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 .