Parametrization of a Portion of a Sphere

[Karnage1993]

Homework Statement

Let $S$ be the portion of the sphere $x^2 + y^2 + z^2 = 9$, where $1 \le x^2 + y^2 \le 4$ and $z \ge 0$. Give a parametrization of $S$ using polar coordinates.

Homework Equations

The Attempt at a Solution

$1 \le r \le 2 \\ 0 \le \theta \le 2\pi \\ 0 \le \phi \le \pi / 2$

We are currently doing 2-dim surfaces in $\mathbb{R}^3$ so I can't do a parametrization of the form $\Phi (r, \theta, \phi)$ so instead, I let $\theta = 4\phi$ so that

$\Phi (r, \phi) = (r\cos(4\phi)\sin(\phi), r\sin(4\phi)\sin(\phi), r\cos(\phi))$.

But when I put it into Mathematica, the graph looks nothing like what it should be. Have I made the correct parametrization? It looks like some sort of spiral shape. What I am thinking it SHOULD look like is the area between the sphere with radius 2 and the unit sphere.

 

[voko]

Can you parametrize $x^2 + y^2 = 1$ on that same sphere in terms of just the polar angle? How about $x^2 + y^2 = 4$ ?

 

[Karnage1993]

I'm not sure what you mean by parametrizing those two circles on the sphere. Are you referring to the projection of the sphere on the $xy$ plane?

 

[LCKurtz]

Homework Statement

Let $S$ be the portion of the sphere $x^2 + y^2 + z^2 = 9$, where $1 \le x^2 + y^2 \le 4$ and $z \ge 0$. Give a parametrization of $S$ using polar coordinates.

Homework Equations

The Attempt at a Solution

$1 \le r \le 2 \\ 0 \le \theta \le 2\pi \\ 0 \le \phi \le \pi / 2$

We are currently doing 2-dim surfaces in $\mathbb{R}^3$ so I can't do a parametrization of the form $\Phi (r, \theta, \phi)$ so instead, I let $\theta = 4\phi$ so that

$\Phi (r, \phi) = (r\cos(4\phi)\sin(\phi), r\sin(4\phi)\sin(\phi), r\cos(\phi))$.

But when I put it into Mathematica, the graph looks nothing like what it should be. Have I made the correct parametrization? It looks like some sort of spiral shape. What I am thinking it SHOULD look like is the area between the sphere with radius 2 and the unit sphere.

You are over-thinking this. Surfaces have two parameters, and in cylindrical coordinates the natural two parameters to use are $r$ and $\theta$. Can you express $x,y,z$ on the sphere in terms of those? And give appropriate limits for them? Your surface should look like a ring shaped piece of the sphere. I note you appear to be trying to use spherical coordinates, is that what you meant to do?

 

[Karnage1993]

Oh, using cylindrical coordinates seems to have made it work. It seems that associating the word sphere with SPHERical coordinates lead me in that wrong direction, haha. And yes, the ring shape you describe is exactly what I was picturing it as. Plugging the new parametrization in, I can confirm that shape is indeed it. Thank you!

 

[voko]

I'm not sure what you mean by parametrizing those two circles on the sphere. Are you referring to the projection of the sphere on the $xy$ plane?

Your sphere's radius is 3. It is definitely possible to have a circle on it with radius 1. $x^2 + y^2 = 1$ is one such circle. You should be able to parametrize this circle with just one parameter, the angle from the XZ plane.

Ditto for $x^2 + y^2 = 4$.

 

[LCKurtz]

Oh, using cylindrical coordinates seems to have made it work. It seems that associating the word sphere with SPHERical coordinates lead me in that wrong direction, haha. And yes, the ring shape you describe is exactly what I was picturing it as. Plugging the new parametrization in, I can confirm that shape is indeed it. Thank you!

Just so you understand, you can do it in spherical coordinates too even though the problem asked for polar (cylindrical) coordinates. You would use $\theta$ and $\phi$ as the parameters and the only part that is slightly trickier is getting the proper limits on $\phi$.