Number of Coordinates for Sphere Positions: Explained

[pivoxa15]

Homework Statement

How many coordinates are needed to know all the different positions of a sphere?

The Attempt at a Solution

Three, two for specifying every point of the sphere and one for rotating it to a different position. Is only one needed to rotate the sphere?

 

[HallsofIvy]

Are there distinguished points on the sphere? And is the sphere of given radius?

If there are distinguished point on the sphere- that is, if you can distinguish one point from another (as on a globe of the earth), then you would require 5 coordinates: three to determine the center of the sphere, two angles to determine the rotations of the sphere in space.

If there are no distinguished points, if, say, the surface of the sphere is blank, then you would need three coordinates, to determine the center of the sphere. The rotations are irrelevant.

That is assuming that you are talking about a given sphere, with given radius. If you mean determine any (blank) sphere, then you will need four coordinates: three to determine the center of the sphere and one to determine the radius.

 

[pivoxa15]

Are there distinguished points on the sphere? And is the sphere of given radius?

If there are distinguished point on the sphere- that is, if you can distinguish one point from another (as on a globe of the earth), then you would require 5 coordinates: three to determine the center of the sphere, two angles to determine the rotations of the sphere in space.

If there are no distinguished points, if, say, the surface of the sphere is blank, then you would need three coordinates, to determine the center of the sphere. The rotations are irrelevant.

That is assuming that you are talking about a given sphere, with given radius. If you mean determine any (blank) sphere, then you will need four coordinates: three to determine the center of the sphere and one to determine the radius.

Let's assume the whole space is the sphere so the centre is the origin. Assume a unit sphere. Then two coordiates for the rotation?

 

[HallsofIvy]

If you are now talking about the possible positions of the unit sphere, then yes, you need to coordinates for the rotations. One way to think about that is to use "spherical coordinates", \rho, \theta, and \phi. Since the position of any point inside the sphere is determined once the points on the surface are fixed, we can take \rho= 1 and have \theta and \phileft as variables.