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Assignment 4

This assignment concerns the topics covered in Parameterization

1. Designing Parametric Curves

Exercise 1 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.

2. Parameterizing Intersections

The cylinder $x^2+y^2=4$ intersects the paraboloid $z=x^2+2x+y^2$ in a curve. Find a parametric equation $\vec{r}(t)$ for this curve.

Hint: use the cylinder to get a parameterization you like for $x(t)$ and $y(t)$. Then use these together with the paraboloid equation to get a parameterization for $z$. Once you’re done, plug the whole thing into Desmos 3D to get a picture, of your two surfaces and the resulting curve to check that you are right!

3. Finding Higher Dimensional Intersections

In the problem above we parameterize the intersection between two surfaces in 3D. The same trick extends to higher dimensions, 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. This is 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 different three dimensional spaces inside of four dimensional space!

Exercise 2 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 reasoning and communication skills!

Solutions

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

Following the hint, we see the cylinder $x^2+y^2=4$ has a circle of radius 2 in the $xy$ plane. That we know how to parameterize, by $(x(t),y(t))=(2\cos t,2\sin t)$. Now on the paraboloid, we can plug this in and get a formula for $z$:

\[\begin{align*} z=x^2+2x+y^2 &= (2\cos t)^2+2\cdot 2\cos t + (2\sin t)^2\\ &= 4\cos^2 t + 4\cos t + 4\sin^2 t\\ &= 4(\cos^2 t+\sin^2 t)+4\cos t\\ &= 4+4\cos t \end{align*}\]

Now we have formulas for $x$,$y$ and $z$ all in terms of $t$. Putting them together gives a parametric equation \[\vec{r}(t)=\left(2\cos t,2\sin t, 4+4\cos t\right)\]

Finding Higher DImensional 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 $x,y,z,w$:

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