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Topic: Mathematical Problem - unsolved
Replies: 2   Last Post: Sep 29, 2017 6:36 AM

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 Blazenko Ivancic Posts: 12 From: Germany Registered: 2/13/16
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 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,)

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/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 :-(

Date Subject Author
9/27/17 Blazenko Ivancic
9/28/17 David Hobby
9/29/17 Blazenko Ivancic