Constraint-Based Surface Processing for Geometric Modeling and Architecture

Abstract

This thesis investigates the application and implementation of geometric constraints to manipulate, approximate, and optimize surfaces for modeling and architecture. In modeling, geometric constraints provide an interface to edit and control the form of a surface. We present a geometry processing framework that enables constraints for positional, metric, and curvature properties anywhere on the surface of a geometric model. Target values for these properties can be specified point-wise or as integrated quantities over curves and surface patches embedded in the shape. For example, the user can draw several curves on the surface and specify desired target lengths, manipulate the normal curvature along these curves, or modify the area or principal curvature distribution of arbitrary surface patches. This user input is converted into a set of non-linear constraints. A global optimization finds the new deformed surface that best satisfies the constraints, while minimizing adaptable measures for metric and curvature distortion that provide explicit control on the deformation semantics. This approach enables flexible surface processing and shape editing operations. In architecture, the emergence of large-scale freeform shapes pose new challenges to the process from design to production. Geometric constraints directly arise from aesthetic, structural, and economical requirements for the fabrication of such structures. A key problem is the approximation of the design surface by a union of patches, so-called panels, that can be manufactured with a selected technology at reasonable cost, while meeting the design intent and achieving the desired aesthetic quality of panel layout and surface smoothness. The production of curved panels is mostly based on molds. Since the cost of mold fabrication often dominates the panel cost, there is strong incentive to use the same mold for multiple panels. Various constraints, such as the limited geometry of mold shapes and tolerances on positional and normal continuity between neighboring panels, have to be considered. We introduce a paneling algorithm that interleaves discrete and continuous optimization steps to minimize production cost while meeting the desired geometric constraints and is able to handle complex arrangements with thousands of panels. The practical relevance of our system is demonstrated by paneling solutions for real, cutting-edge architectural freeform design projects.