Rectifying curves as centrodes and extremal curves

Abstract

Rectifying curves are introduced in [2] as space curves whose position vector always lies in its rectifying plane. In this article, we establish a surprising simple relationship between rectifying curves and the notion of centrodes in mechanics. Furthermore, we show that rectifying curves are indeed the extremal curves which satisfy the equality case of a general inequality. Further geometric properties of rectifying curves are also presented. 1. Rectifying curves. Let E denote Euclidean three-space, with its inner product 〈 , 〉 and let S be the unit sphere in E centered at the origin. Consider a unit-speed space curve x : I → E, where I is a real interval, that has at least four continuous derivatives. Let t denote x. It is possible, in general, that t(s) = 0 for some s; however, we assume that this never happens. Then we can introduce a unique vector field n and positive function κ so that t = κn. We call t the curvature vector field, n the principal normal vector field, and κ the curvature of the given curve. Since t is a constant length vector field, n is orthogonal to t. The binormal vector field is defined by b = t × n. It is a unit vector field orthogonal to both t Received by the editors April 22, 2004. AMS 2000 Subject Classification: Primary: 53A15; Secondary 53C40, 53C42.