Efficient merging of multiple segments of Bézier curves

Abstract

a b s t r a c tThis paper deals with the merging problem of segments of a composite Bézier curve, with the endpoints continuity constraints. We present a novel method which is based on the idea of using constrained dual Bernstein polynomial basis (Woźny and Lewanowicz, 2009) [12] to compute the control points of the merged curve. Thanks to using fast schemes of evaluation of certain connections involving Bernstein and dual Bernstein polynomials, the complexity of our algorithm is significantly less than complexity of other merging methods.

Merging of Bézier curves with box constraints

“…. , D s efficiently, with the complexity O(sm 2 ), using [13,Algorithm 4.1]. Observe that the direct use of (4.2) results in the complexity O(sm 3 ).…”

“…In this section, we apply our method to the composite Bézier curves in R 2 . As in [13], we generalize the approach of [8] and obtain a partition of the interval [t 0 , t s ] = [0, 1] according to the lengths of segments P i :…”

“…However, such an approach increases the error of the approximation as well as the computational cost (see [10, §1]). There are three methods that specialize in merging of more than two Bézier curves at the same time (see [1,10,13]). Regardless of how many curves are merged, the most frequently used strategy is to solve a system of normal equations (see, e.g., [10]).…”

“…Regardless of how many curves are merged, the most frequently used strategy is to solve a system of normal equations (see, e.g., [10]). In [13], one can observe a different approach which is based on the properties of the so-called constrained dual Bernstein basis polynomials (to our knowledge, this method is the fastest one available). The parametric (see, e.g., [1,10,13]) or geometric (see, e.g., [8,10,14]) continuity at the endpoints is preserved.…”

Differential-recurrence properties of dual Bernstein polynomials

“…It is worth noticing that dual Bernstein polynomials introduced in [14], which are associated with the shifted Jacobi inner product, have recently found many applications in numerical analysis and computer graphics (curve intersection using Bézier clipping, degree reduction and merging of Bézier curves, polynomial approximation of rational Bézier curves, etc.). Note that skillful use of these polynomials often results in less costly algorithms which solve some computational problems (see [2,7,8,16,17,21,23,24]).…”