V=8 Cube. Images: A critical points scheme for Generating uniformly distributed points on a sphere.

Author:
Roman Chijner
Topic:
Algebra, Calculus, Circle, Cube, Difference and Slope, Differential Calculus, Differential Equation, Optimization Problems, Geometry, Function Graph, Intersection, Linear Programming or Linear Optimization, Mathematics, Sphere, Surface, Vectors
The applet illustrates the case where 4 vertices of a regular tetrahedron "induce" the vertices of two other polyhedra: V=6 ●Octahedron← V=8 ●Cube →V=12 ●Cuboctahedron. Generating polyhedra is in https://www.geogebra.org/m/rtm56gkb. 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.
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.
[color=#ff0000]max:[/color] Octahedron [color=#0000ff] min:[/color] Cube [color=#6aa84f]sad:[/color] Cuboctahedron
max: Octahedron min: Cube sad: Cuboctahedron
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.
Intersection points of implicit functions over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.
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.
Isolines and Intersection points of implicit functions over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.

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