### Introduction

Curvature is a measure of how sharply a curve is turning. At a particular point along the curve a tangent line can be drawn; this tangent line making an angle $\psi$ with the positive $x$ -axis. Curvature is then defined as the magnitude of the rate of change of $\psi$ with respect to the measure of length on the curve - the arc length $s$ . That is

$\phantom{\rule{2em}{0ex}}\text{Curvature}\phantom{\rule{1em}{0ex}}=\left|\frac{d\psi }{ds}\right|$

In this Section we examine the concept of curvature and, from its definition, obtain more useful expressions for curvature when the equation of the curve is expressed either in Cartesian form $y=f\left(x\right)$ or in parametric form $x=x\left(t\right)\phantom{\rule{1em}{0ex}}y=y\left(t\right)$ . We show that a circle has a constant value for the curvature, which is to be expected, as the tangent line to a circle turns equally quickly irrespective of the position on the circle. For all curves, except circles, other than a circle, the curvature will depend upon position, changing its value as the curve twists and turns.

#### Prerequisites

• understand the geometrical interpretation of the derivative
• be able to differentiate standard functions
• be able to use the parametric description of a curve

#### Learning Outcomes

• understand the concept of curvature
• calculate curvature when the curve is defined in Cartesian form or in parametric form