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 with the positive -axis. Curvature is then defined as the magnitude of the rate of change of with respect to the measure of length on the curve - the arc length . That is
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 or in parametric form . 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