Ribbons and groups: a thin rod theory for catheters and filaments

We use the rotation group and its algebra to provide a novel description of deformations of special Cosserat rods or thin rods that have negligible sh ear. Our treatment was motivated by the problem of the simulation of cathet er navigation in a network of blood vessels, where this description is dire ctly useful. In this context, we derive the Euler differential equations th at characterize equilibrium configurations of stretch-free thin rods. We ap ply perturbation methods, used in time-dependent quantum theory, to the thi n rod equations to describe incremental deformations of partially constrain ed rods. Further, our formalism leads naturally to a new and efficient fini te element method valid for arbitrary deformations of thin rods with neglig ible stretch, Associated computational algorithms are developed and applied to the simulation of catheter motion inside an artery network.