# 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

*Applet*

*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.**P1, P2,...,Pn*

*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={*} of points and*{(x

_{i},y

_{i},z

_{i})∈ℝ

^{3}:i = 1,...,n} -their coordinates. C

*oordinates (x*._{i},y_{i}) are set by moving the P_{i }points and the coordinates z_{i}-by moving the z_{i}-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*

*, but**in the 3-D space**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 p_{i}'s have at**that point***, subject to a constraining equation: g(x,y,z)=x*

*sum of the squares of the distances to the points pi**:*local minimum, maximum or a saddle points. Critical points can be found using**Lagrange multipliers****as**(*Λ(x,y,z,λ)=f(x,y,z)+λ*g(x,y,z))*finding the Extreme values of the function :*f*-_{q}(x,y,z)=^{2}+y

^{2}+z

^{2}-R^{2}-*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*p_{i}*is the**GC:***Geometric Center**(*gravity center,**barycenter, center**of mass,**centroid)***Cm**=**-**its coordinates are the averages of the coordinates of the points from set lP. https://en.wikipedia.org/wiki/Center_of_mass .

*and intersects the sphere***☛**In this problem there are only two critical points (x,y,z)∈**S**. The**Axis**passing through the radius vector of the point**Sum(lP)**passes through the center of mass**Cm****S**at two points: one corresponds to the global maximum, the other- to the global minimum (Fig.6). Their coordinates have**: GC***explicit formulas*_{min}= R* UnitVector(Sum(lP)) and GC_{max}= - R*UnitVector(Sum(lP)) - two antipodal points on sphere. These points can be found using the Steiner theorem too (Fig.1, r_{max}and r_{min}are shown in the Fig.6).

**f(x,y,z):***The solution (critical points of a function f(x,y,z)) of the system of equations can be found as intersection points of the corresponding implicit functions f'**In the spherical coordinate system**we will have**a two-variable function*f(φ,θ) over a rectangular region: - π ≤φ≤ π and π/2≤θ≤π/2.*The two-dimensional surface plot of f(φ,θ) -function of sum of squares of the distances relative to the relative to the Test point 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).*_{φ}=0; f'_{θ}=0, as shown in Fig. 2, or using isolines: Fig.5. The test point is set by moving the "TestP_{2}↔" point in the plane (φ, θ): Fig.2.

*2 critical points**Distribution of points from lP in 3D and corresponding**∈*

*GC*_{min}**S**

*GC*

_{ }*and*_{max}

*shown in*

*∈***S***, directions for them from "resulting of all vectors: ∼Sum(lP)" and position vectors on the surface of a sphere is**Fig. 6(Checkbox-"Points3D"=true)*. For illustration purposes, the vectors are attached to the points.

*applet*

*With*

*you can visually observed and explore*

*2**:*

*geometric centers**GC*

*and**GC*_{min }_{max }on a sphere

*S ,**geometric center Cm in ℝ*^{3}*(Fig. 6)**.*

*changing the positions of the points from lP*