How to generate a 3D spherical symmetric object from a 2D circular graph?

Question

I have a very simple 2d graph.

6 lines separated by equal angle of 60 degrees radiate from the center of a 2d circle, intersecting with the circumference at 6 points. Suppose I know the coordinates of these 6 points, my question is how to turn this graph into a 3d spherical symmetric graph? New lines will be radiating from the center of the sphere, intersecting with the sphere at new points, and how to convert the coordinates of the 6 old points to the coordinates of new vertices on the sphere?

The question seems to be trivial but didn't figure out how to do this. Need some help, thanks.

This question appears to be off-topic because it is about mathematics. The Math StackExchange might be a better place to ask this kind of question. — MvG
I don't think you tell us all you should. Appending a third coordinate with value 0 will turn your 2D points on a circle to 3D coplanar points on a sphere. — Yves Daoust
Cross reference: This question has now been posted on the Math StackExchange. — MvG

MvG MvG · Accepted Answer · 2014-02-19T07:24:01

You might have to elaborate what your symmetry requirements exactly are. But the image I have in my mind when I just read “symmetric” is one where the convex hull of the points on the sphere would be a Platonic solid. And since none of these has a six-fold rotational symmetry, this would be impossible.

But perhaps I'm misreading the question in some other way. Perhaps you don't want to preserve the six-fold planar symmetry, but want six points in total on the sphere. A symmetric arrangement would be the corners of an octahedron, or equivalently the centers of the sides of a cube. Then the simplest coordinates would be:

(1, 0, 0), (-1, 0, 0),
(0, 1, 0), (0, -1, 0),
(0, 0, 1), (0, 0, -1)

How to generate a 3D spherical symmetric object from a 2D circular graph?

1 Answers