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.
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: