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 ψ with the positive x -axis. Curvature is then defined as the magnitude of the rate of change of ψ with respect to the measure of length on the curve - the arc length s . That is

Curvature = d ψ d s

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 ( x ) or in parametric form x = x ( t ) y = y ( t ) . 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.


Learning Outcomes

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