Applet. Geometric centers on a sphere "induced" by moving points in three-dimensional space.

Author:
Roman Chijner
Topic:
Algebra, Calculus, Circle, Difference and Slope, Differential Calculus, Differential Equation, Equations, Optimization Problems, Functions, Geometry, Function Graph, Intersection, Linear Functions, Linear Programming or Linear Optimization, Mathematics, Sphere, Surface, Geometric Transformations, Vectors
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 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.
[size=85]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.[/size]
Distribution of points Pi, Cm, GCmax and GCmin on a sphere. Vectors ∇f and ∇g at these points.
[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] 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] [/size]
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]

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