$$ \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\EE}{\mathbb{E}} \newcommand{\HH}{\mathbb{H}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\length}{\operatorname{length}} \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}} \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}} \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}} \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}} \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} \newcommand{\partitions}[1]{\mathcal{P}_{#1}} \newcommand{\erf}{\operatorname{erf}} \newcommand{\ihat}{\hat{\imath}} \newcommand{\jhat}{\hat{\jmath}} \newcommand{\khat}{\hat{k}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \newcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)} $$

Assignment 3

This assignment concerns the topics covered in Parameterization

1. Parameteric Curves in the Sciences

GPS technology is frequently used by biologists to track organisms and understand their migration habits. This has particularly been helpful for learning about animals that range over large or uninhabited areas - a fun example being the tracking of great white sharks in the Pacific ocean! Here’s a live tracker of some sharks off the coast if you want to see where they are right now. https://www.ocearch.org/tracker/ Below is the tracking data from a particular shark’s locations in the winter of 2014:

GPS tracking data for a Great White off the California coast.

Exercise 1 Part 1: Let $\mathrm{shark}(t)$ be the parametric curve giving the location of the tracked shark. What dimension is the output of this curve? (Is it a curve in 2d, 3d, 4d space?). What are some reasonable coordinates you could write this curve in? (Like, what physical meaning could you attach to the different entires of the parametric curve?)

Part 2: What are some questions that scientists might ask about a great white shark’s life? How could you translate these into questions about the parametric curve $\mathrm{shark}(t)$ so that they could be answered with GPS data?

There are plenty of different correct answers to this question!

2. Designing Parametric Curves

Exercise 2 Create a parametric curve $f(t)=(x(t),y(t))$ in the plane $\mathbb{R}^2$ looks like the following (it doesn’t have to look exactly like this)

More precisely, find a parametric equation with the following properties:

As $t$ increases in $[0,\infty)$, the curve $f(t)$ spirals around the origin at a uniform rate.
At the beginning $f(0)=(0,0)$ and then the spiral approaches a radius of 2 closer and closer, but never quite reaches it as $t\to\infty$.

Hint: draw a graph of what you want the radius function to look like, then try to build such a function! There are infinitely many correct answers to this question, but explain your answer and how you came to it.

3. Finding Intersections

In class we saw how to parameterize the intersection between two surfaces in three dimensional space. At the beginning, it was helpful to have a picture of the surfaces (like a cylinder and a plane) that we were intersecting to get some intuition, but it turned out we didn’t need a picture at all if we just tried methodically to build the parametric equation up one step at a time, starting from the equation with less variables and moving to the one with more.

The same tricks work in higher dimensions and are a powerful tool for mathematicians to understand shapes that we can never hope to see. In this problem, you are going to parameterize the intersection between three differen three dimensional spaces inside of four dimensional space!

Exercise 3 The three spaces are described in the Cartesian coordinate system $(x,y,z,w)$ as follows:

The space $x^2+w^2=4$
The space $w+z^3=1$
The space $x+y+z=1$.

What is a parametric curve $\vec{r}(t)=(x(t),y(t),z(t),w(t))$ that traces out their intersection in four dimensional space?

When you write your answer, do not just give an equation instead, explain each step of your thought process in full sentences: the goal of this homework is to improve mathematical reasioning and communication skills!

Solutions

Parametric Curves in the Sciences

There are tons of correct answers to this question. Perhaps you modeled the sharks position as a curve into $\RR^2$, by tracking the latitude and longitude of the shark. Or maybe you wanted to track its location in 3d by recording latitude longitude and depth: then it would be a function into $\RR^3$. OR - even though time is the input of the curve, maybe you wish to also store the time paramter in our output data: then it would be a curve into 4-dimensional space. Any of these are valid answers, depending on how you want to record things.

Questions you could ask:

What is the farthest west the shark swam this year? This would become a question about “what is the maximum value of $y(t)$” if $y$ is the component of the sharks position measuring longitude (east/west diretion).
What is the fastest the shark swam? This would be a question about the speed $\|\mathrm{shark}^\prime(t)\|$, and the maximum value of this.
Did the shark ever stop swimming? This would be a question about the shark’s velocity: is $\mathrm{shark}^\prime(t)$ ever equal to zero?

Designing Parametric Curves

Because we want the curve to spin around at a constant rate, we know we should start with our usual parameterization of the unit circle $(\cos t,\sin t)$. We want to modify this into a spiral, so we will multiply it by a radius function $r(t)$ to get a curve

\[\vec{c}(t)=\left(r(t)\cos(t),r(t)\sin(t)\right)\]

The problem gives us some constraints on what $r(t)$ should be: it needs to be zero at $t=0$ and needs to asymptote to $2$ as $t\to\infty$. This gives us a good sense of what its graph should look like:

A possible function $\mathrm{radius}(t)$.

Any function whose graph looks sort of like this will do. There are plenty of possibilities! Here are a couple

You could remember from calculus that the function $\arctan(x)$ is zero at $x=0$, has an asymptote at $\pi/2$. Thus, $4/\pi\arctan(x)$ has an asymptote at $2$, and would work.

The curve $\frac{4}{\pi}\arctan(t)(\cos(t),\sin(t))$.

You could build a function with an asymptote at 2 as a rational function, for example $2x/(x+1)$.

The curve $\frac{2t}{t+1}(\cos(t),\sin(t))$.

Finding Intersections

Here we are given three hypersurfaces in four dimensional space, so we cannot picture exactly what is going on, but we can do the mathematics exactly analogous to what we’ve done in 2 and 3 dimensions.

The equations we have available are:

\[ x^2+w^2=4 \hspace{1cm} w+z^3=1\hspace{1cm}x+y+z=1\]

The first two each deal with only two variables at a time, so either of these is a good spot to start. Let’s start with the first one: we see $x^2+w^2=4$, which we recongize as the equation of a circle. Thus we can parameterize these two coordinates in terms of a parameter $t$ as

\[\pmat{x(t)\\ w(t)}=\pmat{2\cos t\\ 2\sin t}\]

Now the second equation involves a $w$, so we can substitute what we learned above into it, getting $2\cos t + z^3=1$. We can then solve this for $z$, to figure out what it must be:

\[z= \sqrt[3]{1-2\sin t}\]

Now we know $x(t),z(t)$ and $w(t)$ so all we need to find is $y(t)$. But the third equation has a $y$ in it, so we can solve for $y$

\[\begin{align*} y&=1-x-z\\ &= 1-2\cos(t)-\sqrt[3]{1-2\sin t} \end{align*}\]

Putting this all together, we get a parametric curve for all of $x,y,z,w$:

\[\vec{c}(t)=\pmat{x(t)\\ y(t)\\ z(t)\\ w(t)}=\pmat{ 2\cos t\\ 1-2\cos(t)-\sqrt[3]{1-2\sin t}\\ \sqrt[3]{1-2\sin t}\\ 2\sin t }\]