An Upper Bound of the Bezout Number for Piecewise Algebraic Curves over a Rectangular Partition

A piecewise algebraic curve is a curve defined by the zero set of a bivariate spline function. Given two bivariate spline spaces (Δ) over a domainDwith a partition Δ, the Bezout number BN(m,r;n,t;Δ) is defined as the maximum finite number of the common intersection points of two arbitrary piecewise algebraic curves (Δ). In this paper, an upper bound of the Bezout number for piecewise algebraic curves over a rectangular partition is obtained.

An Algorithm for Isolating the Real Solutions of Piecewise Algebraic Curves

The piecewise algebraic curve, as the set of zeros of a bivariate spline function, is a generalization of the classical algebraic curve. In this paper, an algorithm is presented to compute the real solutions of two piecewise algebraic curves. It is primarily based on the Krawczyk-Moore iterative algorithm and good initial iterative interval searching algorithm. The proposed algorithm is relatively easy to implement.