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 .