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graphics3d - how to get $n$ equidistributed points on the unit sphere


We can get $n$ equidistributed points in the unit circle using CirclePoints. But how do you get $n$ equidistributed points on the unit sphere(surface of a ball)? The preliminary idea is to suppose that the points are some electric charges with same electric quantity on a sphere. And they have the same effective working area. I think their position is what I want. This is my current solution:


point = RandomPoint[Sphere[], 10^6];
Graphics3D[{Sphere[], Blue, PointSize[.02], 
  Point[Union[point,SameTest -> (EuclideanDistance[#1, #2] < 0.4 &)]]}, 
 Boxed -> False]

points


But this solution cannot draw the specified number of points, and the space is not very equidistributed in some places.



Answer




Aha~ I suppose this question is created while solving this. Am I correct @yode :P


So here's an easy solution, simple, elegant, and may I say even quite fast after some optimization?


pt = With[{p = 
     Table[{x[i], y[i], z[i]}, {i, 80(*number of charges*)}]}, 
   p /. Last@
     NMinimize[
      Total[1/Norm[Normalize[#1] - Normalize[#2]] & @@@ 
        Subsets[p, {2}]], Flatten[p, 1]]];
Graphics3D[{Opacity@.3, Darker@Green, Sphere[], Opacity@1, 
  PointSize@Large, Darker@Blue, Point@*Normalize /@ pt}]


The result is quite good:


result


the setting of the minimization variable is crucial, or the point will not be on surface. But fortunately, our 'kindergarten physics' taught us that when charges are freely scattering in a sphere, they'll always be on surface evenly! Thus this must be some sort of 'most even' form of scattering as it follows physical laws.


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