Frenet曲线和后继曲线：Hectoris和Slant Hectoris的通用参数化
Frenet Curves and Successor Curves: Generic Parametrizations of the Helix and Slant Helix
摘要：
在经典曲线理论中，三维曲线的几何性质本质上由它们的不变量、曲率和挠率来表征。当它们被给定时，找到相应曲线的问题被称为“解自然方程”。已知的显式解仅适用于少数曲线类，特别是包括平面曲线和一般螺旋线。本文构造性地给出了曲线类的无穷级数的自然方程的显式求解方法。对于每一条Frenet曲线，都可以构造出以原曲线的切线为主法线的后继曲线族。螺旋线是平面曲线的后继曲线，将后继变换应用于螺旋线会产生斜螺旋线，斜螺旋线是近年来作为一般螺旋线概念的自然扩张而受到广泛关注的一类曲线。本文首次给出了三维欧氏空间中斜螺旋线的曲率和挠率的一般刻画，并导出了斜螺旋线的切向量的显式弧长参数化。这些结果扩展和投入透视早期的工作Salkowski曲线和曲线的恒定进动，这两者都是子类的倾斜螺旋。本文还为初学者和教师提供了一个新颖的和广义的介绍弗伦内曲线的理论，这是不限于曲线的正曲率。Bishop标架与Frenet标架和Darboux标架一起作为空间曲线理论中的有用工具进行了研究。闭曲线问题也受到关注。
英文摘要：
In classical curve theory, the geometry of a curve in three dimensions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of finding a corresponding curve is known as 'solving natural equations'. Explicit solutions are known only for a handful of curve classes, including notably the plane curves and general helices. This paper shows constructively how to solve the natural equations explicitly for an infinite series of curve classes. For every Frenet curve, a family of successor curves can be constructed which have the tangent of the original curve as principal normal. Helices are exactly the successor curves of plane curves and applying the successor transformation to helices leads to slant helices, a class of curves that has received considerable attention in recent years as a natural extension of the concept of general helices. The present paper gives for the first time a generic characterization of the slant helix in three-dimensional Euclidian space in terms of its curvature and torsion, and derives an explicit arc-length parametrization of its tangent vector. These results expand on and put into perspective earlier work on Salkowski curves and curves of constant precession, both of which are subclasses of the slant helix. The paper also, for the benefit of novices and teachers, provides a novel and generalized presentation of the theory of Frenet curves, which is not restricted to curves with positive curvature. Bishop frames are examined along with Frenet frames and Darboux frames as a useful tool in the theory of space curves. The closed curve problem receives attention as well.
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