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,Š)
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 target-vector P_ta): Target-Vector 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 "unit-cube" is given by the 26 Points (6 Face-, 12 Edge- and 8 Corner-points - 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:
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 vector component of the target-vector P_ta.
Demonstrative description:
Individual vectors (real-vectors) can only be randomly realized on one
of the 26 grid-points due to the probability function in each of the grid-point. 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 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 P_ta i.e. the averaged real
vector is located on the target-vector.
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,Š)
Example: For the target vector P_ta = {+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 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/ˆ2, 1/ˆ2, 0} or P_ta = {cos ¼/8, sin ¼/8, 0} by using symmetry
conditions. Mathematica is calculating since days, without delivering
any solution :-(
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