GeoGebra

n=4 Tetrahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere.

Author:Roman Chijner
The applet illustrates the case where 4 vertices of a regular tetrahedron "induce" the vertices of two other polyhedra: 4 ●Regular Tetrahedron← 4 ●Regular Tetrahedron →6 ●Regular Octahedron. Description are in https://www.geogebra.org/m/y8dnkeuu and https://www.geogebra.org/m/rkpxwceh.
[size=50]A system of points on a sphere S of radius R “induces” on the sphere S0 of radius R0 three different sets of points, which are [color=#93c47d]geometric medians (GM)[/color] -local [color=#ff0000]maxima[/color], [color=#6d9eeb]minima[/color] and [color=#38761d]saddle[/color] points sum of distance function f(x). The angular coordinates of the spherical distribution of a system of points -[color=#0000ff] local minima[/color] coincide with the original system of points.[/size]
A system of points on a sphere S of radius R “induces” on the sphere S0 of radius R0 three different sets of points, which are geometric medians (GM) -local maxima, minima and saddle points sum of distance function f(x). The angular coordinates of the spherical distribution of a system of points - local minima coincide with the original system of points.
Image
Image
Distribution of points Pi, [color=#5b0f00]test Point[/color], [color=#ff0000]Max[/color]/[color=#0000ff]min[/color]/[color=#38761d]saddle[/color] -[color=#333333]Critical points[/color] on a sphere. Vectors ∇f and ∇g at these points.
Distribution of points Pi, test Point, Max/min/saddle -Critical points on a sphere. Vectors ∇f and ∇g at these points.
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[color=#ff0000]max:[/color] Tetrahedron  [color=#0000ff]min: [/color]Tetrahedron [color=#6aa84f]sad:[/color] Octahedron
max: Tetrahedron  min: Tetrahedron sad: Octahedron
Two-variable function f(φ,θ) over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.
Two-variable function f(φ,θ) over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.
Isolines and Intersection points of implicit functions over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.
Isolines and Intersection points of implicit functions over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.

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