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Friday, September 6, 2019 at 3:30pm
Geodesics on the group of rotations SO(3) can be found in fields ranging from optometry to rigid body dynamics and computer graphics. In kinematics of the human eye, where the motion of the eye is assumed to be subject to Listings law, they appear as rotations of constant angular velocity that can be used to interpret saccadic motions of the eye. In computer graphics, they present 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 colloquium talk, we present joint work with Alyssa Novelia on a quaternion-based treatment of geodesics on SO(3). We find a remarkably 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.
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).