[size=85][color=#333333] Let P[/color]i[color=#333333]=(x[/color]i[color=#333333],y[/color]i[color=#333333],z[/color]i[color=#333333]) n moving points in ℝ³ (lP:={[/color]P1,P2,...,Pn[color=#333333]}). I want to find the points P=(x,y,z) [i][u]on the surface of the sphere[/u][/i] -S ([/color][i]radius R[/i][color=#333333]) that are critical (relative [/color][color=#0000ff]min[/color][color=#333333]/[/color][color=#ff0000]max[/color][color=#333333] or [/color][color=#6aa84f]saddle [/color][color=#333333]points at (x,y,z)) of a function [/color][color=#333333][color=#1e84cc]f(x,y,z) is the sum [/color][/color][color=#1e84cc][i]of the distances[/i][/color][color=#333333] from P to the all points from lP. [/color][color=#333333]Critical points can be found using [/color][i]Lagrange multipliers[/i][i]as [/i]finding the Extreme values of the function [color=#1e84cc]f(x,y,z) [/color][i]subject to [/i][i]a g(x,y,z)=0 (constraining equation: [/i][color=#333333]g(x,y,z)=x[/color][sup]2[/sup][color=#333333]+y[/color][sup]2[/sup][color=#333333]+z[/color][sup]2[/sup][color=#333333]-R[/color][sup]2[/sup])[color=#333333]. There is a system of equations: ∇[/color][i][color=#1e84cc]f(x,y,z)[/color][/i][color=#333333]= λ∇g(x,y,z). A local optimum occurs when ∇[/color][i][color=#1e84cc]f(x,y,z)[/color][/i][color=#333333] and ∇g(x,y,z) are parallel[/color], and so ∇[i][color=#1e84cc]f[/color][/i] [color=#000000]is some multiple of ∇[/color][i][color=#000000][i]g[/i][/color][/i][color=#000000]. [br] This applet is used to study the distribution of geometric medians on a sphere of radius R, „induces“ by the discrete sample of 6 movable points in the 3-D space. [br]Description is in [url=https://www.geogebra.org/m/y8dnkeuu]https://www.geogebra.org/m/y8dnkeuu[/url].[/color][/size][br]
[size=85]Fig.1 Settings and control panels, observation windows.[br]a) -Settings and d) -relative position of the the "influence" points Pi around the sphere. [br]b) -Manual settings for finding critical points f (φ, θ) and determining the coefficient m for the saddle point search algorithm.[br]c) -(φ;θ) -plane of the angular coordinates of points on the sphere: Z1, Z2, Z3 -moving points (from b) of iterative solution search; f'_φ=0; f'_θ=0 -implicit functions of equations , their intersection are solutions of the Lagrange equations.[br]e) -graph of the distance sum function f (φ, θ).[/size][br]
[size=85]Fig.2 Results of explorations.[br]a) -Isolines and intersecting the implicit function equations of zeroing partial derivatives.[br]b) - Automatic search for critical points; their values.[br]c) -Distribution of points Pi, test Point, Max/min/saddle -Critical points on a sphere. Vectors ∇f and ∇g at these points.[br]d) -settings and e) -graph of the Two-variable function f(φ,θ) over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.[/size]