Parametrization of a Portion of a Sphere

In summary, the problem asks for a parametrization of a portion of a sphere using polar coordinates. The solution is to use cylindrical coordinates instead, with the parameters ##r## and ##\theta## and appropriate limits. The resulting shape is a ring on the sphere. It is also possible to parametrize the circles with radius 1 and 4 on the sphere using just one parameter, the angle from the XZ plane. This can be done in either cylindrical or spherical coordinates.
  • #1
Karnage1993
133
1

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.
 
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  • #2
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 ## ?
 
  • #3
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?
 
  • #4
Karnage1993 said:

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?
 
  • #5
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!
 
  • #6
Karnage1993 said:
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 ##.
 
  • #7
Karnage1993 said:
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##.
 

1. What is parametrization of a portion of a sphere?

Parametrization of a portion of a sphere is a mathematical process that involves assigning numerical values to the coordinates of points on a portion of a sphere. This allows us to describe the portion of the sphere with a set of equations, making it easier to analyze and study.

2. Why is parametrization of a portion of a sphere important?

Parametrization of a portion of a sphere is important because it allows us to mathematically represent and manipulate complex curved surfaces, such as spheres. This is useful in many fields of science, including physics, engineering, and computer graphics.

3. How is parametrization of a portion of a sphere done?

Parametrization of a portion of a sphere is typically done using spherical coordinates. These coordinates involve using two angles, θ and φ, and a radius r to specify a point on the sphere. The equations used to convert between spherical coordinates and cartesian coordinates are used to parametrize the portion of the sphere.

4. What are some applications of parametrization of a portion of a sphere?

Parametrization of a portion of a sphere has various applications in different fields. In physics, it is used to describe the motion of objects on curved surfaces. In engineering, it is used in the design and analysis of curved structures. In computer graphics, it is used to create 3D models of objects with spherical surfaces.

5. What are the limitations of parametrization of a portion of a sphere?

Parametrization of a portion of a sphere is limited to only portions of the sphere that can be represented using spherical coordinates. This means that it may not be suitable for representing highly irregular or complex portions of a sphere. Additionally, parametrization may become more complex for portions of the sphere with changing curvature or surface features.

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