[size=85]The applet illustrates the case where 4 vertices of a regular tetrahedron "induce" the vertices of two other polyhedra:[br] [b]V=6 ●[color=#ff0000]Octahedron[/color]← V=8 ●[color=#0000ff]Cube[/color] →V=12 ●[color=#38761d]Cuboctahedron.[/color][/b] [br] Generating polyhedra is in [url=https://www.geogebra.org/m/rtm56gkb]https://www.geogebra.org/m/rtm56gkb[/url]. Description are in [url=https://www.geogebra.org/m/y8dnkeuu]https://www.geogebra.org/m/y8dnkeuu[/url] and [url=https://www.geogebra.org/m/rkpxwceh]https://www.geogebra.org/m/rkpxwceh[/url].[/size]
[size=85]A system of points on a sphere S of radius R “induces” on the sphere S0 of radius R0 three different sets of points, which are [color=#93c47d]geometric medians (GM)[/color] -local [color=#ff0000]maxima[/color], [color=#6d9eeb]minima[/color] and [color=#38761d]saddle[/color] points sum of distance function f(x). The angular coordinates of the spherical distribution of a system of points -[color=#0000ff] local minima[/color] coincide with the original system of points.[/size]
Distribution of points Pi, [color=#5b0f00]test Point[/color], [color=#ff0000]Max[/color]/[color=#0000ff]min[/color]/[color=#38761d]saddle[/color] -[color=#333333]Critical points[/color] on a sphere. Vectors ∇f and ∇g at these points.[br]
[color=#ff0000]max:[/color] Octahedron [color=#0000ff] min:[/color] Cube [color=#6aa84f]sad:[/color] Cuboctahedron[br]
Two-variable function f(φ,θ) over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.
Intersection points of implicit functions over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.[br]
Isolines and Intersection points of implicit functions over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.[br]
