Footpoint distance as a measure of distance computation between curves and surfaces

In automotive domain, CAD models and its assemblies are validated for conformance to certain design requirements. Most of these design requirements can be modeled as geometric queries, such as distance to edge, planarity, gap, interference and parallelism. Traditionally these queries are made in discrete domain, such as a faceted model, inducing approximation. Thus, there is a need for modeling and solving these queries in the continuous domain without discretizing the original geometry. In particular, this work presents an approach for distance queries of curves and surfaces, typically represented using NURBS. Typical distance problems that have been solved for curves/surfaces are the minimum distance and the Hausdorff distance. However, the focus in the current work is on computing corresponding portions (patches) between surfaces (or between a curve and a set of surfaces) that satisfy a distance query. Initially, it was shown that the footpoint of the bisector function between two curves can be used as a distance measure between them, establishing points of correspondence. Curve portions that are in correspondence are identified using the antipodal points. It is also identified that the minimum distance in a corresponding pair is bound by the respective antipodal points. Using the established footpoint distance function, the distance between two surfaces was approached. For a query distance, sets of points satisfying the distance measure are identified. The boundary of the surface patch that satisfies the distance is computed using the alpha-shape in the parametric space of the surface. Islands contributing to the distance query are also then computed. A similar approach is then employed for the distance between a curve and a set of surfaces. Initially, the minimum footpoint distance function for a curve to a surface is computed and repeated for all other surfaces. A lower envelope then gives the portions of the curves where the distance is more than the query. (C) 2013 Elsevier Ltd. All rights reserved.