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Intersections in CAD
When using a CAD-system for designing geometric objects, you'll find inside the CAD-system extensive use of intersection algorithms to determine which parts of (curves and) surfaces describe the shells of the volume, and to establish the correct relation between points, curves and surfaces used in the description.

The shells of the object designed are described by composition of surfaces. Sometimes it is possible to define the objects by rectangular surfaces (NURBS) where the surfaces just share common boundaries and there is no need for trimming away parts of a surface. However, often the surfaces will be too large to be able to represent a desired shape and to design it with the constructive tools of the CAD-system. The curve where the surfaces meet will be an edge in the CAD-model to be found by intersecting the surfaces. For representing the relationships between surfaces, curves and surfaces in CAD-models, boundary structures are used.

Low quality of intersection algorithms expensiv for industry

In the Workshop on Mathematical Foundations of CAD (Mathematical Sciences

Research Institute, Berkeley, CA. June 4-5, 1999.) the consensus was that: “The single greatest cause of poor reliability of CAD systems is lack of topologically consistent surface intersection algorithms.” Tom Peters, Computer Science and Engineering, The University of Connecticut, estimated the cost to be $1 Billion/year. For more information consult:

Near singular intersection and non-regular parametrization an intersection challenge

Calculation of intersections is not difficult when the surfaces have a regular parameterization and are not near parallel along intersection curves (transversal intersection). However, if the parameterization is not regular, or if the surfaces intersect in regions where the surfaces are parallel or near parallel, intersection calculation gets challenging. One ambition of the GAIA II project has been to provide more accurate solutions for such intersections than what is currently available.

Self-intersections are a challenge

Another challenge within CAD has been to avoid self-intersections in the shells of the volume described. These self-intersections are of different types:

  • A shell self-intersection is an intersection between different surfaces in the shell of the volume, with the intersection being no part of the adjacency description of the shell.
  • An open self-intersection is a self-intersection that occurs inside a parametric surface and the self-intersection curve does not describe a loop in the parameter domain of the surface. These can be found by first subdividing the surface into smaller pieces that themselves do not have closed self-intersection. The pieces can then be intersected.
  • A closed self-intersection is a self-intersection that occurs inside a parametric surface, where the intersection curve describes a loop in the parameter domain of the surface.

Published June 7, 2005

Example: Transversal intersection.

Example: Near singular (tangential) intersection.

Example: Open self intersection