Approximate implicitization and recursive surface intersection algorithms

Abstract

Most published work on intersection algorithms for Computer Aided Design (CAD) systems addresses transversal intersections [1], situations where the surface normals of the surfaces intersected are well separated along all intersection curves. For transversal intersections the divide and conquer strategy of recursive subdivision, Sinha's theorem [2] and the convex hull property of NonUniform Rational B-Spline surfaces (NURBS) efficiently identify all intersection branches. However, in singular or near singular intersections, situations where the surfaces are parallel or near parallel in an intersection region, along an intersection curve or in an intersection point, even deep levels of subdivision will frequently not sort out the intersection topology. The paper will focus on the novel approach of Approximate Implicitization to address these challenges.

An Evolution-Based Approach for Approximate Parameterization of Implicitly Defined Curves by Polynomial Parametric Spline Curves

“…In particular, approximate techniques for implicitization [9,10,11,15,16,17,24,36] and parameterization [3,4,5,22,30] may be useful for applications in Computer Aided Design, since they are able to deal with input specified by floating point numbers and they are more flexible than exact, symbolic-computation based methods (see, e.g., [12,13,33]). In addition, approximate techniques can be adapted so as to produce a representation within a certain region of interest, which is a very natural requirement in applications.…”