Functional optimization for fair surface design

Abstract

This paper presents a simple-to-use mechanism for the creation of complex smoothly shaped surfaces of any genus or topological type. The surfaces are specified through interpolated geometric constraints consisting of positions and, optionally, surface normals and surface curvatures. From a designer's point of view, this is a very natural way to specify a desired shape, whether free-form or technical. Nonlinear optimization techniques are then used to minimize a fairness functional based on the variation of curvature. This functional produces very high quality surfaces with predictable, intuitive behavior, while generating, where possible, simple shapes, such as cylinders, spheres, or tori, which are commonly used in geometric modeling. While easy to use, this optimization-based approach is computationally quite demanding. With more efficient optimization algorithms and with the ever increasing processing power available on every desk-top, the techniques described here will provide the basis for a new class of practical interactive geometric modeling tools.