Curve intersection using hybrid clipping

Keywords: Bé zier curve Curve intersection Bé zier clipping Hybrid clipping a b s t r a c t This paper presents a novel approach, called hybrid clipping, for computing all intersections between two polynomial Bé zier curves within a given parametric domain in the plane. Like Bé zier clipping, we compute a 'fat line' (a region along a line) to bound one of the curves. Then we compute a 'fat curve' around the optimal low degree approximation curve to the other curve. By clipping the fat curve with the fat line, we obtain a new reduced subdomain enclosing the intersection. The clipping process proceeds iteratively and then a sequence of subdomains that is guaranteed to converge to the corresponding intersection will be obtained. We have proved that the hybrid clipping technique has at least a quadratic convergence rate. Experimental results have been presented to show the performance of the proposed approach with comparison with Bé zier clipping. Computing intersections of two curves has played an important role in many engineering fields, including computer-aided design and manufacturing (CAD/CAM), collision detection, and geometric modeling [1–3]. If the two curves are parametric, the solution is identified by the parameter values of intersection points. Early approaches include the Bé zier subdivision algorithm [4], the interval subdivision method [5], and implicitization [6]. One widely used and robust method is Bé zier clipping, which was developed by [7]. It utilizes the convex hull property of Bé zier curves and proceeds as clipping away regions of the curves that are guaranteed to not intersect. Recently, [8] proved that Bé zier clipping has a quadratic convergence rate. Bé zier clipping can be applied to compute roots of polynomials as well. By extending Bé zier clipping, [9] developed the quadratic clipping technique to compute all the roots of a univariate polynomial equation within an interval. The basic idea is to generate a strip bounded by two quadratic polynomials which encloses the original polynomial via degree reduction. Combined with subdivision, quadratic clipping generates a sequence of intervals that contain the roots of the original polynomial. Recently, [10] extended this technique to cubic clipping and more general cases. Theoretical and experimental results have shown that both quadratic clipping and cubic clipping have at least quadratic convergence rates. In many cases intersection algorithms involve numerical methods for solving systems of bivariate polynomial equations system [11]. Ref. [12] presented an algorithm for …