[size=85] Let L be a circle of radius R around the point O: L:={x∈[size=100]ℝ[/size][sup]2[/sup]: ||x||=R}. [i]There is a set lP={p1, p2,...,pn} of n movable points and [color=#333333]with each p[sub]i[/sub][/color][color=#333333]∈L[/color][color=#333333],[/color][/i][color=#222222]{(xi,yi)∈L:i = 1,...,n} -their coordinates[/color][i]. [/i]Let the points be "repulsive". [br][b][u]Problem[/u][/b]: use the [b][i]method of Lagrange multipliers[/i][/b] to find distribution of "repulsive" points on a circle corresponding to the [i][b][color=#ff7700]maximum[/color][/b][/i] of their [b]sum of mutual distances[/b].[br][i] This means need to find out such [i]mutual arrangement[/i] of "[i][i][color=#ff0066]repulsive" set of particles on a circle,[/color][/i] [/i]when each point of this set is [color=#6aa84f]GM[/color] of the remaining n-1 points. [i]We assume that the equilibrium - [/i][i]stationary state [/i][i][color=#333333]in system of "charges" is reached if the sum of their mutual distances is [/color][color=#ff7700]maximal[/color][color=#333333]. [/color][/i][i][i]Iterative approach[/i][/i] of [i][i]particle placement[/i][/i] is applied for [i]achieving [/i][i]a stationary state.[/i][/i][/size]

[size=85] 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, and so ∇f is some multiple of ∇g. I propose iterative procedures in which each step produces is a more accurate approximation finds the [b][i][color=#ff7700]Maximal Distance Sum[/color][/i][/b].[br][size=85] Relationship between vectors of two consecutive iterations is defined by [/size][size=100][size=200][size=150][math]\vec{r_s}^{\ast}=R\cdot UnitVector\left(\sum^{n,i\ne s}_{i=1}UnitVector\left(\vec{r_s}-\vec{r_i}\right)\right)[/math][/size][/size][/size]. This is the new position of point ps∈L. All points are successively corrected (s = 1,2, ..., n) until the system of points comes to a stationary state. Finally, at the end of iterations they must match: ps=ps* for all s=1,2,...,n. The number of correction steps is carried out until the center of mass of n particles from lP (with a certain degree of accuracy) will not be in the center of the circle.[br] [b]The solution[/b] of the above Lagrange problem means that in" equilibrium " the positional vectors ∇g[sub]s[/sub]:=[math]\vec{r_s}[/math], for each point from lP and the sum of the unit vectors other n-1 points [color=#ff7700]∇f[sub]s[/sub][/color]:=[math]\sum^{n,i\ne s}_{i=1}UnitVector\left(\vec{r_s}-\vec{r_i}\right)[/math][size=85][size=85]must be parallel.[/size] For a similar problem of the [b][i][color=#ff00ff]Maximum squares Distance Sum[/color][/i][/b] the gradients are [color=#ff00ff]∇f[sub]q s[/sub][/color]:=[math]\sum^{n,i\ne s}_{i=1}\left(\vec{r_s}-\vec{r_i}\right)[/math][/size].[/size]

[size=85]-Points are distributed evenly on the circle[br]-It can be seen from table that in the "equilibrium" state is achieved not only the [b][b][i][color=#ff7700]Maximal Distance Sum[/color][/i][/b][/b] but also the [b][b][i][color=#ff00ff]Maximum squares Distance Sum[/color][/i][/b][/b]. This means that each point on the circle is both the [color=#ff7700]geometric median[/color] and the [color=#ff00ff]geometric center[/color] of the other remaining points.[/size]

[size=85]Execute( Join(Sequence(Sequence("SetValue(A"+i+",(0,0)+R UnitVector( CopyFreeObject[Sum[Zip[UnitVector(A"+i+"-lP(r)),r,Sequence[n]\{"+i+"} ]]] ))",i,1,n),ii,1,i0) ) )[br][br]SetValue[Wi_P,Sequence[ Sum[Zip(UnitVector(a-lP(i)), a, lP \ {lP(i)}) ],i,1,n] ][br]SetValue[Wi_Pc,Sequence[Sum(lP \ {lP(i)}) / n,i,1,n] ][br][br]SetValue[Wii_P,Sequence[Angle[Vector[lP(i)], -Vector[Wi_P(i)] ],i,1,n] ][br]SetValue[Wii_Pc,Sequence[Angle[Vector[lP(i)], -Vector[Wi_Pc(i)] ],i,1,n] ][br][br]lVectorGM = Zip(Vector(a, a + 1.5UnitVector(b)), a, lP, Vector(b), Wi_P)[br]lVectorGC = Zip(Vector(a, a + 1.5UnitVector(b)), a, lP, b, Wi_Pc)[br]lVectorP = Zip(Vector(x0, x0 + UnitVector(x0)), x0, lP)[/size]

[b]Applets in a [url=https://www.geogebra.org/m/u7zq6f3e]book[/url]:[/b][br][size=85] [url=https://www.geogebra.org/m/puqnepmv]Description[/url]. Finding Geometric Medians and Geometric Centers on a circle from discrete sample points.[br] [url=https://www.geogebra.org/m/qpmtdkuw]Applet[/url]. Finding Geometric Medians and Geometric Centers on a circle from discrete sample points.[br][url=https://www.geogebra.org/m/zhjadztw]Finding[/url] the location of geometric medians on the circle of discrete sample points depending on the position of the test point.[br] [url=https://www.geogebra.org/m/pjaqednw]Method[/url] of Lagrange multipliers. Relative positioning of repulsive movable points on a circle.[br][url=https://www.geogebra.org/m/d9ytv4wg]Generating[/url] an extreme arrangements of points on a circle[br][url=https://www.geogebra.org/m/uuz6h7xq]Generating[/url] an extreme arrangements of points on a sphere[br][url=https://www.geogebra.org/m/b5zcy52h]Generating[/url] an extreme arrangements of points on a sphere with more structured calculation program-GeoGebra Forum-[br][url=https://help.geogebra.org/topic/geogebra-windows-portable-zip-for-december]https://help.geogebra.org/topic/geogebra-windows-portable-zip-for-december[/url][/size][br][br]