Date: Sep 27, 2017 8:49 AM
Author: Blazenko Ivancic
Subject: Mathematical Problem - unsolved


Problem Description

The target is to describe every point on a unit sphere by a probability
distribution over the 26 points of the grid points of the "unit cube"
which surrounds the unit sphere:

In Detail:
It is given the unit sphere with an arbitrary vector (called here
target-vector Pta):
Target-Vector Pta={X,Y,Z} with |P_ta |=ˆ(X^2+Y^2+Z^2 )=1
X=sin?(?)*cos?(?)
Y=sin?(?)*sin?(?)
Z=cos?(?)

The "unit-cube" is given by the 26 Points (6 Face-, 12 Edge-- and 8
Corner-points - NOT taking into account the origin point {0, 0, 0}).
The cube surrounds the unit sphere. These are the 26 Gridpoints which
defines the cube and in every point a probability value has to be
calculated:

P_01={-1,-1,-1} ; P_02={-1,-1,0} ; P_03={-1,-1,+1}
P_04={-1,0,-1} ; P_05={-1,0,0} ; P_06={-1,0,+1}
P_07={-1,+1,-1} ; P_08={-1,+1,0} ; P_09={-1,+1,+1}

P_10={0,-1,-1} ; P_11={0,-1,0} ; P_12={0,-1,+1}
P_13={0,0,-1} ;   Origin P_05={0,0,0} ; P_14={0,0,+1}
P_15={0,+1,-1} ; P_16={0,+1,0} ; P_17={0,+1,+1}

P_18={+1,-1,-1} ; P_19={+1,-1,0} ; P_20={+1,-1,+1}
P_21={+1,0,-1} ; P_22={+1,0,0} ; P_23={+1,0,+1}
P_24={+1,+1,-1} ; P_25={+1,+1,0} ; P_26={+1,+1,+1}

Every of these 26 Points is afficted with a probability value Wi which
depends on the target vector Pta on the unit sphere.
For these 26 probability values Wi the following equations must be
valid:

W_i?Reals  for i=1 to 26

0 <= W_i <= 1

Sum(W_i) = 1

All probability values Wi are real, every probality value is between
zero and one.
The sum of all 26 probality values is one.
Additionally the following equations must be valid:

X-direction:
W_(+1,+1,+1)+W_(+1,+1,0)+W_(+1,+1,-1)+W_(+1,0,+1)+W_(+1,0,0)+W_(+1,0,-1)
+W_(+1,-1,+1)+W_(+1,-1,0)+W_(+1,-1,-1)
-
(W_(-1,+1,+1)+W_(-1,+1,0)+W_(-1,+1,-1)+W_(-1,0,+1)+W_(-1,0,0)+W_(-1,0,-1)
+W_(-1,-1,+1)+W_(-1,-1,0)+W_(-1,-1,-1) ) = X = sin?(?)*cos?(?)

Y- and Z-direction analogous to the upper equation.

Short form:
…( W_(+1,j,k) - W_(-1,j,k) ) =X=sin?(?)*cos?(?)

…( W_(i,+1,k) - W_(i,-1,k) ) =Y=sin?(?)*sin?(?)

…( W_(i,j,+1) - W_(i,j,-1) ) =Z=cos?(?)

These equations mean that the sum of the probalities in one of the
coordinate direction (x,y or z - taking the positive and negative
direction vector into account) must be the vecor component of the
target-vector Pta.


Demonstrative description:

Individual vectors (real-vectors) can only be randomly realized on the
26 grid-points due to the probability value in each of the grid-point.
Real vectors can not be realized on the unit sphere (there exists only
the target vector Pta). The task is to calcuate the probalities in each
grid-point in that way that within N realizations the averaged real
vector (average over all randomly distributed real vectors on the
grid-points due to their probability values) is exacltly the target
vector Pta i.e. the averaged real vector is located on the
target-vector.

The goal is to find general equations for for all 26 Probability
functions Wi depending on the target vector Pta and other factors
because the solution is not unique. It is manifold!

W_i=f(?,?,C1,C2,Š)

Example:
For the target vector Pta = {+1,0,0} a solution is:

Face-Point: W+1,0,0   = C_Face
Edge-Points: W+1,+1,0   = W+1,-1,0  = W+1,0,+1  = W+1,0,-1  = C_Edge
Corner-Points: W+1,+1,+1  = W+1,+1,-1 = W+1,-1,+1 = W+1,-1,-1 = C_Corner

All other Wi are zero. The following condition must be fullfilled which
represent the manifold of one solution (but probably not the complete
solution manifold):
C_Face + 4*C_Edge + 4*C_Corner = 1    (with C >= 0)

These equation fullfill all demanded conditions and is a solution for
this special case Pta = {+1,0,0}. But The goal is now to find general
equations which gives solution for a any arbitrary target vector Pta on
the unit sphere.

My Problem is that I cant find general equations up to now, I found
only some solutions for special cases like Pta = {+1,0,0}; Pta = {1/ˆ2,
  1/ˆ2, 0} or Pta = {cos ¼/8, sin ¼/8, 0} by using symetry conditions.
Mathematica is calculating since days, without delivering any solution
:-(

Thanks