Interrogation and direct manipulation of inflection properties of curves and surfaces.

Abstract

While inflection properties of curves are readily understood, inflections on surfaces are not uniquely defined. On a curve, the transition from positive to negative curvature creates an inflection point where the curvature is zero. A surface, however, exhibits a variety of different curvatures at any single surface point. Still, a designer perceives intuitively a surface inflection where a convex region changes into a concave shape. As a first step, this dissertation investigates various curvature distributions on surfaces and introduces a design direction that allows for a unique detection of inflection points related to current design practice. An inflection line is defined as the locus of neighboring inflection points on the surface. In addition, the concept of inflection angles representing asymptotic directions in object space is introduced and an inflection angle map is constructed as a new way to inspect inflection properties of surfaces. Several methods for the numerical calculation and graphical display of inflection properties and inflection lines on surfaces are presented. The algorithms developed are fast and robust and provide the designer with tools for the evaluation of surface smoothness and surface character. Finally, methods that allow for the direct manipulation of inflection properties on curves and surfaces are presented. For a given curve or surface with given inflection properties, a designer can specify a desired modification of an inflection point or an inflection line. The algorithms will automatically adjust the curve or surface in a way that the desired properties are obtained. Considering the compound nonlinearity of the inflection properties, the direct manipulation problems are formulated as minimization problems and solved using an existing optimization technique. These methods provide very robust solutions. The unique contributions of this dissertation are a design-oriented definition of surface inflections and innovative methods for the direct manipulation of inflection properties. The results provide a powerful tool for the design of free-form curves and surfaces.