Shape Modification forλ-Bézier Curves Based on Constrained Optimization of Position and Tangent Vector

Abstract

Besides inheriting the properties of classical Bézier curves of degreen, the correspondingλ-Bézier curves have a good performance on adjusting their shapes by changing shape control parameter. Specially, in the case where the shape control parameter equals zero, theλ-Bézier curves degenerate to the classical Bézier curves. In this paper, the shape modification ofλ-Bézier curves by constrained optimization of position and tangent vector is investigated. The definition and properties ofλ-Bézier curves are given in detail, and the shape modification is implemented by optimizing perturbations of control points. At the same time, the explicit formulas of modifying control points and shape parameter are obtained by Lagrange multiplier method. Using this algorithm,λ-Bézier curves are modified to satisfy the specified constraints of position and tangent vector, meanwhile the shape-preserving property is still retained. In order to illustrate its ability on adjusting the shape ofλ-Bézier curves, some curve design applications are discussed, which show that the proposed method is effective and easy to implement.