Linear Time Super-Helices

Abstract

Thin elastic rods such as cables, phone coils, tree branches, or hair, are common objects in the real world but computing their dynamics accurately remains challenging. The recent Super-Helix model, based on the discrete equations of Kirchhoff for a piecewise helical rod, is one of the most promising models for simulating non-stretchable rods that can bend and twist. However, this model suffers from a quadratic complexity in the number of discrete elements, which, in the context of interactive applications, makes it limited to a few number of degrees of freedom - or equivalently to a low number of variations in curvature along the mean curve. This paper proposes a new, recursive scheme for the dynamics of a Super-Helix, inspired by the popular algorithm of Featherstone for serial multibody chains. Similarly to Featherstone s algorithm, we exploit the recursive kinematics of a Super-Helix to propagate elements inertias from the free end to the clamped end of the rod, while the dynamics is solved within a second pass traversing the rod in the reverse way. Besides the gain in linear complexity, which allows us to simulate a rod of complex shape much faster than the original approach, our algorithm makes it straightforward to simulate tree-like structures of Super-Helices, which turns out to be particularly useful for animating trees and plants realistically, under large displacements.