Given all of the difficulties surrounding the parametrization of a curve, it would be good to have some geometrically meaningful way to do it. We want to have some way to choose a parametrization that actually has something to do with the shape of the curve. Fortunately, there is the parametrization by arclength. In this case, we use the distance along the curve from some fixed point as the parameter.
We start with the measurement of length. It is a common enough topic in an Integral Calculus course to study the length of a curve, and after all of the Riemann sums dust settles, you are left with the following:
If $\gamma: [a,b] \to \mathbb{R}^3$ is a regular $C^1$ curve, then the length of $\gamma([a,b])$ is
\[ \mathrm{Len}(\gamma) = \int_a^b ||\gamma'(t)|| \, dt . \]
Now, if we think of the upper limit of integration as variable, then the Fundamental Theorem of Calculus defines a differentiable function for us. Classically, geometers use the letter $s$ for arclength, so we get a function $s = s(t)$
\[s: t \mapsto \int_{t_0}^t ||\gamma'(u)||\, du .\]
This function measures the signed distance along $\gamma$ from $\gamma(t_0)$ to $\gamma(t)$. The sign here will indicate a direction of travel along the curve, so that negative values of this function correspond to moving in the direction opposite the motion described by the original parametrization.
Here is a Sage cell with some relevant computations for a piece of a parabolic arc in the plane. By changing the value of the input $a$, you will get different values of the length. I have also included code to show what that length function looks like.
Now, we will replace the given parameter $t$ with the arclength parameter $s$. The key to this is that the function $s = s(t)$ is differentiable, and the Fundamental Theorem of Calculus tell us that its derivative is
\[ \dfrac{ds}{dt} = \dfrac{d}{dt} \int_{t_0}^t ||\gamma'(u)|| \, du = ||\gamma'(t)|| . \]
Here is where the assumption of regularity comes to help us. Since $\gamma'$ never vanishes, we see that $s$ is a strictly increasing function of $t$. Strictly increasing functions are invertible, so it makes sense to write $t = t(s)$ as a function of $s$. We then plug this inverse function into our curve!
\[ \gamma(s) := \gamma(t(s)) .\]
All of that sounds great, and it is. But there are several troubles with it.
In this picture, you can see that the arclength function $\phi$ assigns a number (really a length) to each value of the parameter $t$.
We start with the measurement of length. It is a common enough topic in an Integral Calculus course to study the length of a curve, and after all of the Riemann sums dust settles, you are left with the following:
If $\gamma: [a,b] \to \mathbb{R}^3$ is a regular $C^1$ curve, then the length of $\gamma([a,b])$ is
\[ \mathrm{Len}(\gamma) = \int_a^b ||\gamma'(t)|| \, dt . \]
Now, if we think of the upper limit of integration as variable, then the Fundamental Theorem of Calculus defines a differentiable function for us. Classically, geometers use the letter $s$ for arclength, so we get a function $s = s(t)$
\[s: t \mapsto \int_{t_0}^t ||\gamma'(u)||\, du .\]
This function measures the signed distance along $\gamma$ from $\gamma(t_0)$ to $\gamma(t)$. The sign here will indicate a direction of travel along the curve, so that negative values of this function correspond to moving in the direction opposite the motion described by the original parametrization.
Here is a Sage cell with some relevant computations for a piece of a parabolic arc in the plane. By changing the value of the input $a$, you will get different values of the length. I have also included code to show what that length function looks like.
Now, we will replace the given parameter $t$ with the arclength parameter $s$. The key to this is that the function $s = s(t)$ is differentiable, and the Fundamental Theorem of Calculus tell us that its derivative is
\[ \dfrac{ds}{dt} = \dfrac{d}{dt} \int_{t_0}^t ||\gamma'(u)|| \, du = ||\gamma'(t)|| . \]
Here is where the assumption of regularity comes to help us. Since $\gamma'$ never vanishes, we see that $s$ is a strictly increasing function of $t$. Strictly increasing functions are invertible, so it makes sense to write $t = t(s)$ as a function of $s$. We then plug this inverse function into our curve!
\[ \gamma(s) := \gamma(t(s)) .\]
All of that sounds great, and it is. But there are several troubles with it.
- There is a little trouble with the meaning of such a reparametrization. In particular, if we want to think physically, we are replacing a something that measures time by something that measures distance. How can we wrap our heads around that?
- Sometimes the classical notation gives people trouble. The notational abuses are pretty bad, so if you are not accustomed to this it can get in the way.
- Even if you understand all the theory, it can be very difficult, or even impossible, to actually compute an arclength reparametrization.
The Meaning and Interpretation of Arclength
First, arclength is a length. There is nothing particularly troublesome about that at first glance. In a way, you are laying some string along your curve, and the length is what you get if you take that string, straighten it out and measure it the usual way. What causes a sense of unease is the mismatch in units.
We made a lot of progress by learning to think of a curve as a parametrized object, and the parameter is then an indication of time and the function tracing the curve then describes the motion of some idealized point through space. But arclength is a distance. So the units are wrong. As all of this has been done as some theoretical mathematics exercise, the units are hiding out of sight, but they do form the basis of our intuition. Getting them all mixed up means we must be careful.
The key here is that our arclength parametrized curve has "unit speed." Using the chain rule
\[ \dfrac{d\gamma}{ds} = \dfrac{d\gamma}{dt}\dfrac{dt}{ds}\]
we see that the speed of our arclength parametrized curve is
\[\left|\left| \dfrac{d\gamma}{ds} \right|\right| = \dfrac{||\gamma'(t)||}{ds/dt} = 1. \]
This means that during the passage of any one unit of time, the motion has proceeded by exactly one unit of distance. It is in this way we can use distance to serve as a proxy for time.
If you are from the vast central part of the United States of America, you grow up learning to speak in exactly this way! Well, almost. Rather than the conversion being done at "unit speed," it is done at highway speed, and the conversion is done backwards. Ask someone from Iowa how far it is from Des Moines to the Quad Cities (a distance), and you will get an answer in hours! The implicit conversion is that you will be driving on a major interstate highway at about 70 miles per hour. This is important because it helps you plan your travel more efficiently. Even the folks at Google have bought in to this way of talking. I just performed a Google Search for "Des Moines to Quad Cities" and the search engine is programmed to give me the useful information first: A small map and a declaration of approximate travel time.
Arclength parametrization works the same way, but the conversion is from time to distance. This is geometrically natural since one might change speed during travel, but the distance is a fixed quantity. Using a unit speed parametrization (i.e. by arclength) we remove this bit of uncertainty by evening out the flow of traffic to some standardized speed: one unit of distance per unit of time. Driving across the american midwest this kind of conversion is quite handy: 60 miles per hour is the same as one mile per minute. (That is a bit slower than most Americans drive those highways, but is not bad for a rough approximation.)
Better Notation
It is a bit confusing for the new learner to handle the fact that $s$ is sometimes an independent variable, and sometimes a function $s = s(t)$. It is better to make a new name for the length function that makes the conversion
\[ \phi(t) = \mathrm{Len}(\gamma(t)) = \int_{t_0}^t ||\gamma'(u)|| \, du , \]
and then use $s$ as a generic name for an output of that function: $s= \phi(t)$. Then the inversion process is a bit clearer, and we can make sense of the reparametrization process by introducing a new name for the new function
\[ t = \phi^{-1}(s), \qquad \alpha(s) = \gamma\circ \phi^{-1}(s) . \]
And $\alpha$ is genuinely a different function! The image traced out by $\alpha$ is the same as that traced by $\gamma$, but as functions they are distinct, and should merit distinct names.
I find it useful to keep this kind of diagram in my head:
In this picture, you can see that the arclength function $\phi$ assigns a number (really a length) to each value of the parameter $t$.
Technical Troubles
Arclength is a wonderful tool, and the Fundamental Theorem of Calculus ensures that we should be able to use it for any regular curve. But each of the steps involved might be impractical to do: performing the integral could be cumbersome, or even impossible in terms of elementary functions; inverting an arbitrary expression written terms of elementary functions is also very difficult, the inverse may not have a convenient expression in terms of elementary functions.
It is no trouble to assume unit speed parametrization when working theoretically. If we want to prove a theorem it is an assumption we can make, and it is an assumption we should make, because it simplifies many things. But given a curve "from the wild," it might be very hard to make the conversion.
In a later post, we will explore an example which turns out to have some historical significance.
In a later post, we will explore an example which turns out to have some historical significance.
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