Geometry of Collective Steering
P. S. Krishnaprasad
University of Maryland
Electrical and Computer Engineering
The idea of studying the Frenet–Serret equations as a control system dates back to the 1970s and perhaps earlier. From this perspective, they provide a fundamental example of a control system on a Lie group. In this talk, we consider these and other moving frames adapted to the trajectories of a collection of unit-speed particles.
We examine the geometry of interactions between the adapted frames through coupling laws relating the frame invariants (curvatures). We show that certain choices of such coupling laws lead to coherent bundling of the curves. Extensions of these laws to include non-collisional interactions between particles and surfaces are also discussed.
Possible applications to modeling shape deformations in imaging science are suggested.
This work is joint with Eric Justh and Fumin Zhang.