We find a new description of curves that trivializes arc length computations.

For any given a curve in space, there are many different vector-valued functions that draw this curve. For example, consider a circle of radius centered at the origin. Each of the following vector-valued functions will draw this circle: Each of these functions is a different parameterization of the circle. This means that while these vector-valued functions draw the same circle, they do so at different rates.
Considering , , and , which draws the circle of radius quickest?
Which draws the circle of radius slowest?

In this section, we are going to be interested in parameterizations of curves where there is a one-to-one ratio between the parameter (the variable) and distance drawn (the arc length) from the start of the curve. Recall that if is a continuous vector-valued function where the curve drawn by is traversed once for , then the arc length of the curve from to is given by This is all good and well, but the integral could be quite difficult to compute. On the other hand, if were an arc length parameterization, this would be simple to compute, because then the arc length is in a one-to-one ratio with the variables. Hence Let’s state this as a definition.

It is nice to work with functions parameterized by arc length, because computing the arc length is easy. If is parameterized by arc length, then the length of when , is simply . No integral computations need to be done. Also we should point out that is typically (though not necessarily) the name of the variable when a function is parameterized by arc length, as often represents “distance.”
Suppose the curve below has an arc length parameterization given by . Compute: , , and

Consider the following example:

From your own experience and the work above, we think the next theorem should be quite sensible.

If we imagine our vector-valued function as giving the position of a particle, then this theorem says that the path is parameterized by arc length exactly when the particle is moving at a speed of .

Which of the following vector-valued functions are parameterized by arc length?
Consider for . Find that makes this parameterized by arc length.
Set and solve for .

Often given a curve one wishes to have an arc length parameterization of the curve. We proceed by discussing several special cases, and then by giving a general method.

Disguised lines

Sometimes you have a vector-valued function that is merely a line in disguise. How could this be? Well consider the vector-valued function: This doesn’t look very much like a line, for one thing it has the function in each component. On the other hand, if we look at , we see Ah, we can now factor a out of each component to get: this is a scalar-function times a constant vector. The fact that we can “pull-out” the scalar function, and are left with a constant vector tells us that the line segment plotted by for is identical to the line segment plotted by:

Which of the following are line segments in disguise?
for for for for for for
Once we identify a vector-valued function as a disguised line, we can rewrite it as and we have an arc length parameterization. Note, we need a unit vector to ensure that the magnitude of the derivative is one!
Give an arc length parameterization of for . for

Try your hand at this one now:

Give an arc length parameterization of for .
Check the values of and .

Disguised circles

Sometimes the curve we are given is a circle in disguise.

Consider for . Parameterize this curve by arc length. for

A general method

While we are about to present a general method for finding representations of functions parameterized by arc length, one must not overestimate its strength.

Regardless, if you want an arc length parameterization of starting at here is the idea:

Now write and solve for . In this case you will have
The function will be parameterized by arc length.

Try your hand at it.