Checking for a point inside a region of a spherical surface

[egvega]

Hi everybody,

I am working on a 3D Sphere rotation algorithm.

I have a point X lying on a unit sphere in the spherical coordinate system. I have divided the sphere in regions or areas delimited by 3 (polar areas) and 4 points which are also on the unit sphere. The regions have different labels.

I need to find out on which spherical surface (region 1,2,...) lies point X. Then, when I randomly rotate the sphere but kept fixed point X, which will be the new region? I need something general to handle any rotation of the sphere.

Can anyone tell me how to do this?

Thanks a lot in advance.

Esteban

 

[fresh_42]

You could use $\mathbb{S}^2\simeq SO(3,\mathbb{R})/SO(2,\mathbb{R}) \simeq \mathbb{P}(1,\mathbb{C})$ where you have a natural operation for the rotations, or $\mathbb{S}^3\simeq SO(4,\mathbb{R})/SO(3,\mathbb{R}) \simeq U(1,\mathbb{H}) \simeq SU(2,\mathbb{C})$ since I'm not sure whether you meant the 2-sphere or the 3-sphere.

Cp. https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/ for other presentations.