保形分段三次多项式曲线的形状分析

构造了一种保形并且形状可调的分段三次多项式曲线，并分析其形状特征与控制多边形之间的关系.首先，通过预设基函数的性质再解方程组，构造了一组带2个形状参数的多项式基函数，其包含三次均匀B样条基函数作为特例.然后，借助基函数与三次Bernstein基函数之间的关系证明了基函数的全正性，由这组基函数定义了一种分段三次多项式曲线，使该曲线拥有一个局部和一个全局形状参数.最后，分析了控制多边形边变量之间的相对位置关系对曲线段形状特征的影响，得到了曲线段拥有1个或2个拐点，1个二重点或1个尖点，为局部凸或全局凸时的充要条件.该结论为曲线段的形状调整提供了理论基础.

Construction and analysis of a new class of shape-preserving piecewise cubic polynomial curves

This paper proposes a new class of shape-preserving piecewise cubic polynomial curves with both local and global shape control parameters. By presetting the properties of its basis functions and then solving equations, a set of polynomial basis functions with two shape parameters are derived, including the cubic uniform B-spline basis functions as a special case. Based on the relationship between the new basis functions and the cubic Bernstein basis functions, the totally positive property of the new basis functions is proved and a new class of piecewise cubic polynomial curves is therefore defined. The effect of the relative position of the control polygons' side vectors onto the shape characteristic of the corresponding curve segments is analyzed. Necessary and sufficient conditions are obtained for the curve segments containing single or double inflection points, a loop or a cusp, or be locally or globally convex, which provide a theoretical guide for adjusting the shape of curve segments.