[size=85][i][i][i][url=https://www.geogebra.org/m/bm76gdkb]Applet[/url] is used to study the distribution of [/i][color=#333333]geometric [/color][color=#ff7700]medians [/color][/i][color=#000000][i]on a sphere of radius R, „induces“ by the discrete sample of movable points in the 3-D space[/i][/color][color=#333333]. On a circle- in [url=https://www.geogebra.org/m/puqnepmv]Applet[/url].[/color] [br] There is a set lP={P[/i]1, P2,...,Pn[i]} of points and [/i][color=#222222]{(xi,yi[/color],z[color=#222222]i)∈ℝ3:i = 1,...,n} -their coordinates[/color][i]. [/i][i]Let a point (x,y,z) minimize a function of the sum of distances from this point to points lP. It is called [/i][i][color=#0000ff]Fermat's point [/color][/i][i][url=https://en.wikipedia.org/wiki/Fermat_point]https://en.wikipedia.org/wiki/Fermat_point[/url] [/i][i]or the [/i][color=#ff8080][i]Geometric median [/i][/color][color=#6557d2][url=https://en.wikipedia.org/wiki/Geometric_median]https://en.wikipedia.org/wiki/Geometric_median[/url] [/color][i]of the set lP. [/i][br][i][u]Problem[/u]: [/i][i]find the positions of [color=#ff7700]critical [/color][color=#222222]points[/color] (global [color=#0000ff]minima[/color] and [color=#ff0000]maxima[/color] too) of the function of [/i][color=#222222][i]sum of distances [/i][/color][i]not only in the entire [i][color=#000000][i]in the 3-D space[/i][/color][/i], but[/i][color=#222222][i] on its bounded area[/i][/color][i].[br][/i] Definitions of geometric [color=#ff7700]medians[/color] can be generalized[color=#222222], in the sense that the [/color][color=#ff7700]critical[/color] [color=#222222]points of the [/color][color=#222222][i]sum [/i][/color][color=#222222][i]of distances function searched[color=#222222][i] on a bounded area [i]of [i][color=#000000][i]3-D space[/i][/color][/i][/i][/i][/color][/i][/color][/size][color=#222222][i][color=#222222][i][i][i][color=#333333]. [/color] [/i][/i][/i][/color][/i][/color]

[size=85][color=#222222][color=#222222][color=#222222][color=#222222][color=#222222][color=#222222]The [color=#ff7700]Geometric Median [/color]is defined here as [/color][color=#333333][b]point [/b][/color]on [i]a sphere[/i] from where the [/color][color=#1e84cc][i]function of [/i][/color][color=#1e84cc]sum of the distances [/color]to each point [/color][/color][/color][/color][b]P[/b][b]i[/b][color=#222222][color=#222222]'s have at [color=#222222][b]that point[/b][/color][/color][/color][i][color=#333333]: [/color][/i]local [color=#0000ff]minimums[/color], [color=#ff0000]maximums[/color] or [color=#6aa84f]saddle[/color] points.[color=#1e84cc][i][br][color=#333333] Critical points can be found using [/color][/i][/color][i][b]Lagrange multipliers [/b]as (Λ(x,y,z,λ)=f(x,y,z)+λ*g(x,y,z)) finding the Extreme values of the function :[/i][color=#222222][color=#1e84cc][i][i][i][i] f(x,y,z)=[math]\sum_{i=1}^n\sqrt{\left(x-x_i\right)^2+\left(y-y_i\right)^2+\left(z-z_i\right)^2}[/math][/i] [/i]-[color=#1e84cc]sum of the distances [/color][color=#333333]from[/color] ([/i][/i][/color][/color][i]x,y,z) to each point p[sub]i[/sub]'s, subject to a constraining equation: [/i][i][i][i][i][color=#333333]g(x,y,z)[/color][/i][/i]=[math]\sqrt{\left(x\right)^2+\left(y\right)^2+\left(z\right)^2}-R[/math][/i] [/i][color=#222222][color=#1e84cc][i][i][color=#333333].[/color][/i][/i][/color][/color][i] Let [b]S[/b] be a sphere of radius R around the point O: [b]S[/b]:[/i][color=#222222][color=#1e84cc][i]={[/i][/color][/color][math]\vec{r}[/math][i][color=#333333]∈ℝ[sup]3[/sup][/color][color=#222222]: ||[/color][/i][math]\vec{r}[/math][i][color=#222222]||[/color][color=#333333]=R[/color][color=#222222]}. [i][i][color=#222222][color=#1e84cc][i][i][color=#333333]I.e. it is necessary to find the critical points f(x,y,z) subject to: g(x,y,z)=0. [/color][/i][/i][/color][/color][/i][/i][/color][color=#333333]Further we will use positional vectors [/color][/i][math]\vec{r}[/math][color=#222222][color=#1e84cc][i][color=#333333]:=(x,y,z)[sup]T [/sup][/color]: f(x,y,z)=[/i][/color][/color][math]\sum_{i=1}^n\parallel\vec{r}-\vec{r_i}\parallel_2[/math][i][color=#333333]and g(x,y,z):=[/color][/i][math]\parallel\vec{r}\parallel[/math][color=#333333][i]-R=0. The gradients are: ∇f=[/i][/color][math]\sum_{i=1}^nUnitVector\left(\vec{r}-\vec{r_i}\right)[/math][i][color=#333333]and ∇g[/color][color=#222222] =[/color][/i][math]\vec{r}[/math][i][color=#222222][i][color=#222222]/R -gradient of the magnitude of a position vector r[/color][/i]. [/color][color=#333333]There is a system of equations: ∇f(x,y,z)= λ*∇g(x,y,z). A local optimum occurs when ∇f(x,y,z) and ∇g(x,y,z) are parallel[/color][color=#000000], and so ∇f [/color][color=#000000]is some multiple of ∇[/color][i][color=#000000][i]g[/i][/color][/i][color=#000000]. The condition that ∇f is parallel to ∇g means ∇f = λ∇g[/color][color=#333333]. [br] [/color][color=#333333][b] [size=150]☛[/size] [/b]Thus, condition for a point [/color][color=#ff7700]c[/color][color=#333333] on [i]a restricted region [/i][b]S[/b] to be [/color][color=#ff7700]GM[/color][color=#333333] is:[/color][color=#333333] [br] [b]the [/b][/color][b][i][color=#333333]resultant of all unit vectors[/color][/i][color=#222222] from the set of points lP to this [/color][color=#ff7700]point[/color][/b][i][color=#333333] ([/color][color=#1e84cc] [math]\sum_{i=1}^nUnitVector\left(\vec{c}-\vec{r_i}\right)[/math][/color][/i][color=#333333]) [b]is[/b][/color][color=#333333][b][i] parallel[/i] to [/b][/color][i][color=#333333][b]its positional vector [/b][/color][color=#ff7700][math]\vec{c}[/math][/color][/i][color=#333333][color=#333333] , i.e. the angle Δφ between these two vectors is 0 or π (Fig. 2c and Fig. 2b in column Δα≟ π ).[br][/color][/color][/i][/size] [size=85][i][color=#333333]There are no [i]explicit[/i] [color=#ff7700]Geometric Medians[/color] formulas, in contrast to [color=#ff00ff]Geometric Centers[/color] explicit formulas. The solution of the system of equations can be found use iterative procedures. I propose iterative procedures in which each step produces is a more accurate approximation:[br][/color][/i][math]\vec{r}_s\text{^✱}[/math][color=#222222][color=#1e84cc][i][color=#333333]=R*[i][size=100]UnitVector[/size][/i]([/color][/i][/color][/color][math]\sum_{i=1}^{n,i\text{≠}s}UnitVector\left(\vec{r_s}-\vec{r_i}\right)[/math][color=#222222][color=#1e84cc][i][color=#333333]) finds the [/color][color=#ff0000]Maxima -[/color][color=#333333] points and [/color][/i][/color][/color][math]\vec{r}_s^{\text{✱}}[/math][color=#222222][color=#1e84cc][i][color=#333333]=R*[i][size=100]UnitVector[/size][/i]([/color][/i][/color][/color][math]\sum_{i=1}^{n,i\text{≠}s}\frac{\left(\vec{r_i}\right)}{\parallel\vec{r_i}-\vec{r_s}\parallel}[/math][color=#222222][color=#1e84cc][i][color=#333333]) [/color][/i][/color][/color][i][color=#333333]finds the [/color][color=#0000ff]Minim[/color][color=#333333]a -[/color][color=#222222] points. Iterative procedure to find [/color][color=#93c47d]Saddle[/color][color=#222222] points has a two-step combination: [br][/color][math]\vec{r}_s\text{^✱}[/math][color=#222222]=R*[/color][i][size=100]UnitVector[/size][/i][color=#222222]([/color][math]\sum_{i=1}^{n,i\text{≠}s}UnitVector\left(\vec{r_s}-\vec{r_i}\right)[/math][color=#222222]+[/color][b]m/R[/b][color=#222222]*[/color][math]\vec{r_s}[/math][color=#222222]); [/color][math]\vec{r}_s\text{ }^{\text{✱}\text{✱}}\text{ }[/math][color=#222222]=R*[/color][i][size=100]UnitVector[/size][/i][color=#222222]([/color][math]\sum_{i=1}^{n,i\text{≠}s}\frac{\left(\vec{r_i}\right)}{\parallel\vec{r_i}-\vec{r_s}^{\text{✱}}\parallel}[/math][color=#222222]), where [/color][b][color=#6aa84f]m[/color][/b][color=#222222] is the parameter that sometimes need to be configured in manual settings mode (Fig.1a and Fig.1b). These iterative formulas has the advantages of fast convergence speed for all initial positions.[/color][/i] [br] You can visually observe (Fig. 1c) and explore single solutions [color=#ff0000]max-Z1[/color]/[color=#0000ff]min-Z2[/color]/[color=#6aa84f]saddle-Z3[/color] in the [b]manual settings mode[/b] (Fig. 1b) at points from the lP. T[i][color=#333333]heir coordinates (x[sub]i[/sub],y[sub]i[/sub]) are set by moving the [color=#ff00ff](x,y)[sub]Pi[/sub][/color] points and the coordinates z[sub]i[/sub] -by moving the [color=#ff00ff]z[sub]i[/sub][/color]-points (Fig. 1a)[/color][/i]. [br] In the [b]automatic settings mode[/b] [u][b][color=#980000]click on[/color][/b] [/u]Buttons: [color=#ff0000]max[/color] [color=#0000ff]min[/color] [color=#93c47d]sad[/color] (in [i]4. Automatic search for critical points -[/i]Fig. 1c) and will get all the critical points.[/size]

[size=85][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 the distances for a 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 [i][color=#333333]Fig. 1e and Fig 2e[/color][/i]. [/color][/i][br] [/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 too as the intersection points of the corresponding implicit functions and [/color][color=#333333][i]is shown in Fig. 1c and Fig. 2a.[br] [i][i][color=#333333] By changing the positions of the points from lP (fig. 1a) you can find the positions of the [i][size=85][color=#333333]geometric [/color][color=#ff7700]medians[/color][color=#333333] in ℝ[sup]3[/sup][/color][/size][/i] on the bounded surface S.[/color][/i][/i][/i] Distribution of points from lP in 3D and corresponding critical points [/color][b][color=#ff0000]max[/color][color=#333333]/[/color][color=#0000ff]min[/color][color=#333333]/[/color][color=#45818e]sad[/color][/b][color=#333333] (from Table -Fig. 2b) and directions for them "resulting all unit vectors" and position vectors on the surface of a sphere is[i] shown in Fig. 2c. For illustration purposes, the vectors are attached to the points.[/i][/color][/size]