Discrete elastic rods

We present a discrete treatment of adapted framed curves, parallel transport, and holonomy, thus establishing the language for a discrete geometric model of thin flexible rods with arbitrary cross section and undeformed configuration. Our approach differs from existing simulation techniques in the graphics and mechanics literature both in the kinematic description—we represent the material frame by its angulardeviationfromthenaturalBishopframe— as well as in the dynamical treatment—we treat the centerline as dynamic and the material frame as quasi static. Additionally, we describe a manifold projection method for coupling rods to rigidbodiesandsimultaneouslyenforcingrodinextensibility. Theuse of quasi statics and constraints provides an efficient treatment for stiff twisting and stretching modes; at the same time, we retain the dynamic bending of the centerline and accurately reproduce the coupling between bending and twisting modes. We validate the discrete rod model via quantitative buckling, stability, and coupled-mode experiments, and via qualitative knot-tying comparisons.