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V=12 Cuboctahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere
Autor:
Roman Chijner
Thema:
Algebra
,
Analysis
,
Kreis
,
Differenzenquotient und Steigung
,
Differentialrechnung
,
Differentialgleichungen
,
Gleichungen
,
Extremwertaufgaben oder Extremwertprobleme
,
Geometrie
,
Graph
,
Schnittmenge
,
Lineare Optimierung
,
Mathematik
,
Kugel
,
Oberfläche
,
Vektoren
A system of points on a sphere S of radius R “induces” on the sphere S
0
of radius R
0
three different sets of points, which are
geometric medians (GM)
-local
maxima
,
minima
and
saddle
points sum of distance function f(x). The angular coordinates of the spherical distribution of a system of points -
local minima
coincide with the original system of points.
Distribution of points Pi
,
test Point
,
Max
/
min
/
saddle
-
Critical points
on a sphere. Vectors ∇f and ∇g at these points. ● max Tetrakis hexahedron:
n=14
●
min Cuboctahedron:
n=12
●
sad Rhombicuboctahedron:
n=24
Distribution of points Pi
,
test Point
,
Max
/
min
/
saddle
-
Critical points
on a sphere. Vectors ∇f and ∇g at these points. ● max Tetrakis hexahedron:
n=14
●
min Cuboctahedron:
n=12
●
sad Rhombicuboctahedron:
n=24
Two-variable function f(φ,θ) over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.
Isolines and Intersection points of implicit functions over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.
Critical Points
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