The idea of studying the Frenet-Serret equations as a control system goes back to the seventies and perhaps earlier. From this vantage point, they constitute a fundamental example of a control system on a Lie group. In this talk we consider these and other moving frames adapted to trajectories of a collective of unit speed particles. We discuss the geometry of interactions between the adapted frames via coupling laws between the frame invariants (curvatures). We show that there exist interesting choices of such coupling laws that cause coherent bundling of the curves. We also discuss extensions of such laws to include non-collisional interactions of particles with surfaces. We suggest some possible applications in modeling shape deformations in imaging science.
This is joint work with Eric Justh, and Fumin Zhang.
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