Curvature measures the rate at which a space curve r(t) changes direction. The direction of curve is given by the unit tangent vector
which has length 1 and is tangent to r(t). The picture below shows the unit tangent vector T(t) to the curve r(t)=<2cos(t),sin(t)> at several points.
Obviously, if r(t) is a straight line, the curvature is 0. Otherwise the curvature is non-zero. To be precise, curvature is defined to be the magnitude of the rate of change of the unit vector with respect to arc length:
The reason that arc length comes into the definition is that arc length is independent of parameterization.
In most cases a curve is described by a particular parameterization and we have the unit tangent vector as a function of t: T(t). We can compute the curvature using the chain rule
Recall that ds/dt=|r'(t)|. Hence, it follows that
What is the curvature of the ellipse r(t)=<2cos(t),sin(t)>? We have several computations to perform. First, we have
and
implying
Differentiating again, we have
Finally, we have
Look at the graph of the ellipse above. The curvature is greatest near x=2 and y=0 and x=-2 and y=0. These points correspond to t=0 and t=pi. In the above expression for the curvature, the denominator is at its minimum when t=0 or t=pi, implying the curvature is at a maximum. The plot of ellipse indicates that the curvature is at is lowest value when y=1 and x=0 and y=-1 and x=0. These points correspond to t=pi/2 and 3*pi/2. For these values of t the curvature takes on its minimum value in the formula above.
Curvature of a Circle
A circle of radius a can be described by the parameterization r(t)=a<cos(t),sin(t)>. It can be shown that
and
implying
Differentiating again, we have
Finally, we have
Alternative formula for the Curvature
It can be shown that the curvature is also given by the formula
Copyright © 1996 Department of Mathematics, Oregon State University
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