This applet is used to study the distribution of geometric centers on a sphere of radius R, „induces“ by the discrete sample of movable points in the 3-D space.
Description is in https://www.geogebra.org/m/nge6gawt

GeoGebra

-Settings plane, equalities from the Steiner theorem
-Two-variable function f(φ,θ) over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.
- Intersection Implicit Curves f'φ(φ, θ)=0; f'θ(φ,θ)=0 over a rectangular region φ∈[-π,π], θ∈[-0.5π,0.5π]
-Distribution of points Pi and their local maxima/minima and saddle -critical points of distance sum function f(φ,θ) on a sphere + test Point. Vectors ∇f and ∇g at these points.

Distribution of points Pi, Cm, GCmax and GCmin on a sphere. Vectors ∇f and ∇g at these points.

Fig1. Distribution of points Pi and their local maxima/minima -critical points of distance sum function f(φ,θ) on a sphere + test Point. Vectors ∇f and ∇g at these points.
Fig.2 Two-variable function f(φ,θ) over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.
Fig.3 Intersection Implicit Curves f'φ(φ, θ)=0; f'θ(φ,θ)=0 and Isolines over a rectangular region φ∈[-π,π], θ∈[-0.5π,0.5π]