张力平面参数曲线的几何性态

给出了一种分析平面参数曲线奇拐点和凸性的方法。这种方法在几何上直观，对于四阶代数曲线和非代数曲线的几何性态分析都是有效的。将它应用于基于指数函数的张力平面参数曲线，以及基于双曲Ｂ－样条函数的张力Ｂ－样条平面参数曲线的研究，解决了这些曲线的奇拐点和凸性等几何性态问题。

On Properties of Planar Parametric Curve under Tension

Barsky, well known authority on CAD/CAM, presented in 1984 a useful modellingtool-Planar parameter curve under tension 6]. But, up to now, almost nothing is knownabout the convexity properties of Barsky's curve and the properties of its singular and inflection points. Thus, in design using Barsky's curve, undesirable features, such as cusps, maybe inadvertently brought in. In this paper we present all these needed properties.After thinking and rethinking about this difficult problem for several years, the first author finally hit upon a powerful tool in topology-topology map. Happily, a discovery indifferential geometry, given as theorem 1 in this paper, was made by the first author almostsimultaneously. With the aid of the said topology map and the said discovery, we obtain allthe needed properties concerning Barsky's curve.Barsky's curve is just one particular curve with two adjustable parameters. It should beemphasized that the said topology map and the said discovery may by employed to obtain allthe needed properties of any planar curve with two adjustable parameters.