A New Extension of the Cubic Bézier Curve

Abstract

An extended cubic Bézier curve is constructed, which can realize <italic>FC</italic>³ continuity under the condition of <italic>C</italic>² continuity, and ensure that the shape of both single-segment curve and composite curve can be freely adjusted without changing the control points. For this reason, a new curve with the same structure as the cubic Bézier curve is constructed. Firstly, the preliminary expression of the blending function is given, including undetermined coefficients. Then, according to the preset properties of the curves during joining, the endpoint properties of the blending function are deduced in reverse, so as to establish the equations that the undetermined coefficients in the blending function should satisfy. By solving the equations, a set of quintic polynomial blending functions with four parameters is obtained, which can be reduced to quartic when special parameters are taken. At last, a new curve with four shape parameters determined by four control points is defined by linear combination of the blending function and the control points. It has the basic properties of Bézier curve, such as convex hull property, geometric invariance and affine invariance. Thanks to the introduction of multiple shape parameters, curves of different shapes can be defined by the same control polygon, and the change of each shape parameter will drive the points on the curve to move linearly along a fixed direction. When constructing composite curve, the <italic>FC</italic>³ continuity conditions between adjacent curve segments are the same as the <italic>C</italic>² continuity conditions, and the shape of composite curve can be adjusted freely on the premise of maintaining continuity, control points and parameter segmentation unchanged. The new method can expand the processing ability of CAD system to industrial products, improve the speed of product design, modification and optimization, and save the investment of manpower and material resources in design.