Is there any mathematical formula to find the coordinates of equidistant poins on the surface of a sphere?t

Question

The general form of the coordinates is:

coords = r $\{$$\cos\theta$ $\sin\phi$, $\sin\theta$ $\sin\phi$, $\cos\phi$}

I've considered the radius $r=1$. Now varying the angles $\theta$ and $\phi$, I can generate equidistant points on the surface of the sphere.

But, I want those points to represent the vertices of a regular polyhedron (not in a true sense always; for example, I will consider a triangular bipyramid for $n=5$) for $n\ge4$, where $n$ is the number of points so that the vertices lie on the surface of the sphere. In this way, probably (please correct me if I am wrong) I can have polyhedron vertices for any $n\ge4$.

Can I have any formula to perform this task?

Answer 1

I don't find your question very clear.

coords = r {cos⁡θ sin⁡ϕ, sin⁡θ sin⁡ϕ, cos⁡ϕ} = {x,y,z}

For (r, θ, ⁡ϕ) = (radial, azimuthal, polar) (!)

So why not write just (x,y,z)?

Maybe you mean to use spherical coordinates

coords = {r, θ, ⁡ϕ} 

instead?

You say you can already generate equidistant points, by varying theta and phi. If you vary theta by 360/i degrees where i is an integer in the range [1,360]. (Or alternatively phi), you have your equidistant points already, haven't you?

I would think just calculating the Euclidian distance between 2 points on the sphere (I would use cartesian coordinates instead to this end) is enough as the size of the sphere is fixed. d = sqrt(xx + yy + z*z)

Because you asked for a formula, you should be able to derive it from here: http://mathworld.wolfram.com/SphericalCoordinates.html