插值曲率线的直纹面可展设计

Design of a ruled developable surface through the line of curvature

目的曲率线在微分几何中起着非常重要的作用,它在曲面分析中是一个很有用的工具。可展曲面是曲面造型中最简单也最常用的一类曲面,目前大部分工作都是研究在给定曲面上寻找或者计算曲率线,而其反问题研究工作较少,为此,提出一种插值曲率线的可展曲面构造方法,并进一步将它应用到曲面造型中。方法利用Frenet标架表示直纹面的母线,根据曲线为曲面曲率线以及曲面可展的充要条件,得到直纹面的母线需要满足的关系式。并引入控制函数控制曲面的形状。结果给出了以给定曲线为曲率线的直纹面可展的具体表达式,根据可展曲面分类分析了设计曲面为柱面、锥面和空间曲线切线面的充要条件,并给出了两个代表性的实例验证该方法的有效性,实例结果表明,该方法不仅适用于一般参数曲线,对分段参数曲线也是有效的。结论利用构造性的方法给出了插值曲率线的可展曲面的具体表达形式,并通过具体实例验证了该方法的有效性。

Objective Line of curvature plays an important role in differential geometry. It is a useful tool in surface analy- sis. A developable surface is simple and frequently used in surface modeling. Many studies have investigated how to find or compute the line of curvature from a given surface ; its anti-problem is rare. This paper presents an approach of designing a developable surface by interpolating the line of curvature and applying it to surface modeling. Method The Frenet frame is employed to express the ruling of the ruled surface. According to the necessary and sufficient conditions of line of curvature and developable surface, the equations that satisfy the ruling can be obtained. The developable surface can be adjusted flexibly by introducing control functions. Result The paper presents the concrete expression of a developable surface. The necessary and sufficient conditions are analyzed by classifying the developable surface. The method is validated by two rep- resentative examples. Results show that the method is suitable not only for general parametric curves but also for piecewise parametric curves. Conclusion The concrete expression of a developable surface is constructed by interpolating the line of curvature through a constructive method. The proposed method is efficient and convenient, as verified by several represent-ative examples.