Advanced Engineering Mathematics, 7th Edition

9.3 Curvature

INTRODUCTION

Let C be a smooth curve in either 2- or 3-space traced out by a vector function r(t). In this section we are going to consider in greater detail the acceleration vector a(t) = r″(t) introduced in the last section. But before doing this, we need to examine a scalar quantity called the curvature of a curve.

A Definition

We know that r′(t) is a tangent vector to the curve C, and consequently

(1)

is a unit tangent. But recall from the end of Section 9.1 that if C is parameterized by arc length s, then a unit tangent to the ...