[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 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]=0. 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]. [/color][color=#333333][br] If we substitute the variable z=R cos(θ), then we will have a two-variable function [/color][i][color=#1e84cc]f(φ,θ[/color][/i][color=#333333]) over a rectangular region: - π ≤φ≤ π; -π/2≤θ≤π/2. The solution of the system of equations can be found as the intersection points of the corresponding implicit functions. The point P=(x,y,z)∈ ℝ³ [/color][i]that minimize a function of the [color=#1e84cc]sum of distances[/color] from this point to all points lP is called their [/i][url=https://en.wikipedia.org/wiki/Fermat_point][color=#ff00ff]Fermat's point[/color][/url] [i]or the [/i][i][i][color=#333333][color=#ff7700]Geometric [url=https://en.wikipedia.org/wiki/Geometric_median]Median[/url][/color][/color][/i][/i][color=#ff7700]. [br][/color][color=#333333][i] In our case P=(x,y,z)∈S! [/i][/color][color=#333333]There are no [i]explicit[/i] [color=#ff7700]Geometric Medians [/color]formulas, in contrast to [color=#ff00ff]Geometric Centers[/color] explicit [url=https://www.geogebra.org/m/nge6gawt]formulas[/url]. The solution of the system of equations can be found use iterative procedures. [/color]Here you can visually observe and explore these solutions in different positions of 3 moving points from the lP set[color=#333333]. Description is in [url=https://www.geogebra.org/m/y8dnkeuu] https://www.geogebra.org/m/y8dnkeuu[/url][/color][color=#9900ff].[/color][/size]
[size=85] - Intersection Implicit Curves f'φ(φ, θ)=0; f'θ(φ,θ)=0 over a rectangular region φ∈[-π,π], θ∈[-0.5π,0.5π] [br][color=#ff7700]- Critical[/color] points on the surface of [color=#1e84cc]distance sum function [/color]f(φ,θ) over a rectangular region φ∈[-π,π], θ∈[-0.5π,0.5π] [br]- Distribution of [color=#1e84cc]points Pi[/color] and their local [color=#ff0000]maxima[/color]/[color=#0000ff]minima[/color] and [color=#6aa84f]saddle[/color] -[color=#ff7700]critical[/color] points of distance sum function f(φ,θ) on a sphere + [color=#b45f06]test Point[/color]. Vectors ∇f and ∇g at these points.[br][/size]