Curves of Constant Precession

TLDR: In this paper, an arclength-parametrized closed-form solution of the natural equations for curves of constant precession was obtained through direct geometric analysis. But it is not a necessary condition that integral curves be closed.

Abstract: 1. INTRODUCTION. Given initial position and direction, the flight-path of a ship in Euclidean space is completely determined by how much it turns and how much it twists at each odometer reading. This is an intuitive interpretation of the Fundamental Theorem for Space Curves, which states that curvature K and torsion , as functions of arclength s, determine a space curve uniquely up to rigid motion. This statement of the Fundamental Theorem ([14], §1-8) should be tempered with the reservations expressed by Nomizu [12] and Wong & Lai [15]. Given a parametric space curve, there are well-known formulae for the arclength, curvature, and torsion (as functions of the parameter). Given two functions of one parameter (potentially curvature and torsion parametrized by arc-length) one might like to find a parametrized space curve for which the two functions are the curvature and torsion. This activity, called "solving natural equations" ([14], §1-10), is generally achieved by solving Riccati equations like dw/ds = -iz/2 iKW + i7W /2. Although the solution generally exists, it usually cannot be obtained explicitly. Euler [6] found explicit integral formulae for plane curves (where z - O) through direct geometric analysis. Hoppe [9] developed a general method for solving the natural equations for space curves by solving Riccati equations through a complicated sequence of integral transformations. He digressed to obtain formulae for the tangent, normal, and binormal indicatrices for general helices and essentially for curves of constant precession. Enneper [5] obtained explicit closed-form solutions for helices on revolved conic sections through direct geometric analysis. A curve of constant precession is defined by the property that as the curve is traversed with unit speed, its centrode revolves about a fixed axis with constant angle and constant speed. In this paper we obtain an arclength-parametrized closed-form solution of the natural equations for curves of constant precession through direct geometric analysis. As part of this analysis, we obtain a new theorem for curves of constant precession analogous with Lancret's Theorem for general helices. We provide the first rendering of a curve of constant precession. We also note for the first time that curves of constant precession lie on circular hyperboloids of one sheet and have closure conditions that are simply related to their arclength, curvature, and torsion. These are 3-type curves, except one family of closed 2-type curves (when Z = 4,u; see [2], [3], and [1]). Given a closed C3 curve in space, it is rather obvious that the curvature and torsion functions will be periodic functions of the arclength, with period equal the total arclength. This is a necessary condition but, as the circular helices (K and z both constant) show, not a sufficient condition that integral curves be closed. Efimov [4] and Fenchel [7] independently formulated The Closed Curve Problem. Find (explicit) necessary and sufficient conditions that determine when, given two periodic functions K(S) and z(s) with the same period L, the integral curve is closed.