Weaving Kirchhoff Ribbons
Wednesday, June 26, 2019 - 10:30am - 11:30am
Humans have been using weaving to construct baskets, mats, and other two and three-dimensional aspects of the built environment for at least 10,000 years; weaving remains attractive today as a fabrication technique thanks to its cost-effectiveness and the increasing capability of automated looms, with applications in architecture, art, chemical engineering, and medicine. A weave can be viewed as a coupled network of thin elastic ribbons, with a strong connection between the physics of the rod network and the geometry of the woven surface. I will discuss how we discretize the kinematics and statics of Kirchhoff ribbons to simulate the behavior of woven structures, and several new variational algorithms for designing a weave of a given target shape. Our key insight is that the ribbon layout is determined by a geodesic foliation---a foliation whose leaves are all approximately geodesic curves---on a sixfold branched cover of the target surface, and that finding such a foliation can be posed as a discrete vector field design problem.