Description. Geometric Centers on a Sphere.
Previously, we considered the case of finding geometric centers on a circle generated by a system of points that are somehow distributed in space.
Description. Finding Geometric Medians and Geometric Centers on a circle from discrete sample points.
Applet. Finding Geometric Medians and Geometric Centers on a circle from discrete sample points.
Finding the location of geometric medians/centers on the circle from discrete sample points depending on the position of the test point.
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.
There is a set lP={P1, P2,...,Pn} of points and {(xi,yi,zi)∈ℝ3:i = 1,...,n} -their coordinates. Coordinates (xi,yi) are set by moving the Pi points and the coordinates zi -by moving the zi-points (Fig. 3). The point Cm minimizes a function of the sum of squared distances and is called the center of mass https://en.wikipedia.org/wiki/Center_of_mass, centroid or Geometric Center https://en.wikipedia.org/wiki/Centroid of the set lP.
Problem: find the positions of critical points (global minima and maxima too) of the function of squares of the sum of distances not only in the entire in the 3-D space, but on its bounded area.
Definitions of geometric centers can be generalized, in the sense that the critical points of the squares of the sum of distances function searched on a bounded area of 3-D space.
Geometric Centers on a bounded area (on a sphere), „induced“ by the points from 3-D space
The Geometric Center is defined here as point on sphere from where the sum of the squares of all Euclidean distances to each point pi's have at that point: local minimum, maximum or a saddle points.
Critical points can be found using Lagrange multipliersas (Λ(x,y,z,λ)=f(x,y,z)+λ*g(x,y,z)) finding the Extreme values of the function :
fq(x,y,z)= -sum of the squares of the distances to the points pi, subject to a constraining equation:
g(x,y,z)=x2+y2+z2-R2 - points (x,y,z)∈S. I.e. it is necessary to find the critical points fq(x,y,z) subject to: g(x,y,z)=0. Let's denote the resulting point as Sum(lP):= , then ∇f/2=n*(x,y,z)- or ∇f/2=n*(x,y,z)-Sum(lP) and ∇g/2=(x,y,z): ⇒(x,y,z)∼Sum(lP).
The point (x,y,z) (Fig.6) that minimize the sum of the squares of the distances to the points pi is the Geometric Center (gravity center, barycenter, center of mass, centroid) GC: Cm= -its coordinates are the averages of the coordinates of the points from set lP. https://en.wikipedia.org/wiki/Center_of_mass .
![[size=85][i][i][i][i][b][size=150]☛ [/size][/b]In this problem there are only two [color=#ff7700]critical points[/color][color=#333333] (x,y,z)∈[b]S[/b]. The [b]Axis[/b][/color][color=#333333] passing through the radius vector of the point[/color][b] Sum(lP)[/b][color=#333333] passes through the center of mass [/color][/i][b][color=#ff00ff]Cm [/color][/b][/i][color=#333333]and intersects the sphere [b]S[/b] at two points: one corresponds to the [/color][color=#ff0000]global maximum[/color][color=#333333], the other- to the [/color][color=#0000ff]global minimum (Fig.6)[/color][color=#333333].[/color][color=#333333] Their coordinates have [/color][b][i]explicit formulas[/i][/b][color=#333333]:[/color][color=#0000ff] GC[sub]min[/sub]= R* UnitVector(Sum(lP))[/color][color=#333333] and [/color][color=#ff0000]GC[sub]max[/sub]= - R*UnitVector(Sum(lP)[/color][color=#333333]) [/color][color=#1e84cc]- [/color][color=#d5a6bd]two antipodal points on sphere. These points can be found using the Steiner theorem too (Fig.1, [/color][color=#ff0000]r[sub]max[/sub][/color][color=#d5a6bd] and [/color][color=#1e84cc]r[sub]min[/sub][/color][color=#d5a6bd] are shown in the Fig.6).
[/color][/i][/i][i][i][b][color=#1e84cc] f(x,y,z)[/color][color=#333333]: [/color][/b][i][i]In the spherical coordinate system [/i][i][color=#333333][color=#333333]we will have [/color][/color][/i][i]a two-variable function [/i][color=#1e84cc]f(φ,θ)[/color][color=#333333] over a rectangular region: - π ≤φ≤ π and π/2≤θ≤π/2. [/color][i][color=#333333]The two-dimensional surface plot of f(φ,θ) -function of sum of squares of the distances relative to the relative to the [/color][color=#980000]Test point[/color][color=#333333] with angular position (φ,θ) on a sphere of radius R for a given location of points from the set lP is shown in Fig. 4(Checkbox-"f(φ,θ)"=true). [/color][/i][/i][color=#333333] The solution ([/color][color=#ff7700]critical points[/color][color=#333333] of a function f(x,y,z)) of the system of equations can be found [/color]as intersection points of the corresponding implicit functions f'[sub]φ[/sub]=0; f'[sub]θ[/sub]=0, as shown in Fig. 2, or using isolines: Fig.5. The [color=#333333]test point[/color] is set by moving the "[color=#980000]TestP[sub]2[/sub]↔[/color]" point in the plane (φ, θ): Fig.2.[/i][/i][/size]](https://beta.geogebra.org/resource/e26kpuct/jBQJ9teSmZmccNkT/material-e26kpuct.png)
![[size=85][size=85][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333] Distribution of points from lP in 3D and corresponding [/color][/color][/color][/i][/i][color=#333333]2 [/color][color=#ff7700]critical points [/color][/i][/i][size=85][i][i][color=#333333][i][i][color=#0000ff]GC[sub]min[/sub][/color][/i][/i][/color][/i][/i]∈[b]S[/b][i][i][color=#333333][i][i][color=#0000ff][sub] [/sub][/color][/i][/i][/color][color=#d5a6bd][color=#333333][color=#333333][i][i][i][color=#d5a6bd][color=#333333][color=#d5a6bd][color=#333333][color=#333333][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][color=#d5a6bd][color=#333333][color=#d5a6bd][color=#333333][color=#333333]and [/color][/color][/color][/color][/color][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][/color][/color][/color][/color][/color][/i][/i][/i][/color][/color][/color][/i][/i][color=#333333][color=#ff0000]GC[sub]max[/sub][/color][/color][/size][i][i][color=#333333]∈[b]S[/b][/color][i][i][i][i][i][i][i][i][i][i][i][i][i][i][i][i][color=#333333], directions for them from "resulting of all vectors: ∼Sum(lP)" and position vectors on the surface of a sphere is[/color][/i][/i][/i][/i][/i][/i][/i][/i][/i][/i][/i][/i][/i][/i][/i][/i][/i][/i][color=#333333] shown in [/color][size=85][color=#d5a6bd][color=#333333][color=#333333][i]Fig. 6(Checkbox-"Points3D"=true)[/i][/color][/color][/color][/size]. For illustration purposes, the vectors are attached to the points.
[/size][/size][size=85][size=85][size=85][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][color=#d5a6bd][color=#333333][i][i][color=#d5a6bd] [img]data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOCIgaGVpZ2h0PSIxOCIgdmlld0JveD0iMCAwIDI0IDI0Ij48cGF0aCBmaWxsPSJub25lIiBkPSJNMCAwaDI0djI0SDB6Ii8+PHBhdGggZD0iTTcgMTRINXY1aDV2LTJIN3YtM3ptLTItNGgyVjdoM1Y1SDV2NXptMTIgN2gtM3YyaDV2LTVoLTJ2M3pNMTQgNXYyaDN2M2gyVjVoLTV6Ii8+PC9zdmc+[/img][/color][/i][/i]With [/color][/color][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][url=https://www.geogebra.org/m/smfgsshb]applet[/url][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][color=#d5a6bd][color=#333333] you can visually observed and explore [/color][/color][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][size=85][size=85][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][color=#d5a6bd][color=#333333][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][i][i][color=#333333][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][color=#d5a6bd][color=#333333][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][i][i][color=#333333]2 [/color][/i][/i][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][/color][/color][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][/color][/i][/i][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][/color][/color][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][size=85][size=85][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][color=#d5a6bd][color=#333333][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][i][i][color=#333333][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][color=#d5a6bd][color=#333333][color=#d5a6bd][color=#333333][color=#333333][i][i][i][i][color=#333333][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][color=#d5a6bd][color=#333333][i][color=#333333][i][color=#333333]geometric [/color][color=#ff00ff]centers[/color][/i][/color][/i][/color][/color][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][/color][/i][/i][/i][/i][/color][/color][/color][/color][/color][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][/color][/i][/i][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][/color][/color][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][color=#333333]:[/color][/size][/size][/size][/size][/size][/size][/size] [size=85][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][color=#d5a6bd][color=#333333][color=#d5a6bd][color=#333333][color=#333333][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][color=#d5a6bd][color=#333333][color=#d5a6bd][color=#333333][color=#333333][i][i][i][i][color=#333333][i][i][color=#0000ff]GC[sub]min [/sub][/color][/i][/i][/color][/i][/i][/i][/i]and [/color][/color][/color][/color][/color][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][/color][/color][/color][/color][/color][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][color=#ff0000]GC[sub]max [/sub][/color][color=#333333]on a sphere[/color][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][color=#d5a6bd][color=#333333][color=#d5a6bd][color=#333333][color=#333333][i][i][i][i][color=#333333] S , [i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][color=#d5a6bd][color=#333333][i][color=#333333] [i][color=#333333]geometric [/color][color=#ff00ff]center Cm[/color][color=#333333] in ℝ[sup]3[/sup][/color][/i] [i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][color=#d5a6bd][color=#333333][color=#d5a6bd][color=#333333][color=#333333][i][i][i][i][color=#333333](Fig. 6) [/color][/i][/i][/i][/i][/color][/color][/color][/color][/color][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][/color][/i][/color][/color][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][/color][/i][/i][/i][/i][/color][/color][/color][/color][/color][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i][/size][size=85][i][i][i][i][color=#d5a6bd][color=#333333][color=#333333][i][i][i][color=#d5a6bd][color=#333333][i][color=#333333]changing the positions of the points from lP[/color][/i][/color][/color][/i][/i][/i][/color][/color][/color][/i][/i][/i][/i]. [/size]](https://beta.geogebra.org/resource/gcbsgk56/WrIgeyvJUAHAfPB6/material-gcbsgk56.png)
With applet you can visually observed and explore 2 geometric centers: GCmin and GCmax on a sphere S , geometric center Cm in ℝ3 (Fig. 6) changing the positions of the points from lP.