高阶连续的形状可调三角多项式曲线曲面

目的目前使用的B样条曲线曲面存在着高连续阶与高局部调整性两者无法兼而有之的不足,且B样条曲线曲面的形状被控制顶点和节点向量唯一确定,这些因素影响着B样条方法的几何设计效果与方便性。本文旨在克服这种局限,以期构造具有高次B样条方法的高连续阶,低次B样条方法的高局部调整性,以及有理B样条方法权因子决定的形状调整性的曲线曲面。方法在三角函数空间上构造了一组含参数的调配函数,进而定义具有与3次B样条曲线曲面相同结构的新曲线与张量积曲面。结果新曲线曲面继承了B样条方法的凸包性、对称性、几何不变性等诸多性质。不同的是,同样是基于4点分段,3次均匀B样条曲线C2连续,而对于等距节点,在一般情况下,新曲线C5连续,当参数取特殊值时可达C7连续。新曲线在C5连续的情况下存在1个形状参数,能较好地调整曲线的形状同时又无须改变控制顶点。另外,将形状参数设为特定值,新曲线可以自动插值给定点列。新曲面具有与新曲线相应的优点。结论在强局部性下实现高阶连续性的形状可调分段组合曲线曲面,为高阶光滑曲线曲面的设计提供了可能,并且新曲线实现了逼近与插值的统一表示,能较好地应用于工程实际。调配函数的构造方法具有一般性,可用相同方式构造其他具有类似性质的调配函数。

Higher-order continuous shape adjustable trigonometric polynomial curve and surface

Objective The widely used B-spline method is limited by its incapability to realize high-order continuity and high local adjustment simultaneously. The shape of the B-spline curve and surface is uniquely determined by the control points and knot vector. These factors influence the design effects and convenience of the B-spline method. This study aims to overcome these limitations. We construct a new curve and surface that possess the high order continuity of the high-order B-spline method, the high local adjustment of the low-order B-spline method, and the shape adjustment property of the rational B-spline method as determined by the weight factor. Method We first construct a set of blending functions with a parameter on the trigonometric function space. We then define the new curve and tensor product surface, which have the same structure as those of the cubic B-spline curve and surface, respectively. Result The new curve and surface inherit many of the B-spline method properties, such as convex hull, symmetry, and geometric invariability. The difference is that although both the new and cubic B-spline curves are based on a four-point segment, the latter is C² continuous, while for equidistant knots, the new curve is generally C⁵ continuous and can reach C⁷ continuity when taking a special parameter. A new curve that is C⁵ continuous has one shape parameter, which can be used to adjust the shape of the curve without changing the control points. In addition, taking the special parameter, the new curve can automatically interpolate the given set of points. The new surface has properties corresponding to the new curve. Conclusion This study presents a kind of shape-adjustable piecewise combination curve and surface, which can achieve high-order continuity under strong local control capability. These features can be used for the possible design of a high-order smooth curve and surface. The new curve realizes the uniform representation of approximation and interpolation. The construction method of the blending function is general and can be used to construct other functions with similar properties.