From: mailbox@dmitry-kazakov.de   
      
   On 2023-01-09 21:37, Ben Bacarisse wrote:   
   > "Dmitry A. Kazakov"  writes:   
   >   
   >> On 2023-01-09 04:25, Ben Bacarisse wrote:   
   >>> "Dmitry A. Kazakov"  writes:   
   >>>   
   >>>> On 2023-01-08 21:41, Ben Bacarisse wrote:   
   >>>>   
   >>>>> But   
   >>>>> the problem is /finding/ a specific average -- the point (or angle) that   
   >>>>> minimises the sum of squares of the distances (or angles) from that   
   >>>>> average point (or angle).   
   >>>>   
   >>>> That is not a programming problem. It is a problem space's one. You   
   >>>> must go to the corresponding part of science or engineering or   
   >>>> economics etc and formulate it there in terms of that space.   
   >>> I don't follow.  It's a mathematical problem for which an algorithm is   
   >>> sought.   
   >>   
   >> Firstly, this is comp.programming group.   
   >   
   > Do you consider discussion algorithms off topic here?  Seriously?   
      
   No, I consider off topic discussing mathematical problems /= algorithms.   
   You did not discuss any algorithms so far.   
      
   >> Secondly, if algorithm is sought then there must be a mathematical   
   >> formulation of what a numeric solution is sought for.   
   >   
   > There needs to be a problem statement that can be discussed and refined.   
   > The original post about circular averages has led to two different   
   > refinements -- the vector average (projected onto the unit circle) and   
   > the arc-length average.  Both generalise to more dimensions and I   
   > happened to mention in passing that the great-circle average would be a   
   > natural fit in some situations.   
      
   I missed generalization to 3D. Sorry.   
   [You kept on writing about unit *circle* and an angle, where it actually   
   meant unit sphere and polar angles/spherical coordinates]   
      
   > Tim mentioned (again in passing) that it looks hard.  You seemed to   
   > suggest it was not, but I am not sure you knew what the problem was at   
   > that time.   
   >   
   > All of this seems to be to be exactly what comp.programming should be   
   > about.   
      
   Differential geometry and spherical trigonometry are certainly off topic.   
      
   >> (Finding a minimum is an optimization problem, there are thousands of   
   >> algorithms already)   
   >   
   > Programmers should know that specifications are not algorithm designs.   
      
   Who said they are?   
      
   >>> I don't see any connection to some science or engineering   
   >>> space.  If it /came/ from some scientific study, it has already been   
   >>> turned into what the programmer needs come up with.   
   >>>   
   >>>> Average is just such formulation of finding a representative member   
   >>>> from some set having some specific properties. E.g. least squares in   
   >>>> physics usually has the meaning of least energy etc. So, it goes in   
   >>>> this order (not in reverse):   
   >>>>   
   >>>> 1. You need the least energy representative.   
   >>>> 2. An average gives you that.   
   >>>> 3. You measure the inputs.   
   >>>>   
   >>>> After that you ask yourself given these measures how to compute the   
   >>>> average? And only now it becomes a programming issue.   
   >>> So if I asked you for code to calculate the arithmetic mean of an array   
   >>> of numbers you would ask for all this first, declaring it not yet a   
   >>> programming issue?   
   >>   
   >> Then I would ask you, if this is a solution, what was the problem?   
   >   
   > This is a solution to the simpler problem.   
      
   And what was that the problem? [Adjective adds/clarifies nothing]   
      
   > But we are not in that situation.  A few knowledgeable and experienced   
   > people are discussing a programming problem.   
      
   I saw no discussion on programming so either.   
      
   > You are welcome to join   
   > in, but the best way for you to do that is to ask whatever questions you   
   > think would help you understand what's being discussed.  So far, your   
   > replies have alternated between "it's trivial" and "it's unstated".   
      
   Yes, because you wrote about minimum of arc length on a *circle*. And   
   that is trivial.   
      
   >>>>> You say you need a formula, so I'll try...  Let P_n be a collection of n   
   >>>>> unit vectors specifying n points on a unit sphere.  Find the unit vector   
   >>>>> A that minimises   
   >>>>>      Sum_{i=1,n} ( arctan( |A x P_n|  /  A . P_n ) )^2   
   >>>>> (arctan is the "all quadrant" version that is often called atan2 in   
   >>>>> programming languages.)   
   >>>>   
   >>>> 1. atan2 has two arguments (the adjacent and the opposite legs);   
   >>> Obviously I can't guess how much I can safely assume so I expected there   
   >>> might be questions.  arctan(p / q) is usually coded as atan2(p, q).   
   >>>   
   >>>> 2. What is "A";   
   >>> The unit vector that is being sought -- the result of the algorithm.  It   
   >>> represents a point on the unit sphere that is the desired average of the   
   >>> input points.   
   >>>   
   >>>> 3. What operations(?) "x" and "." denote?   
   >>> x denotes the cross product, and . the dot product.   
   >>> Do you agree that this is not a trivial problem?  If not, please give   
   >>> the trivial algorithm!   
   >>   
   >> If I correctly understood your explanation (and with a few obvious   
   >> corrections):   
   >>   
   >>     Sum atan2 (|A x Pi|, A · Pi)  --> min   
   >>    i=1..n                       {A | A on the circle}   
   >   
   > No.  A is a vector denoting a point on the unit sphere.  And you have   
   > missed out the power of two.  Were these changes what you call "obvious   
   > corrections"?   
   >   
   >> |A x Pi| = |A| |Pi| |sin θi|  (θi is the angle between the vectors A and   
   Pi)   
   >>   
   >> A · Pi = |A| |Pi| cos θi   
   >>   
   >> atan2 (|A x Pi|, A · Pi) = atan2 (|sin θi|, cos θi) = θi'   
   >>   
   >> Where θi' is θi with lower semicircle mapped to the upper one (for   
   >> whatever reason).   
   >>   
   >> (If mapping was unintended you must have taken the direction of the   
   >> cross product for the sign for |A x Pi|)   
   >>   
   >> On a circle varying the angle between A and (R, 0), where R is the   
   >> radius, you find that the mean of the angles between Pi and (R, 0)   
   >> yields the minimum.   
   >   
   > This does not even solve the related case of finding the average point   
   > on the unit circle.   
      
   It elementary does for both formula you gave (after corrections) and the   
   circumference:   
      
       Sum Length_Of_Arc (Pi, A)^2 =   
      i=1..n   
      
   =  Sum (R Angle (Pi, A))^2 =   
      i=1..n   
      
   =  R^2 Sum ((Angle (Pi, (R, 0)) - Angle (A, (R, 0)))^2   
          i=1..n   
      
   (R, 0) denotes vector x = R, y = 0. Let θi denote Angle (Pi, (R, 0)) and   
   α do Angle (A, (R, 0)) then   
      
   =  R^2 Sum (θi - α)^2 --> min   
          i=1..n   
      
   Derivative d/dα:   
      
      2 R^2 Sum (θi - α) = 0 <=> (R /= 0)   
           i=1..n   
      
   <=> Sum θi - nα = 0 <=>   
       i=1..n   
      
   <=> α = Sum θi / n = mean ({θi})   
           i=1..n   
      
   The angle between A and (R, 0) is the mean of angles of points.   
      
   I can't help you with spherical trigonometry, which is an utter off   
   topic anyway. I do not know if there is a direct solution of least   
   squares. For two points it is like for a circle. For three points it   
   would be the circumcenter of the spherical triangle.   
      
   However even if the direct solution existed, it could be numerically   
      
   [continued in next message]   
      
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