1.3.5 Arc length parameterization
- Author:
- Sarah Harrelson
The way I displayed the distance traveled in the last activity implied that arc length can be used to define a (scalar-valued) function. (BTW scalar-valued is a way of saying the codomain of a function is , as opposed to paths which are vector-valued functions).
If is a differentiable path we can define a scalar-valued function as follows:
By the Fundamental Theorem of Calculus, what is ?
Since always (since speed by definition cannot be negative), we know that is an increasing function. If we further impose the requirement that is a regular curve (recall regular means the velocity never vanishes), then we find that the arc length function is injective. This allows us to use the arc length function to reparameterize the image curve to achieve a unit-speed (aka arc length) parameterization - that is a parameterization whose speed is constantly 1.
Consider the example Compute the speed of this path as a function of .
Now find an expression for
is invertible as desired. Find an inverse function so that and .
Finally, find an arc length parameterization of the image curve of
In the GeoGebra applet below you can see the two parameterizations of the image curve of . How can you tell the second parameterization is an arc length parameterization?