P-42 A Curve Satisfying τ/κ=1/s With Constant κ > 0

According to the Fundamental Theorem of Curves, any regular curve with a smooth positive curvature and smooth torsion can be completely determined up to its position. Helices have the property that the ratio of torsion to curvature is a constant. For rectifying curves, the ratio of torsion to curvature is a linear function. In this paper, we study a space curve whose ratio of torsion to curvature is given by 1/s, where s is an arc length. For this problem, we consider the curvature is constant. After reparametrization, we use a series solution to solve a third-order differential equation and obtain the general equation of the curve.