一种新的曲线曲面

A New Kind of Curves and Surfaces

给出了由任意n(n≥3)个函数构成的混合函数组,这些函数组具有非负性、规范性、对称性,以及特殊的端点性质。由这些函数组定义的曲线具有凸包性、几何不变性、对称性等基本性质。曲线的起点、终点分别为控制多边形首、末边的中点,曲线在起点处的一阶、二阶导矢都平行于控制多边形的首边,在终点处的一阶、二阶导矢都平行于控制多边形的末边。对于任意给定的m(m?3)个控制顶点,可以由之定义一条曲线段,也可以由之定义由多条曲线段构成的组合曲线,而各条曲线段可以由不同数量的控制顶点来定义,因此由同一组控制顶点可以定义出多种不同的形状。另外,组合曲线在分段连接点处均G2连续,可以满足工程实际中大多数的需求。由函数组定义的张量积曲面具有类似于曲线的诸多良好性质。

Some groups of blending functions consisting of n （n ≥ 3） functions are given. They have non-negativity, normalization, symmetry and special endpoint property. The curves defined by them possess convex hull property, geometric invariance, symmetry. The starting and ending points of the curves are the midpoints of the first and last edges of their control polygons. At the starting points of the curves, both the first order and the second order derivative vectors are parallel to the first edge of the control polygon. And at the ending points, the first order and the second order derivative vectors are parallel to the last edge of the control polygon. Given m （rn 〉 3） control points, we define one curve segment, and a composite curve consists of several segments. Each segment can be determined by different number of control points. So the same control points can be used to define a variety of shapes. The composite curve is G2 continuous at the piecewise points. The tensor product surfaces defined by the function groups also have many good properties similar to the curves.