AN ENERGY-BASED FRAMEWORK FOR CONTACT-AWARE MESH DEFORMATION IN CREATIVE SOFTWARE

Abstract

This thesis develops a customizable framework for a class of mesh deformation problems increative software. In particular, it concerns problems where a mesh, here called the conforming surface or conforming mesh, is to be deformed in a way that (i) respects geometric contact or near-contact constraints defined by a fixed target surface and (ii) preserves the character of the mesh in a way that can be tuned by a set of interpretable parameters. This fixed target surface is represented by a higher-order algebraic discrete signed distance function (ADSDF), while the conforming mesh carries Euclidean vertex positions together with auxiliary unit-quaternion rotational data. We refer to this process as retargeting. Retargeting is formulated as the minimization of an energy built from signed-distance evaluations and their derivatives, combining contact-aware energy terms with an energy term inspired by the as-rigid-as-possible (ARAP) technique [1] that preserves designer intent by promoting near-isometric deformations. Optional terms for normal alignment, conformity to prescribed geometric landmarks, smoothing, and related constraints can be added. The resulting optimization problem is posed intrinsically on a product space of Euclidean and manifold-valued variables. The thesis makes three main contributions. First, it organizes mesh retargeting against signed distance targets as a customizable energy optimization framework – the choice of energy terms determines the character of the final retargeted mesh. Second, it proves a local theorem for ADSDFs: on regular regions, least-squares (LSQ) polynomial samples inherit the same approximation order as Taylor polynomial samples, providing a mathematical justification for the practically important LSQ construction given by Valasek and Bán [2]. Third, it extends subdivision to manifold valued data by replacing affine averages with Karcher means and proves continuous convergence by proximity to the underlying linear scheme. The framework is illustrated through problem formulations for fitting a shirt around an aligned body, pose-retargeting a shirt with landmarked geometric features, and armature-driven contact resolution against a synthetic obstacle field. Together these examples show how higher-order signed-distance fields, customizable energies, and intrinsic optimization can be combined to preserve design intent while resolving geometric constraints.