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Section 6.2 Parametrization

Subsection 6.2.1 Varied Parametrizations

Figure 6.2.1. The graph of a parabola as a parametric curve

There are many ways to describe the same shape by a parametric curve.

Consider the following curves.

\begin{align*} \gamma_1(t) \amp = \left( t, t^2 \right) \amp \amp t \in [0, 16]\\ \gamma_2(t) \amp = \left( t^2, t^4 \right) \amp \amp t \in [0, 4]\\ \gamma_3(t) \amp = \left( \sqrt{t}, t \right) \amp \amp t \in [0, 256]\\ \gamma_4(t) \amp = \left( 5t, 25t^2 \right) \amp \amp t \in \left[0, \frac{16}{5} \right] \end{align*}

All four of these have exactly the same parabolic image. They all describe the same curve, shown in Figure 6.2.1.

Subsection 6.2.2 Reparametrization

Since the same shape can have many different parametrizations, we want a process to switch between them. This process is called reparametrization.

Definition 6.2.3.

Let \(\gamma(t): [a,b] \rightarrow \RR^n\) be a parametric curve with coordinates \((\gamma_1(t), \ldots, \gamma_n(t))\text{.}\) A reparametrization of \(\gamma\) is a monotonic increasing function \(t = t(u)\) expressing the parameter \(t\) in terms of a new parameter \(u\text{.}\) We replace \(t\) by the function \(t(u)\) to give a parametric curve in terms of \(u\text{:}\) \(\gamma(u) = (\gamma_1(t(u)), \gamma_2(t(u)), \ldots, \gamma_n(t(u)))\text{.}\)

The unit circle in \(\RR^2\) is parametrized by \(\gamma(t) = (\cos t, \sin t)\text{.}\) If \(t = 3u\) then \(\gamma(u) = (\cos 3u, \sin 3u)\) is a reparametrization of the same circle. The first parametrization finishes a revolution in \(t \in [0, 2\pi]\text{,}\) but multiplication by \(3\) in the second parametrization means that a full revolution is completed in \(u \in [0, 2\pi/3]\) — that is, the second parametrization moves along the circle three times as fast.

Many reparametrizations of the circle are possible.

\begin{align*} \text{ If } t = \frac{u}{3} \amp \text{ then } \gamma(u) = \left( \cos \frac{u}{3}, \sin \frac{u}{3} \right)\\ \text{ If } t = u^2 \amp \text{ then } \gamma(u) = \left( \cos u^2, \sin u^2 \right)\\ \text{ If } t = \sqrt{u} \amp \text{ then } \gamma(u) = \left( \cos \sqrt{u}, \sin \sqrt{u} \right) \end{align*}

Even though the shape of the curve is the same, the parametrization affects the rate movement along the curve.

Subsection 6.2.3 Arc Length

Figure 6.2.5. Arclength setup by approximation

Our goal in this section is to produce a formula describing the length of a parametric curve. We approximate this length by breaking up a curve \(\gamma\) into a series of straight lines. In order to visualize the process, we work in \(\RR^2\) for the moment, as in Figure 6.2.5.

For each straight line segment, if we know \(\Delta y\) and \(\Delta x\text{,}\) then the length of the segment is \(\sqrt{\Delta x^2 + \Delta y^2}\text{.}\) We approximate the total length of the curve by adding up the lengths of these segments.

\begin{equation*} L \cong \sum \sqrt{\Delta x^2 + \Delta y^2} \end{equation*}

If we take the limit of this process, breaking the curve into smaller and smaller pieces, we get a Riemann sum which defines a definite integral. In the limit, the \(\Delta\) terms become the infinitesimals \(dy\) and \(dx\text{.}\)

\begin{equation*} L = \int_a^b \sqrt{dx^2 + dy^2} \end{equation*}

Both \(x\) and \(y\) depend on \(t\text{,}\) so we can change this to a integral in \(t\text{.}\)

\begin{equation*} L = \int_a^b \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } dt \end{equation*}

The same arguments work in any \(\RR^n\text{,}\) leading to the general result.

In \(\RR^3\text{,}\) consider the helix \(\gamma(t) = (\cos t, \sin t, t)\text{.}\) On the domain \(t \in [0, 8\pi]\text{,}\) the helix makes four revolutions. We calculate its arclength.

\begin{equation*} L = \int_0^{8\pi} \sqrt{ \sin^2 t + \cos^2 t + 1} dt = \int_0^{8\pi} \sqrt{2} dt = 8 \sqrt{2} \pi \end{equation*}

The asteroid is the parametric curve \(\gamma(t) = (\cos^3 t, \sin^3 t)\) for \(t \in [0,2\pi]\text{,}\) shown in Figure 6.2.9. We want to calculate its arclength.

Figure 6.2.9. The Asteroid
\begin{align*} L \amp = \int_a^b \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } dt\\ \amp = \int_0^{2\pi} \sqrt{ (-3 \cos^2 t (\sin t))^2 + (3 \sin^2 t (\cos t))^2} dt\\ \amp = \int_0^{2\pi} \sqrt{ 9 \cos^4 t \sin^2 t + 9 \sin^4 t \cos^2 t} dt\\ \amp = \int_0^{2\pi} |3 \sin t \cos t | \sqrt{ \sin^2 + \cos^2 t} dt\\ \amp = \int_0^{2\pi} |3 \sin t \cos t | dt \end{align*}

The absolute value here causes trouble. A convenient way to drop it is to notice that both \(\sin t\) and \(\cos t\) are positive on \([0, \pi/2]\text{.}\) That range covers a quarter of the asteroid and the asteroid is symetric, so we can calculate the length of that quarter and multiply by 4.

\begin{equation*} L = 4\int_0^{\frac{\pi}{2}} 3 \sin t \cos t dt = 6 \int_0^{\frac{\pi}{2}} \sin 2t = -3\cos 2t \Big|_0^{\frac{\pi}{2}} = 6 \end{equation*}

Subsection 6.2.4 Parametrization by Arclength

Our arclength calculation determins the length of the whole curve. However, we can also ask for the length of pieces of the curve.

Definition 6.2.10.

Let \(\gamma(t): [a,b] \rightarrow \RR^n\) be a parametric curve with components \((\gamma_1(t), \ldots, \gamma_n(t))\text{.}\) The arclength function \(s(t): [a,b] \rightarrow [0, \infty)\) is defined by this integral.

\begin{equation*} s(t) = \int_a^t \sqrt{ \left( \frac{d\gamma_1(u)}{du} \right)^2 + \left( \frac{d\gamma_2(u)}{du} \right)^2 + \ldots + + \left( \frac{d\gamma_n(u)}{du} \right)^2 } du \end{equation*}

The letter \(s\) is conventional notation for the arclength function.

The arclength function measures the length of the curve as a function of the parameter; it is simply the distance travelled along the curve. For example, if \(t \in [0,10]\text{,}\) then \(s(3)\) is the distance along the curve from \(\gamma(0)\) to \(\gamma(3)\text{,}\) \(s(5)\) is the distance the curve has covered from \(\gamma(0)\) to \(\gamma(5)\text{,}\) \(s(8)\) is the distance along the curve from \(\gamma(0)\) to \(\gamma(8)\) and so on. Since \(s\) depends on \(t\) outside the integral, we have to choose a temporary variable \(u\) for inside the integral; in the integral, \(t\) is simply repalced with \(u\text{.}\)

We want to use the arclength function to reparametrize curve. The process has three steps.

  1. We calculate the arclength fuction \(s(t)\) by integration.

  2. We invert the arclength function. Arclength is always an increasing function, so it is always invertible. We write the inverse as \(t(s)\text{.}\)

  3. We use the inverse of the arclength function to reparametrize by replacing \(t\) with \(t(s)\text{.}\)

Definition 6.2.11.

Let \(\gamma(t) : [a,b] \rightarrow \RR^n\) be a parametric curve. Let \(s(t)\) be the arclength function, with \(t(s)\) its inverse. The reparametrization \(\gamma(t(s))\) is called the parametrization by arclength. It is the unique parametrization where the parameter is the distance along the curve.

Consider the helix \(\gamma(t) = (2 \cos t, 2 \sin t, 4t)\) defined for \(t > 0\text{,}\) shown in Figure 6.2.13.

Figure 6.2.13. The helix \(\gamma(t) = (2 \cos t, 2 \sin t, 4t)\)

We calculate the arclength function.

\begin{align*} s(t) \amp = \int_0^t \sqrt{x^\prime(u)^2 + y^\prime(u)^2 + z^\prime(u)^2} du\\ \amp = \int_0^t \sqrt{4\sin^2 u + 4 \cos^2 u + 4^2} du = \int_0^t \sqrt{20} du = t \sqrt{20} \end{align*}

The arclength function is \(s = t \sqrt{20}\text{;}\) the inverse is \(t = \frac{s}{\sqrt{20}}\text{.}\) We replace \(t\) with \(t(s)\) in \(\gamma(t)\text{.}\)

\begin{equation*} \gamma(s) = \left( 2 \cos \left( \frac{s}{\sqrt{20}} \right), 2 \sin \left( \frac{s}{\sqrt{20}} \right), \frac{4s}{\sqrt{20}} \right) \end{equation*}