
Re: Mathematical Problem  unsolved
Posted:
Sep 29, 2017 6:36 AM


Am Donnerstag, 28. September 2017 19:24:18 UTC+2 schrieb David Hobby: > > > > 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,) > > > One simplification is to note that every solution consists of a > "minimal vector" M of W values that meet the other conditions but have > the sum of the W values less than 1, added to a "balanced vector" B of > W values that make X = Y = Z = 0, where the sum of values in B is > chosen so that the sum for B + M is 1. The possible B are easier to > characterize. (Or if you just want one solution, take a simple choice > of B, like W_(1,0,0) = W_(1,0,0) with the rest of the W values zero.) > As for the minimal vectors M, they will have all their W values zero, > except for those "pointing the same way" as the target vector. This > should make finding M much easier, since it cuts the number of > variables needed.
Sorry, there were some mistakes in the description, here is the correct one:
Problem Description
The target is to describe every point on a unit sphere by a probability distribution over the 26 grid points of the "unit cube grid" which surrounds the unit sphere:
In Detail: It is given the unit sphere with an arbitrary vector (called here targetvector P_ta): TargetVector P_ta={X,Y,Z} with P_ta = Sqrt(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  in a first Step 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 function 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_00={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 fuction Wi which depends on the target vector P_ta on the unit sphere. For these 26 probability values Wi the following equations must be valid:
W_i ? R for i = 1 to 26
0 ¾ W_i ¾ 1
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 vector component of the targetvector P_ta.
Demonstrative description:
Individual vectors (realvectors) can only be randomly realized on one of the 26 gridpoints due to the probability function in each of the gridpoint. Real vectors can not be realized on the unit sphere (there exists only the target vector P_ta). The task is to calcuate the probality functions 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 P_ta i.e. the averaged real vector is located on the targetvector.
The goal is to find general equations for all 26 Probability functions Wi depending on the target vector P_ta and other factors, because the solution is not unique. It is manifold (i.e. manifold solution space)!
W_i=f(?,?,C1,C2,)
Example: For the target vector P_ta = {+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 maybe not the complete manifold solution):
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 P_ta = {+1,0,0}. But The goal is now to find general equations which gives solution for a any arbitrary target vector P_ta on the unit sphere.
My Problem is that I cant find general equations up to now. Only some solutions have been found for special cases like P_ta = {+1,0,0} ; P_ta = {1/2, 1/2, 0} or P_ta = {cos ¼/8, sin ¼/8, 0} by using symmetry conditions. Mathematica is calculating since days, without delivering any solution :(

