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Applet. 10 moving points „induces“ two Geometric Centers -two antipodal points on a sphere.

Author:Roman Chijner
This applet is used to study the distribution of geometric centers on a sphere of radius R, „induces“ by the discrete sample of movable points in the 3-D space. Description is in https://www.geogebra.org/m/nge6gawt
[size=85]-Settings plane, equalities from the Steiner theorem -Two-variable function f(φ,θ) over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2. - Intersection Implicit Curves f'φ(φ, θ)=0; f'θ(φ,θ)=0 over a rectangular region φ∈[-π,π], θ∈[-0.5π,0.5π] -Distribution of [color=#1e84cc]points Pi[/color] and their local [color=#ff0000]maxima[/color]/[color=#0000ff]minima[/color] and [color=#6aa84f]saddle[/color] -[color=#ff7700]critical[/color] points of distance sum function f(φ,θ) on a sphere + [color=#b45f06]test Point[/color]. Vectors ∇f and ∇g at these points.[/size]
-Settings plane, equalities from the Steiner theorem -Two-variable function f(φ,θ) over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2. - Intersection Implicit Curves f'φ(φ, θ)=0; f'θ(φ,θ)=0 over a rectangular region φ∈[-π,π], θ∈[-0.5π,0.5π] -Distribution of points Pi and their local maxima/minima and saddle -critical points of distance sum function f(φ,θ) on a sphere + test Point. Vectors ∇f and ∇g at these points.
Distribution of points Pi, [color=#ff00ff]Cm[/color], [color=#ff0000]GCmax[/color] and [color=#0000ff]GC[/color][color=#0000ff]min[/color] on a sphere. Vectors ∇f and ∇g at these points.
Distribution of points Pi, Cm, GCmax and GCmin on a sphere. Vectors ∇f and ∇g at these points.
[size=85] Fig1. Distribution of [color=#1e84cc]points Pi[/color] and their local [color=#ff0000]maxima[/color]/[color=#0000ff]minima[/color] -[color=#ff7700]critical[/color] points of distance sum function f(φ,θ) on a sphere + [color=#b45f06]test Point[/color]. Vectors ∇f and ∇g at these points. Fig.2 Two-variable function f(φ,θ) over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2. Fig.3 Intersection Implicit Curves f'φ(φ, θ)=0; f'θ(φ,θ)=0 and Isolines over a rectangular region φ∈[-π,π], θ∈[-0.5π,0.5π] [/size]
Fig1. Distribution of points Pi and their local maxima/minima -critical points of distance sum function f(φ,θ) on a sphere + test Point. Vectors ∇f and ∇g at these points. Fig.2 Two-variable function f(φ,θ) over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2. Fig.3 Intersection Implicit Curves f'φ(φ, θ)=0; f'θ(φ,θ)=0 and Isolines over a rectangular region φ∈[-π,π], θ∈[-0.5π,0.5π]

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