三次均匀B样条曲线的保形扩展

Conformal extension of cubic uniform B-spline curve

为了在不增加计算复杂度的前提下,构造既具有凸包性又具有保形性的类三次均匀B样条曲线,首先采用逆向思维法,通过预设的曲线性质来反推调配函数的性质,进而计算出调配函数的表达式;然后采用定性分析法,分别讨论当曲线具备凸包性、保单调性、保凸性、变差缩减性时,曲线中参数的取值范围,图例显示了分析结果的正确性。不同情况下所得参数取值范围的交集,即为最终确定的曲线中形状参数的可行域,在可行域内改变形状参数,可以在不破坏曲线保形性的前提下调整曲线对控制多边形的逼近程度。简要讨论了与曲线对应的张量积曲面,并给出了图例。

This article aimed to construct an extended cubic uniform B-spline curve, which had both convex hull property and shape-preserving property, under the premise of without increasing the computational complexity. First, by adopting reverse thinking method, this paper obtained the properties of the blending function from the preset properties of the curve, and then calculated the expression of the blending function. Then, by qualitative analysis, it discussed the parameter intervals of the curve which made it has convex hull property, monotonicity-preserving, convexity-preserving, variation diminishing property respectively. The legends show the correctness of the analysis result. The intersection of the different parameter value range is the finalized feasible region of the shape parameter. Altering the shape parameter in the feasible region, it can adjust the approximation degree of the curve to the control polygon under the premise that do not destroy the shape-preserving property of the curve. Finally, this paper briefly discussed the tensor product surface corresponding to the curve, and gave some legends.