₊Finding the optimal relative position of the" repulsive " set of particles on the sphere
Generating a uniform distribution of points on a sphere. Let S be a sphere of radius R around the point O: S:={x∈ℝ³: ||x||=R}. There is a set lP={A1, A2,...,An} of n movable free points on a sphere.
Problem: use the method of Lagrange multipliers find such their distribution corresponding to the maximum sum of all their mutual distances.
This means need to find out such mutual arrangement of "repulsive" set of particles on a sphere, when each point of this set is Geometric median (GM) of the remaining n-1 points. We assume that the equilibrium - stationary state in system of "charges" is reached if the sum of their mutual distances is maximal. Iterative approach of particle placement is applied for achieving a stationary state.
An iterative procedure for calculating a polyhedron with an extreme vertex arrangement allows one point to be fixed at any point on the sphere. In the applet, point A can be set at any point in the sphere. For orientation, you can set Check box "sphere" - true. Click on the "Click→Input,Initial settings"-button and other starting points will be randomly set. Click the "Start"-button and wait for the iteration process to finish!
Note:
1.Theoretical instructions can be found in https://www.geogebra.org/m/rcm4ayek
2.Images and explanations to them in https://www.geogebra.org/m/uegb5ym3
3.This applet might not work properly in the online worksheet, but it works well in a ggb file.
*old version