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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta
name="viewport"
content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no"
/>
<title>APMA E2000 - Arc Length & Curvature</title>
<link rel="stylesheet" href="reveal.js/dist/reset.css" />
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rel="stylesheet"
href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/all.min.css"
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<!-- local styles -->
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</head>
<body>
<div class="reveal">
<div class="slides">
<section>
<section>
<h3 class="framelabel">
Lecture 06
<a
href="https://link.excalidraw.com/l/2VYuQqjhg2J/7Ta4gazeSlR"
target="_blank"
rel="noopener noreferrer"
> </a
>
</h3>
<h1>Arc Length & Curvature</h1>
<h2>APMA E2000</h2>
<div class="r-stretch"></div>
<p style="text-align: right">
Drew Youngren <code>[email protected]</code>
</p>
</section>
<section>
<!-- Set up standard LaTeX macros -->
$\gdef\RR{\mathbb{R}}$ $\gdef\vec{\mathbf}$
$\gdef\bv#1{\begin{bmatrix} #1 \end{bmatrix}}$
$\gdef\proj{\operatorname{proj}}$ $\gdef\comp{\operatorname{comp}}$
<h2>Announcements</h2>
<ul>
<li class="fragment">Recitation 03 this week.</li>
<li class="fragment">Quiz 2 (HW2 topics) this week.</li>
<li class="fragment">HW3 due next Tues</li>
</ul>
</section>
</section>
<section>
<section>
<h1>1-minute review</h1>
</section>
<section>
<h6 class="framelabel">Motion</h6>
<ul>
<li><strong>position</strong>: $\vec r(t)$</li>
<li><strong>velocity</strong>: $\vec r'(t)$</li>
<li><strong>acceleration</strong>: $\vec r''(t)$</li>
<li><strong>speed</strong>: $|\vec r'(t)|$</li>
</ul>
</section>
<section>
<h3 class="framelabel">Example</h3>
<p>
Suppose a particle starts at the origin with initial velocity
$\left\langle 1, 0 \right\rangle$ and acceleration \[ \vec a(t) =
\begin{cases} \left\langle -t, 1 \right\rangle, \text{ if } t \leq
1 \\ \left\langle 1, -t-1 \right\rangle, \text{ if } t > 1 \\
\end{cases}. \] Find its position at $t = 2$.
</p>
</section>
<section>
<h6 class="framelabel">Solution</h6>
<p>
\[ \vec v(t) = \begin{cases} \left\langle -\frac12 t^2 + 1, t
\right\rangle, \text{ if } t \leq 1 \\ \left\langle t - \frac12,
-t - \frac12 t^2 + \frac52 \right\rangle, \text{ if } t > 1 \\
\end{cases}. \]
</p>
<p>
\[ \vec x(t) = \begin{cases} \left\langle -\frac16 t^3 + t,
\frac12 t^2 \right\rangle, \text{ if } t \leq 1 \\ \left\langle
\frac12 t^2 - \frac12 t + \frac56, \frac12 t^2 - \frac16 t^3 +
\frac52 t - \frac43 \right\rangle, \text{ if } t > 1 \\
\end{cases}. \]
</p>
<p></p>
<p>
<a
href="https://3demos.ctl.columbia.edu/?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"
target="_blank"
rel="noopener noreferrer"
>Graph of solution</a
>
</p>
</section>
<section>
<h6 class="framelabel">Definition</h6>
<p>
A parametrized curve $\vec r(t)$ is called
<strong>smooth</strong> provided \[\vec r'(t) \neq \vec 0\] for
all $t$. A smooth curve has a well-defined
<strong>unit tangent vector</strong> \[\vec T(t) = \frac{\vec
r'(t)}{|\vec r'(t)|}\]
</p>
</section>
<section>
<p>
A counterexample might express this more easily. \[ \vec r(t) =
\left\langle t^2, t^3 \right\rangle \]
</p>
<iframe
data-src="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SG93K1RvJnNjYWxlPTAmc2hvd1BhbmVsPWZhbHNlJm9iajBfa2luZD1jdXJ2ZSZvYmowX2NvbG9yPSUyMzBkMDg4NyZvYmowX3BhcmFtc19hPS0xJm9iajBfcGFyYW1zX2I9MSZvYmowX3BhcmFtc194PXQlNUUyJm9iajBfcGFyYW1zX3k9dCU1RTMmb2JqMF9wYXJhbXNfej0w"
frameborder="0"
width="800px"
height="400px"
></iframe>
</section>
</section>
<section>
<section>
<h1>Arc Length</h1>
</section>
<section>
<h6 class="framelabel">Idea</h6>
<p>
To measure the <strong>arc length</strong> of a path in space, we
would like to just "straighten it out" (without stretching) and
use the distance formula<span class="fragment"
>, but this is difficult, mathematically.</span
>
</p>
<p class="fragment">
Instead, measure distances between sample points along the curve.
</p>
</section>
<section data-auto-animate>
<h6 class="framelabel">Formula</h6>
<p class="r-fit-text">
For $\vec r(t)$ with $a \leq t \leq b$, we define $t_i = a + i
\frac{b - a}{N}$ and then <strong>arc length</strong> of the curve
is
</p>
<p>
\[ s = \lim\limits_{N\to\infty}\sum_{i = 1}^N |\vec r(t_i) - \vec
r(t_{i - 1})|\]
</p>
<div class="r-stretch"></div>
</section>
<section data-auto-animate>
<h6 class="framelabel">Formula</h6>
<p class="r-fit-text">
For $\vec r(t)$ with $a \leq t \leq b$, we define $t_i = a + i
\frac{b - a}{N}$ and then <strong>arc length</strong> of the curve
is
</p>
<p>
\[ s = \lim\limits_{N\to\infty}\sum_{i = 1}^N \frac{|\vec r(t_i) -
\vec r(t_{i - 1})|}{\Delta t} \Delta t \]
</p>
<div class="r-stretch fragment">
\[ \to \int_a^b |\vec r'(t)|\,dt \]
</div>
</section>
<section>
<h3 class="framelabel">Example</h3>
<p>
Find the arc length of one coil of a
<a
href="https://3demos.ctl.columbia.edu/?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"
target="_blank"
rel="noopener noreferrer"
>helix</a
>. \[\vec r(t) =\left\langle \cos t, \sin t, t \right\rangle \]
</p>
<p class="fragment">
\[s = \int_0^{2\pi} |\left\langle -\sin t, \cos t, 1
\right\rangle|\,dt \] \[ = \int_0^{2\pi} \sqrt{\sin^2 t + \cos^2 t
+ 1}\,dt = \int_0^{2\pi} \sqrt2\,dt = 2\sqrt 2 \pi \]
</p>
</section>
</section>
<section>
<section>
<h1 class="r-fit-text">Reparametrization</h1>
</section>
<section>
The same path in space can be traced in any number of ways—fast,
slow, backwards.
<div class="fragment">
<h6 class="framelabel">Quick example</h6>
<p>
$\langle \cos t, \sin t\rangle$ traces out the
<a
href="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SG93K1RvJnNjYWxlPTAmc2hvd1BhbmVsPWZhbHNlJm9iajBfa2luZD1jdXJ2ZSZvYmowX2NvbG9yPSUyM2RjNWU2NiZvYmowX2FuaW1hdGlvbj10cnVlJm9iajBfcGFyYW1zX2E9MCZvYmowX3BhcmFtc19iPTIqcGkmb2JqMF9wYXJhbXNfeD1jb3MlMjh0JTI5Jm9iajBfcGFyYW1zX3k9c2luJTI4dCUyOSZvYmowX3BhcmFtc196PTAmb2JqMF9wYXJhbXNfYTA9MCZvYmowX3BhcmFtc19hMT0x"
target="_blank"
rel="noopener noreferrer"
>unit circle</a
>
over the interval $[0,2\pi]$.
</p>
<p class="fragment">
$\langle \cos (6\pi t), \sin (6\pi t)\rangle$ traces out the
unit circle much "<a
href="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SG93K1RvJnNjYWxlPTAmc2hvd1BhbmVsPWZhbHNlJm9iajBfa2luZD1jdXJ2ZSZvYmowX2NvbG9yPSUyM2RjNWU2NiZvYmowX2FuaW1hdGlvbj10cnVlJm9iajBfcGFyYW1zX2E9MCZvYmowX3BhcmFtc19iPTElMkYzJm9iajBfcGFyYW1zX3g9Y29zJTI4NitwaSt0JTI5Jm9iajBfcGFyYW1zX3k9c2luJTI4NitwaSt0JTI5Jm9iajBfcGFyYW1zX3o9MCZvYmowX3BhcmFtc19hMD0wJm9iajBfcGFyYW1zX2ExPTE="
target="_blank"
rel="noopener noreferrer"
>faster</a
>", over the interval $[0,\frac13]$. Take the derivatives of
each of the above to see this.
</p>
<p class="fragment">
$\langle \cos (\pi \sin t), \sin (\pi \sin t)\rangle$ traces out
the unit circle "<a
href="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SG93K1RvJnNjYWxlPTAmc2hvd1BhbmVsPWZhbHNlJm9iajBfa2luZD1jdXJ2ZSZvYmowX2NvbG9yPSUyM2RjNWU2NiZvYmowX2FuaW1hdGlvbj10cnVlJm9iajBfcGFyYW1zX2E9MCZvYmowX3BhcmFtc19iPTIrcGkmb2JqMF9wYXJhbXNfeD1jb3MlMjhwaStzaW4lMjh0JTI5JTI5Jm9iajBfcGFyYW1zX3k9c2luJTI4cGkrc2luJTI4dCUyOSUyOSZvYmowX3BhcmFtc196PTAmb2JqMF9wYXJhbXNfYTA9MCZvYmowX3BhcmFtc19hMT0x"
target="_blank"
rel="noopener noreferrer"
>back and forth</a
>."
</p>
</div>
</section>
<section>
<h3 class="framelabel">Definition</h3>
<p>
Consider a smooth curve $\vec r(t)$ for $a \leq t \leq b$. A
smooth
<strong>reparametrization</strong> is a choice of scalar function
$f(s)$ with $f'(s) > 0$, $f(c) = a$, and $f(d) = b$ such that
\[\vec q(s) = \vec r(f(s))\] traces out the same path.
</p>
</section>
<section>
<h6 class="framelabel">Theorem</h6>
<p style="font-size: smaller">
Arc length is independent of parametrization. Using the notation
of previous slide,
</p>
<p class="fragment r-fit-text">
\[ \int_c^d \left| \vec q '(s) \right|\,ds = \int_c^d \left|
\frac{d}{ds} \vec r (f(s)) \right|\,ds \] \[ = \int_c^d \left|
\vec r '(f(s)) \right| f'(s)\,ds \qquad \begin{cases} w =f(s) \\
dw = f'(s)\,ds \\ \end{cases} \] \[ = \int_a^b |\vec r'(t) |\,dt
\]
</p>
</section>
<section>
<h6 class="framelabel">Arc Length as a Function</h6>
<p>\[ s = f(t) = \int_a^t |\vec r'(w)|\,dw. \]</p>
<video src="assets/arc_length.mp4" controls></video>
</section>
<section>
<h6 class="framelabel">Arc Length as a Function</h6>
<p>\[ s = f(t) = \int_a^t |\vec r'(w)|\,dw. \]</p>
<ul>
<li class="fragment">$\frac{ds}{dt} = |\vec r'(t)|$</li>
<li class="fragment">
For smooth $\vec r$, $f$ is invertible, $t = f^{-1}(s)$.
</li>
</ul>
<p class="fragment">
\[\frac{d}{ds} \vec r(f^{-1}(s)) = \vec r '(f^{-1}(s))
(f^{-1})'(s) = \vec r'(t) \frac{1}{f'(f^{-1}(s))}\]
</p>
<p class="fragment">
\[ = \frac{\vec r'(t)}{|\vec r'(t)|} = \vec T \]
</p>
</section>
<section>
<h6 class="framelabel">Reparametrization by arc length</h6>
<p>
Using the definition from the previous slide, a curve is
paramtrized by arc length $ \vec q(s) $ for $0 \leq s \leq L$ if
$|\vec q'(s)| = 1$.
</p>
</section>
<section>
<h3 class="framelabel">Example</h3>
<p>
Parametrize the
<a
href="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SW50cm8mc2NhbGU9MCZzaG93UGFuZWw9ZmFsc2Umb2JqMF9raW5kPWN1cnZlJm9iajBfY29sb3I9JTIzZmYxNTAwJm9iajBfcGFyYW1zX2E9MCZvYmowX3BhcmFtc19iPTUmb2JqMF9wYXJhbXNfeD1leHAlMjgtdCUyOSZvYmowX3BhcmFtc195PTErLSsyKmV4cCUyOC10JTI5Jm9iajBfcGFyYW1zX3o9MipleHAlMjgtdCUyOSZvYmowX3BhcmFtc190YXU9MC43NDAwNzU2NTg5NzM1MjQ1Jm9iajBfcGFyYW1zX2NvbG9yPSUyMzAwNWFhNyZvYmowX3BhcmFtc19hMD0wJm9iajBfcGFyYW1zX2ExPTE="
target="_blank"
>curve</a
>
\[ \langle e^{-t}, 1-2e^{-t}, 2e^{-t} \rangle \] for $t \geq 0$ by
arc length.
</p>
<p class="fragemnt r-stretch"></p>
</section>
<section>
<h1>Curvature</h1>
</section>
<section
data-background-iframe="https://3demos.ctl.columbia.edu/?currentChapter=Intro&scale=0&showPanel=false&obj0_kind=curve&obj0_color=%234133ff&obj0_params_a=-2&obj0_params_b=2&obj0_params_x=cos%28pi*t%29+-+1&obj0_params_y=sin%28pi*t%29&obj0_params_z=t&obj0_params_tau=0&obj0_params_color=%23ff0000&obj1_kind=curve&obj1_color=%23ff3333&obj1_params_a=-1&obj1_params_b=1&obj1_params_x=t+cos%282+pi+t%29&obj1_params_y=t+sin%282+pi+t%29&obj1_params_z=t+%2F+4&obj1_params_tau=0&obj1_params_color=%2300ff00&obj2_kind=curve&obj2_color=%2300d65d&obj2_params_a=-1&obj2_params_b=1&obj2_params_x=t&obj2_params_y=2+t&obj2_params_z=-t&obj2_params_tau=0&obj2_params_color=%230000ff"
data-background-opacity="0.3"
>
<p>We aim to quantify how <strong>curvy</strong> a curve can be.</p>
<div class="r-stretch"></div>
<p>
Which of these
<a
href="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SW50cm8mc2NhbGU9MCZzaG93UGFuZWw9ZmFsc2Umb2JqMF9raW5kPWN1cnZlJm9iajBfY29sb3I9JTIzNDEzM2ZmJm9iajBfcGFyYW1zX2E9LTImb2JqMF9wYXJhbXNfYj0yJm9iajBfcGFyYW1zX3g9Y29zJTI4cGkqdCUyOSstKzEmb2JqMF9wYXJhbXNfeT1zaW4lMjhwaSp0JTI5Jm9iajBfcGFyYW1zX3o9dCZvYmowX3BhcmFtc190YXU9MCZvYmowX3BhcmFtc19jb2xvcj0lMjNmZjAwMDAmb2JqMV9raW5kPWN1cnZlJm9iajFfY29sb3I9JTIzZmYzMzMzJm9iajFfcGFyYW1zX2E9LTEmb2JqMV9wYXJhbXNfYj0xJm9iajFfcGFyYW1zX3g9dCtjb3MlMjgyK3BpK3QlMjkmb2JqMV9wYXJhbXNfeT10K3NpbiUyODIrcGkrdCUyOSZvYmoxX3BhcmFtc196PXQrJTJGKzQmb2JqMV9wYXJhbXNfdGF1PTAmb2JqMV9wYXJhbXNfY29sb3I9JTIzMDBmZjAwJm9iajJfa2luZD1jdXJ2ZSZvYmoyX2NvbG9yPSUyMzAwZDY1ZCZvYmoyX3BhcmFtc19hPS0xJm9iajJfcGFyYW1zX2I9MSZvYmoyX3BhcmFtc194PXQmb2JqMl9wYXJhbXNfeT0yK3Qmb2JqMl9wYXJhbXNfej0tdCZvYmoyX3BhcmFtc190YXU9MCZvYmoyX3BhcmFtc19jb2xvcj0lMjMwMDAwZmY="
target="_blank"
rel="noopener noreferrer"
>paths</a
>
is "curviest" at the origin?
</p>
</section>
<section>
<h6 class="framelabel">Definition</h6>
<p>
Recall that $\vec T$ is the
<strong>unit tangent vector</strong> to a curve $\vec r(t)$. The
<strong>curvature</strong> $\kappa$ of $\vec r$ at a given point
is given by \[ \kappa = \left|\frac{d\vec T}{ds} \right|. \]
</p>
<p class="fragment">
That is, we measure <em>how much the direction changes</em>
<span class="fragment"><em>per unit arc length</em>.</span>
<span class="fragment">
Alternatively, \[\kappa = \frac{|\vec T'(t)|}{|\vec r'(t)|}\]
</span>
</p>
</section>
<section>
<h6 class="framelabel">Alternative Formula</h6>
<p>\[\kappa = \frac{|\vec T'(t)|}{|\vec r'(t)|}\]</p>
<p class="fragment">
Because \[\left| \vec T'(t) \right| = \left|\frac{d\vec
T}{dt}\right| = \left|\frac{d\vec T}{ds}\frac{ds}{dt} \right|=
\kappa |\vec r'(t)|. \]
</p>
</section>
<section>
<h3 class="framelabel">Example</h3>
<p>
Find the relation between the radius $R$ of a
<a
href="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SG93K1RvJnNjYWxlPTAuMzYmc2hvd1BhbmVsPXRydWUmb2JqMF9raW5kPWN1cnZlJm9iajBfY29sb3I9JTIzMGQwODg3Jm9iajBfYW5pbWF0aW9uPXRydWUmb2JqMF9wYXJhbXNfYT0wJm9iajBfcGFyYW1zX2I9MitwaSZvYmowX3BhcmFtc194PWErY29zJTI4dCUyOSZvYmowX3BhcmFtc195PWErc2luJTI4dCUyOSZvYmowX3BhcmFtc196PTAmb2JqMF9wYXJhbXNfYTA9MSUyRjImb2JqMF9wYXJhbXNfYTE9Mw=="
target="_blank"
rel="noopener noreferrer"
>circle</a
>
and its curvature.
</p>
<div class="fragment">
<p>
$\vec r(t) = \left\langle R\cos t, R\sin t \right\rangle$ has
arclength $s = Rt$. Thus we parametrize by arc length \[\vec
q(s) = \left\langle R\cos \frac{s}{R}, R\sin \frac{s}{R}
\right\rangle \]
</p>
<p class="fragment">\[\kappa = |\vec q''(s) |= \frac{1}{R} \]</p>
</div>
</section>
<section>
<h3 class="framelabel">Demo</h3>
<p>
Use a curve plot in
<a
href="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SG93K1RvJnNjYWxlPTAmc2hvd1BhbmVsPXRydWUmb2JqMF9raW5kPWN1cnZlJm9iajBfY29sb3I9JTIzMGQwODg3Jm9iajBfcGFyYW1zX2E9MCZvYmowX3BhcmFtc19iPTIqcGkmb2JqMF9wYXJhbXNfeD1jb3MlMjh0JTI5Jm9iajBfcGFyYW1zX3k9c2luJTI4dCUyOSZvYmowX3BhcmFtc196PTAuNTUrKitjb3MlMjg1KnQlMjkrKnNpbiUyOHQrKyUyRisyKyUyOSU1RTI="
target="_blank"
rel="noopener noreferrer"
>3Demos</a
>
to explore a curve, its reparametrization by arc length, and the
so-called <strong>osculating circle</strong>, which best
approximates the curve at a point with a circle with a matching
$\kappa$ value.
</p>
</section>
</section>
<section>
<section><h1>Learning Outcomes</h1></section>
<section id="learning-outcomes">
<h6 class="framelabel">You should be able to...</h6>
<ul>
<li>
Parametrize simple curves such as lines, graphs, and circles.
</li>
<li>Match plots of curves with their parametrizations.</li>
<li>
Apply and interpret rules of differential calculus to
vector-valued functions.
</li>
</ul>
</section>
</section>
</div>
</div>
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