$G^{k,l}$-constrained multi-degree reduction of B\'ezier curves

TL;DR

This paper introduces a new efficient method for multi-degree reduction of Bézier curves with geometric continuity constraints, minimizing least squares error using novel parameter determination techniques.

Contribution

It proposes two methods for determining geometric continuity parameters and provides an efficient algorithm with complexity O(mn) for control point computation.

Findings

The methods effectively minimize least squares error in curve reduction.

The proposed algorithm has lower computational complexity than existing methods.

Examples demonstrate the accuracy and efficiency of the algorithms.

Abstract

We present a new approach to the problem of $G^{k,l}$-constrained ($k,l \leq 3$) multi-degree reduction of B\'{e}zier curves with respect to the least squares norm. First, to minimize the least squares error, we consider two methods of determining the values of geometric continuity parameters. One of them is based on quadratic and nonlinear programming, while the other uses some simplifying assumptions and solves a system of linear equations. Next, for prescribed values of these parameters, we obtain control points of the multi-degree reduced curve, using the properties of constrained dual Bernstein basis polynomials. Assuming that the input and output curves are of degree $n$ and $m$, respectively, we determine these points with the complexity $O(mn)$, which is significantly less than the cost of other known methods. Finally, we give several examples to demonstrate the effectiveness of our algorithms.