
Mathematical Problem  unsolved
Posted:
Sep 27, 2017 8:49 AM


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 targetvector Pta): TargetVector Pta={X,Y,Z} with P_ta =(X^2+Y^2+Z^2 )=1 X=sin?(?)*cos?(?) Y=sin?(?)*sin?(?) Z=cos?(?)
The "unitcube" is given by the 26 Points (6 Face, 12 Edge and 8 Cornerpoints  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:
Xdirection: 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 Zdirection 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 targetvector Pta.
Demonstrative description:
Individual vectors (realvectors) can only be randomly realized on the 26 gridpoints due to the probability value in each of the gridpoint. 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 gridpoint in that way that within N realizations the averaged real vector (average over all randomly distributed real vectors on the gridpoints due to their probability values) is exacltly the target vector Pta i.e. the averaged real vector is located on the targetvector.
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,)
Example: For the target vector Pta = {+1,0,0} a solution is:
FacePoint: W+1,0,0 = C_Face EdgePoints: W+1,+1,0 = W+1,1,0 = W+1,0,+1 = W+1,0,1 = C_Edge CornerPoints: 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 = {1/2, 1/2, 0} or Pta = {cos ¼/8, sin ¼/8, 0} by using symetry conditions. Mathematica is calculating since days, without delivering any solution :(
Thanks

