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

Problem DescriptionThe target is to describe every point on a unit sphere by a probabilitydistribution 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 heretarget-vector Pta):Target-Vector Pta={X,Y,Z} with |P_ta |=(X^2+Y^2+Z^2 )=1X=sin?(?)*cos?(?)Y=sin?(?)*sin?(?)Z=cos?(?)The "unit-cube" is given by the 26 Points (6 Face-, 12 Edge-- and 8Corner-points - NOT taking into account the origin point {0, 0, 0}).The cube surrounds the unit sphere. These are the 26 Gridpoints whichdefines the cube and in every point a probability value has to becalculated: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 whichdepends on the target vector Pta on the unit sphere. For these 26 probability values Wi the following equations must bevalid:W_i?Reals  for i=1 to 260 <= W_i <= 1Sum(W_i) = 1All probability values Wi are real, every probality value is betweenzero 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 thecoordinate direction (x,y or z - taking the positive and negativedirection vector into account) must be the vecor component of thetarget-vector Pta.Demonstrative description: Individual vectors (real-vectors) can only be randomly realized on the26 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 onlythe target vector Pta). The task is to calcuate the probalities in eachgrid-point in that way that within N realizations the averaged realvector (average over all randomly distributed real vectors on thegrid-points due to their probability values) is exacltly the targetvector Pta i.e. the averaged real vector is located on thetarget-vector.The goal is to find general equations for for all 26 Probabilityfunctions Wi depending on the target vector Pta and other factorsbecause 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:Face-Point: W+1,0,0    = C_FaceEdge-Points:   W+1,+1,0   = W+1,-1,0  = W+1,0,+1  = W+1,0,-1  = C_EdgeCorner-Points: W+1,+1,+1  = W+1,+1,-1 = W+1,-1,+1 = W+1,-1,-1 = C_CornerAll other Wi are zero. The following condition must be fullfilled whichrepresent the manifold of one solution (but probably not the completesolution manifold):C_Face + 4*C_Edge + 4*C_Corner = 1    (with C >= 0)These equation fullfill all demanded conditions and is a solution forthis special case Pta = {+1,0,0}. But The goal is now to find generalequations which gives solution for a any arbitrary target vector Pta onthe unit sphere.My Problem is that I cant find general equations up to now, I foundonly 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
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