Spherical Geometry Question

[SaxonViolence]

If I have a Sphere—like the Earth, or whatever—and I have "X" number of point that I want to distribute in an equidistant manner over the surface of the Sphere...

 

Is there a formula that will tell me how far apart the points are—both straight line and along great circle arcs?

 

And will it tell me the angle between the Radii connecting each point to the center of the Sphere?

 

{Obviously the vertices of the Regular Polygons and many _(all?) of the Regular Stellated Polyhedra are special cases.}

 

Also, can I generalize my formula to Regular Ellipsoids?

 

How about "X" number of points to be distributed equidistantly in a closed space—like the interior of a Sphere or what have you?

 

Are there numbers that cannot be arranged Equidistantly?

 

 

Saxon Violence

 

 

BIG PS: Upon reflection, with anything with more points than a Tetrahedron, you won't be able to get all the points Equidistant—I don't think.

 

Substitute "Evenly Distributed."

 

Thanks.

Edited May 31, 2013 by SaxonViolence

[LaurieAG]

To get the most accurate result for a sphere you would use an Icosahedron, divide X by 20 and use that number (ideally a square) to work out how much further you have to subdivide each facet to create your geodesic grid. i.e. each point will be in the center of an individual face (equilateral triangle) and is therefore equidistant.

 

http://en.wikipedia.org/wiki/Geodesic_grid

 

A geodesic grid is a technique used to model the surface of a sphere (such as the Earth) with a subdivided polyhedron, usually an icosahedron.

http://en.wikipedia.org/wiki/Icosahedron