Repulsive Energies and their Applications

Functionals that penalize bending or stretching of a surface play a key role in geometric and scientific computing, but to date have ignored a very basic requirement: in many situations, shapes must not pass through themselves or each other. This condition is critical, for instance, when shapes represent physical membranes (e.g. in biological simulation), physical products (e.g. in digital manufacturing), or certain mathematical objects (e.g. isotopy classes of embeddings). This thesis develops a numerical framework for the intersection-free optimization of curves and surfaces. The starting point is the tangent-point energy, a “repulsive energy” that effectively pushes apart pairs of points that are close in space but distant along the domain. We develop discretization of this energy for curves and surfaces, and introduce a novel acceleration scheme based on a fractional Sobolev inner product. We further accelerate this scheme via hierarchical approximation, and describe how to incorporate a variety of constraints (lengths, areas, volumes, etc,). Finally, we explore how this machinery might be applied to problems in mathematical visualization, geometric modeling, and geometry processing.