V=14 Tetrakis hexahedron 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 S[sub]0[/sub] of radius R[sub]0[/sub] 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.

[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] n=14 [color=#0000ff]min:[/color] n=12 [color=#6aa84f]sad:[/color] n=24 — Distribution of points Pi, test Point, Max/min/saddle -Critical points on a sphere. Vectors ∇f and ∇g at these points. max: n=14 min: n=12 sad: n=24

[size=85]Two-variable function f(φ,θ) over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.[/size] — Two-variable function f(φ,θ) 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]Critial Points, Intersections[/size] — Critial Points, Intersections

Critial Points

[size=85]a,b,c Typen Rhombicuboctahedron[/size] — a,b,c Typen Rhombicuboctahedron

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