The bisector of a point and a plane parametric curve
Computer Aided Geometric Design
Geometric constraint solver using multivariate rational spline functions
Proceedings of the sixth ACM symposium on Solid modeling and applications
Precise Voronoi cell extraction of free-form rational planar closed curves
Proceedings of the 2005 ACM symposium on Solid and physical modeling
Geometric computations in parameter space
Proceedings of the 21st spring conference on Computer graphics
A higher dimensional formulation for robust and interactive distance queries
Proceedings of the 2006 ACM symposium on Solid and physical modeling
An Efficient Solution to Systems of Multivariate Polynomial Using Expression Trees
IEEE Transactions on Visualization and Computer Graphics
Critical point analysis using domain lifting for fast geometry queries
Computer-Aided Design
Precise Hausdorff distance computation for planar freeform curves using biarcs and depth buffer
The Visual Computer: International Journal of Computer Graphics
Interior Medial Axis Transform computation of 3D objects bound by free-form surfaces
Computer-Aided Design
Global curve analysis via a dimensionality lifting scheme
IMA'05 Proceedings of the 11th IMA international conference on Mathematics of Surfaces
On the shape of a set of points in the plane
IEEE Transactions on Information Theory
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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 @a-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.