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

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.

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.

**parameterized by arc length**if Such a parameterization is called an

**arc length parameterization**.

*not necessarily*) the name of the variable when a function is parameterized by arc length, as often represents “distance.”

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 .

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:

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!*identically*in each component of . Indeed a quick check with a graph will show that a graph of produces the same graph as a graph of when . Ah, so this is a line in disguise! To parameterize a line by arc length you need to write something like: So let’s find two points on the line. Setting , we see that is on the line. Setting we see that is also on the line. The unit vector that runs from to is: Thus as runs from to , draws the same curve as as runs from to .

Try your hand at this one now:

### Disguised circles

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

- The Earth will be at the origin.
- At the starting time, , the Moon will be at the point in the -plane.
- The Moon will travel in a counterclockwise direction around the Earth.

Give a parameterization of the Moon’s orbit that will model the Moon’s position in terms of , the distance traveled in thousands of miles.

### 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:

- (a)
- Compute
- (b)
- Now write and solve for . In this case you will have
- (c)
- The function will be parameterized by arc length.

Try your hand at it.