Intersection points algorithm for piecewise algebraic curves based on Groebner bases.(English)Zbl 1176.65019

Summary: A piecewise algebraic curve is defined as the zero set of a bivariate spline. In this paper, we mainly study the intersection points algorithm for two given piecewise algebraic curves based on Groebner bases. Given a domain $$D$$ and a partition $$\Delta$$, we present a flow and introduce the truncated signs, and then represent the two piecewise algebraic curves in the global form. We get their Groebner bases with respect to a lexicographic order and adopt the interval arithmetic in the back-substitution process, which makes the algorithm numerically precise. An example is also presented to show the algorithm’s feasibility and effectiveness.

MSC:

 65D18 Numerical aspects of computer graphics, image analysis, and computational geometry 65D07 Numerical computation using splines 14Q05 Computational aspects of algebraic curves 68W25 Approximation algorithms 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 65G30 Interval and finite arithmetic
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