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n=8 Square antiprism(anticube). Images: A critical points scheme for Generating uniformly distributed points on a sphere
Author:
Roman Chijner
Topic:
Algebra
,
Calculus
,
Circle
,
Difference and Slope
,
Differential Calculus
,
Differential Equation
,
Equations
,
Optimization Problems
,
Geometry
,
Function Graph
,
Intersection
,
Linear Programming or Linear Optimization
,
Mathematics
,
Sphere
,
Square
,
Surface
,
Vectors
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
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 P
i,
test Point
,
Max
/
min
/
saddle
-
Critical points
on a sphere. Vectors ∇f and ∇g at these points.
max:
Gyroelongated square bipyramid
min:
Anticube
sad:
Octagonal Antiprism
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.
Isolines and Intersection points of implicit functions over a rectangular region: - π ≤φ ≤ π; -π/2≤θ≤π/2.
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