subject to Listings law, they manifest as motions of constant angular velocity rotations which can be used to interpret saccadic motions of the eye. In computer graphics, they appear as great circles on the 3-sphere that are used as the basis for realistic interpolations using Shomake's SLERP algorithm. In rigid body dynamics, geodesics on SO(3) can be used to provide the simplest realization of the dependency of the geodesic on the metric used for the configuration manifold.
In this talk, we present a quaternion-based treatment of geodesics on SO(3). We find a simple set of differential equations that characterize these motions. The solutions to these integrable equations are readily interpreted as great circles on the 3-sphere. We also show how they can be projected onto Steiner's Roman Surface using a transformation developed by Apery. Applying our results to the human eye shows some remarkable consequences of Listing's law for the dynamics of this system.
The work presented in this talk is based on two papers that were coauthored with Alyssa Novelia: one appeared in Nonlinear Dynamics (https://doi.org/10.1007/s11071-015-1945-0) and the other in Regular and Chaotic Dynamics (https://doi.org/10.1134/S1560354715060088)
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Oliver M. O’Reilly is a professor in the Department of Mechanical Engineering at the University of California at Berkeley. His research and teaching feature a wide range of problems in the dynamics of mechanical systems. He received his B.E. in Mechanical Engineering from the National University of Ireland, Galway (NUIG). Subsequently, he received his M.S. and Ph.D. degrees in Theoretical and Applied Mechanics from Cornell University. O’Reilly has received multiple teaching awards, including the Distinguished Teaching Award from U.C. Berkeley, published over 90 archival journal articles, written three textbooks and is a co-inventor on two patents. His latest book, coauthored with Alyssa Novelia and Khalid Jawed is a Primer on the Kinematics of Discrete Elastic Rods (Springer, 2018).