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Ryu, JoongHyun,Cho, Dongsoo,Cho, Youngsong,Kim, Deok-Soo 한국경영과학회 1999 한국경영과학회 학술대회논문집 Vol.- No.1
Intersection problem occurs in various engineering application areas, such as CAD/CAM, GIS, computer graphics, etc. Most of all, intersection algorithms are fundamental to CAD/CAM. Parametric curves have been frequently used in CAGD and thus intersection algorithm between parametric curves been studied intensively in several respects such as the speed, the robustness and the efficiency. Although many intersection algorithms have been published, there exists no algorithm that is satisfactory in all the above three aspects. The intersection techniques that appear in the literature can be classified into three categories; Newton-Raphson iteration method, subdivision method and implicitization method. Newton-Raphson iteration-wise method shows a good convergence rate in case that a good initial seed is given. Otherwise, it provides a wrong solution or diverges. Bezier clipping algorithm copes with intersection problem like an intelligent Newton method. Though it is faster than Implicitization algorithm and Interval subdivision for curve of degree less than 5. Intersection algorithm based on subdivision method divides the original intersection problem into easier ones and then conquers the each divided problem. Be´zier subdivision and interval subdivision algorithm is included in this category. Implicitization method transforms intersection problem to the problem of finding a single polynomial root by substitution a parametric from curve into the implicitized curves. This approach is known to be fastest in computing the intersections between curves of degrees less than quintics. In this paper, an algorithm for intersecting Bezier curves is provided and is extended to an algorithm for intersections between NURBS curves. The algorithm characterized both curves to be intersected and approximates them in lower degree curves. Implicitization technique is applied to the intersections between approximated low degree curves for locating initial solution. Then a good initial solution is obtained and Newton-Raphson iteration converge to a true intersection quickly abs robustly. Tangential case overlapping case are not considered in the pro^+posed algorithm.