V=20 Dodecahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere

Author:Roman Chijner

Topic:Algebra, Calculus, Circle, Difference and Slope, Differential Calculus, Differential Equation, Equations, Optimization Problems, Geometry, Function Graph, Intersection, Linear Programming or Linear Optimization, Mathematics, Sphere, Surface, Vectors

[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.

[size=85][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] Icosahedron
[color=#0000ff]min:[/color] Dodecahedron
[color=#6aa84f]sad:[/color] Icosidodecahedron[/size] — Distribution of points Pi, test Point, Max/min/saddle -Critical points on a sphere. Vectors ∇f and ∇g at these points. max: Icosahedron min: Dodecahedron sad: Icosidodecahedron

[size=85][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] Icosahedron
[color=#0000ff]min:[/color] Dodecahedron
[color=#6aa84f]sad:[/color] Icosidodecahedron[/size] — Distribution of points Pi, test Point, Max/min/saddle -Critical points on a sphere. Vectors ∇f and ∇g at these points. max: Icosahedron min: Dodecahedron sad: Icosidodecahedron

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.

[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.

[size=85]Critical Points[/size] — Critical Points

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