给定曲率和挠率为常数的空间曲线方程

Equations of Space Curves with Constant Curvature and Torsion

通过解向量微分方程组的特解的方法,给出了曲率函数和挠率函数为常数的空间曲线的方程,并通过分析说明具有该特征的曲线就是圆柱螺线,在此基础上进一步探讨了曲率和挠率为非常数但它们的比值为常数的空间曲线的方程.

Curvature and torsion are two internal parameters of a space curve. From the basic theorems of space curves, it is known theoretically that the shape and function of a space curve can be uniquely determined if its curvature function and torsion function are given. Practically, however, it is very hard to do so because of the difficulties in the solution of a set of differential equations. With the method for the solution of vector differential equations, the equation of a space curve with constant curvature and torsion is given. Through analysis, it is pointed out that such a curve is of circular helix. Based on this, equations of space curves with variable curvature and torsion but constant ratio are studied.