Parametrizing a Self-Intersecting Rectangle

[Karnage1993]

Homework Statement

Let S be the self-intersecting rectangle in $\mathbb{R}^3$ given by the implicit equation $x^2−y^2z = 0$. Find a parametrization for S.

Homework Equations

The Attempt at a Solution

This is my first encounter with a surface like this. The first thing that came to my mind was letting $z = f(x,y)$ so that the parametrization can be given as:

$\Phi(u,v) = (u,v,\displaystyle \frac{u^2}{v^2})$

The problem I'm having is finding the limits for $u$ and $v$. When I plugged the implicit equation into Mathematica, the surface looks like an X shape from above, though I had to expand the range for the axes by a very large amount for it to look like that. I do know that $v$ has to be non-zero, but that's pretty much it.

 

[LCKurtz]

Homework Statement

Let S be the self-intersecting rectangle in $\mathbb{R}^3$ given by the implicit equation $x^2−y^2z = 0$. Find a parametrization for S.

Homework Equations

The Attempt at a Solution

This is my first encounter with a surface like this. The first thing that came to my mind was letting $z = f(x,y)$ so that the parametrization can be given as:

$\Phi(u,v) = (u,v,\displaystyle \frac{u^2}{v^2})$

The problem I'm having is finding the limits for $u$ and $v$. When I plugged the implicit equation into Mathematica, the surface looks like an X shape from above, though I had to expand the range for the axes by a very large amount for it to look like that. I do know that $v$ has to be non-zero, but that's pretty much it.

If you look at the traces when $z = c^2$ for you get $x^2-y^2c^2=0$ so the level curves of that surface are $(x+cy)(x-cy)=0$, which is two intersecting straight lines. I would think about letting one of the parameters be $z=c^2,\ y = y$ and doing it as two pieces. That would certainly solve your range problem and also eliminates your problem when $c=0$.

 

[Karnage1993]

I don't quite get why you set $z = c^2 \ge 0$. For example, if $x=0,y=0$ and $z$ negative, that also satisfies the equation, right?

 

[LCKurtz]

I don't quite get why you set $z = c^2 \ge 0$. For example, if $x=0,y=0$ and $z$ negative, that also satisfies the equation, right?

Yes. It's a bit strange though. The $z$ axis solves the equation alright, but there is no surface if $z<0$. You just have the bare axis. I wouldn't include it as part of the parameterization, but your mileage may vary.

 

[Karnage1993]

Okay, so suppose one piece of the surface is when $x = cy$. How would I go about finding the limits for $c,y$?

 

[LCKurtz]

Okay, so suppose one piece of the surface is when $x = cy$. How would I go about finding the limits for $c,y$?

You know $c$ parameterizes the $z$ axis which is $z\ge 0$ for our purposes, and we are using $c^2$ for convience, so $c\ge 0$ is easy. Then those straight lines go forever so you wouldn't have any limit on $y$. Have you tried a parametric plot to compare with your original plot?