目的 为了同时解决传统多项式B样条曲线在形状调控、精确表示常见工程曲线以及构造插值曲线时的不足,提出了一类集多种特性的三次三角伪B样条。方法 首先构造了一组带两个参数的三次三角伪B样条基函数,然后在此基础上定义了相应的参数伪B样条曲线,并讨论了该曲线的特性及光顺性问题,最后研究了相应的代数伪B样条,并给出了最优代数伪B样条的确定方法。结果 参数伪B样条曲线不仅满足C²连续,而且无需求解方程系统即可自动插值于给定的型值点。当型值点保持不变时,插值曲线的形状还可通过自带的两个参数进行调控。在适当条件下,该参数伪B样条曲线可精确表示圆弧、椭圆弧、星形线等常见的工程曲线。相应的代数伪B样条具有参数伪B样条曲线类似的性质,利用最优代数伪B样条可获得满意的插值效果。结论 所提出的伪B样条同时解决了传统多项式B样条曲线在形状调控、精确表示常见工程曲线以及构造插值曲线时的不足,是一种实用的曲线造型方法。

Objective A class of cubic trigonometric pseudo B-spline functions combining multiple characteristics is proposed to address the deficiency of normal polynomial B-spline curves in shape control, the accurate representation of several engineering curves, and the construction of interpolation curves. Method A class of cubic trigonometric pseudo B-spline basis functions with two parameters is constructed. The parametric pseudo B-spline curve is defined based on the proposed basis functions, and the characteristics and fairness of the curve are discussed. Lastly, the corresponding algebraic pseudo B-spline is studied, and a method of determining the optimal algebraic pseudo B-spline is provided. Result The parametric pseudo B-spline curves can not only be C² continuous, but can also automatically interpolate the given data points without solving the system equations. When the data points are fixed, the shape of the interpolation curves can be adjusted by the two parameters. Under appropriate conditions, the parametric pseudo B-spline curves can be precisely expressed as several engineering curves, such as circular arc, elliptical arc, and star line. The corresponding algebraic pseudo B-spline has properties similar to those of the parametric pseudo B-spline curve. The optimal algebraic pseudo B-spline can obtain satisfactory interpolation effects. Conclusion The proposed pseudo B-spline simultaneously solves the deficiency of normal polynomial B-spline in shape control, the accurate representation of several engineering curves, and the construction of interpolation curves. Thus, it is a practical method for curve modeling.