Description. Geometric Centers on a Sphere.
- 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
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.