[size=50][size=100][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 circle from where the [/color][color=#1e84cc][i]function of [/i][/color][color=#1e84cc]sum of the distances [/color]to each point p[sub]i[/sub]'s have at [/color][b]that point [/b][/color][/color][/color][i][color=#333333]either the [/color][/i][color=#0000ff]local[/color] [color=#0000ff]minimum or[/color] [color=#ff0000]local[/color] [color=#ff0000]maximum [/color][color=#222222]or [/color][color=#222222]a [/color][color=#93c47d]stationary point of inflection[/color][color=#93c47d]. [br][/color][color=#1e84cc][i] [color=#333333]Critical points can be found using [/color][/i][/color][i][b]Lagrange multipliers [/b]as (Λ(x,y ,λ)=f(x,y )+λ*g(x,y )) finding the Extreme values of the function :[/i][color=#222222][color=#1e84cc][i][i][i][br] f(x,y)=[/i][/i][/i][/color][/color][math]\sum_{i=1}^n\sqrt{\left(x-x_i\right)^2+\left(y-y_i\right)^2}[/math][color=#222222][color=#1e84cc][i][i] -[color=#1e84cc]sum of the distances [/color][color=#333333]from[/color] ([/i][/i][/color][/color][i]x,y) to each point p[sub]i[/sub]'s, subject to a constraining equation: [/i][color=#222222][color=#1e84cc][i][i] [br][i][i][i][i][color=#333333]g(x,y)[/color][/i][/i]=[math]\sqrt{\left(x\right)^2+\left(y\right)^2}-R[/math][/i] [/i][color=#333333] -[i] point (x,y)∈[b]L[/b][/i].[/color][/i][/i][/color][/color][i] Let [b]L[/b] be a circle of radius R around the point O: [b]L[/b]:[/i][color=#222222][color=#1e84cc][i]={[/i][/color][/color][math]\vec{r}[/math][i][color=#333333]∈ℝ²[/color][color=#222222]: ||[/color][/i][math]\vec{r}[/math][i][color=#222222]||[/color][color=#333333]=R[/color][color=#222222]}. [i][i][color=#333333] I.e. it is necessary to find the critical points f(x,y) subject to: g(x,y)=0. [/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 )[sup]T [/sup][/color]: f(x,y)=[/i][/color][/color][math]\sum_{i=1}^n\parallel\vec{r}-\vec{r_i}\parallel_2[/math][i][color=#333333]and g(x,y)=[/color][/i][math]\parallel\vec{r}\parallel[/math][color=#333333][i]-R. The gradients are: [math]\vec{f_r}[/math]:=∇f=[/i][/color][math]\sum_{i=1}^nUnitVector\left(\vec{r}-\vec{r_i}\right)[/math][i][color=#333333]and [i] [math]\vec{c_r}[/math]:=[/i]∇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)= λ*∇g(x,y). A local optimum occurs when ∇f(x,y) and ∇g(x,y) 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]. [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]L[/b] to be [/color][color=#ff7700]GM[/color][color=#333333] is:[/color][color=#333333] [br] [b]the [i][i][color=#333333][i][color=#333333][b]the resultant vector [/b][/color][/i][/color][/i][/i][/b][/color][/i][i][b][i][color=#333333][i][math]\vec{f_r}[/math][/i] -sum of all unit vectors[/color][/i][color=#222222] from the set of points lP to this [/color][color=#ff7700]point c[/color][/b][/i][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][/i][i][color=#333333], i.e. the angle Δφ [i]between these two vectors is multiple of π .[/i][/color][/i][i]There are no explicit [/i][color=#ff7700]Geometric Medians[/color][i] formulas, in contrast to [/i][color=#ff00ff]Geometric Centers[/color][i] explicit formulas. The solution of the system of equations can be found use iterative procedures. [/i][/size][/size][/size]

[size=50][size=100][size=85][color=#222222] The [/color][color=#ff00ff]Geometric Center [/color][color=#222222]is defined here as [/color][color=#333333][b]point [/b][/color]on circle from where[color=#333333] the [/color][color=#1e84cc]sum of the [i]squares[/i] of all Euclidean distances[/color][color=#333333] [color=#222222][color=#222222][color=#222222][color=#222222][color=#222222]to each point p[sub]i[/sub]'s have at [/color][b]that point[/b][/color][/color][/color][/color][/color][i][color=#333333]: [/color][/i]local [color=#0000ff]minimum[/color], [color=#ff0000]maximum [/color][color=#93c47d][color=#222222]or [/color][color=#222222]a [/color][color=#93c47d]stationary point of inflection[/color].[br] [/color][color=#333333] Critical points can be found using [/color][i][b][color=#333333]Lagrange multipliers[/color][/b][/i][i][b][color=#333333]as [/color][/b]([i]Λ(x,y,λ)=f(x,y)+[color=#ff7700]λ[/color]*g(x,y))[/i] finding the Extreme values of the function : [br][i] f[sub]q[/sub](x,y)=[math]\sum_{i=1}^n\left[\left(x-x_i\right)^2+\left(y-y_i\right)^2\right][/math][/i] -[/i][i][color=#1e84cc]sum of the squares of the distances[/color] to the points p[sub]i[/sub][/i][color=#333333], [/color][i]subject to a constraining equation: [br][color=#333333]g(x,y)=x[/color][sup]2[/sup][color=#333333]+y[/color][sup]2[/sup][color=#333333]-R[/color][sup]2[/sup][color=#333333] -[i] points (x,y)∈[b]L[/b][/i].[/color][/i] [i][color=#222222][i][i][color=#333333]I.e. it is necessary to find the critical points f(x,y) subject to: g(x,y)=0. [/color][/i][/i][/color][/i][color=#6a6a6a]Let's denote the resulting point as Sum(lP):[color=#333333]=[math]\sum_{i=1}^np_i[/math] , then [/color][/color][i]∇f/2=n*(x,y)-[/i][math]\left(\sum_{i=1}^nx_i,\sum_{i=1}^ny_i\right)[/math][i] or ∇f/2=n*(x,y)-Sum(lP)[/i][color=#333333] and [/color][i][color=#333333]∇g/2=(x,y): ⇒[/color][b](x,y)∼Sum(lP)[/b][color=#333333].[br] The point (x,y) [/color][i]that minimize the [color=#1e84cc]sum of the squares of the distances[/color] to the points [/i][color=#333333]p[/color][color=#333333]i [/color][i][color=#333333]is the [/color][i][b][color=#ff00ff]Geometric Center[/color][/b] ([i][color=#333333]gravity center, [/color][/i][i]barycenter, center[/i][i] [/i][i]of mass, [/i][i]centroid)[/i] [/i][color=#ff00ff]GC[/color][color=#333333]: [/color][color=#ff00ff][b]Cm[/b][/color][color=#333333]=[/color][math]\frac{\sum_{i=1}^np_i}{n}[/math][/i][i][color=#333333][b] -[/b]its coordinates are the averages of the coordinates of the points from set lP. [/color][url=https://en.wikipedia.org/wiki/Center_of_mass]https://en.wikipedia.org/wiki/Center_of_mass[/url] .[/i][b] [br][size=150] ☛ [/size][/b]In this problem there are only two critical points[color=#333333] (x,y)∈[b]L[/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 circle [b]L[/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[/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=#333333][b]two antipodal points on circle[/b]. These points can be found using the Steiner theorem too[/color][color=#d5a6bd].[br][/color][color=#333333] With [url=https://www.geogebra.org/m/qpmtdkuw]applet[/url] you can visually observed and explore these solutions at different positions of the points from set lP. By changing the positions of points from lP, click on the "F-Ferma Point" and "critical points: max/min" buttons to find the new positions of geometric [/color][color=#ff7700]medians[/color][color=#333333] and [/color][color=#ff00ff]centers[/color][color=#333333] in ℝ² and on the selected circle L. To evaluate the accuracy of the proposed iterative methods for finding critical points click on checkbox "Extremum_f".[br] In polar coordinates functions[/color][color=#1e84cc] f(x,y)[/color][color=#333333], [/color][color=#ff00ff]f[sub]q[/sub] (x,y)[/color][color=#333333] (in Cartesian coordinates x,y) are functions of one variable [/color][color=#1e84cc]f(x)[/color][color=#333333], [/color][color=#ff00ff]f[sub]q[/sub](x)[/color][color=#333333] depending only on the angular position (polar variable x) of the point on the circle of radius R.[/color][color=#333333] The dependencies of these functions on the angular position x of a point on a circle for a given location of points from lP are shown on the right side of the [url=https://www.geogebra.org/m/qpmtdkuw]applet[/url].[/color][/i][/size][/size][/size]