n=30 Icosidodecahedron. 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, Optimization Problems, Geometry, Function Graph, Intersection, Linear Programming or Linear Optimization, Mathematics, Sphere, Surface, Vectors

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.

[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.

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