n=30 Icosidodecahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere.

The applet illustrates the case where 30 vertices of a Icosidodecahedron "induce" the vertices of two other polyhedra: V=32 ●Pentakis Dodecahedron← V=30 ●Icosidodecahedron →V=60 ☐Rhombicosidodecahedron. Generating polyhedra is in https://www.geogebra.org/m/hymcebuw. Description are in https://www.geogebra.org/m/y8dnkeuu and https://www.geogebra.org/m/rkpxwceh.
[size=85]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.
A uniform distribution of points on the surface of a sphere induces two other uniform distributions. Two-variable  function f(φ,θ) over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.
A uniform distribution of points on the surface of a sphere induces two other uniform distributions. Two-variable  function f(φ,θ) over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.
Image
[color=#333333]Distribution of points Pi[/color][color=#ff0000], [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.
max:[/color] Pentakis Dodecahedron
[color=#0000ff]min:[/color]  Icosidodecahedron  
[color=#6aa84f]sad:[/color]  Rhombicosidodecahedron(c)
Distribution of points Pi, test Point, Max/min/saddle -Critical points on a sphere. Vectors ∇f and ∇g at these points. max: Pentakis Dodecahedron min: Icosidodecahedron sad: Rhombicosidodecahedron(c)
[size=85][color=#ff0000]max:[/color] Pentakis Dodecahedron
[color=#0000ff]min:[/color]  Icosidodecahedron  
[color=#6aa84f]sad:[/color]  Rhombicosidodecahedron(c)[/size]
max: Pentakis Dodecahedron min: Icosidodecahedron sad: Rhombicosidodecahedron(c)

Isolines

Isolines
[size=85]Intersection points of implicit functions over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2. A [color=#980000][b]Test[/b][/color] [b][color=#980000]point[/color][/b] -color indicator of the critical point   [/size]
Intersection points of implicit functions over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2. A Test point -color indicator of the critical point
[size=85]Isolines and Intersection points of implicit functions over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.[/size]
Isolines and Intersection points of implicit functions over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.
 Automatic calculation of critical: [color=#ff0000]max[/color], [color=#0000ff]min[/color], [color=#6aa84f]saddle[/color] points -Solutions of the Lagrange equations.
Automatic calculation of critical: max, min, saddle points -Solutions of the Lagrange equations.