From: tr.17687@z991.linuxsc.com   
      
   "Dmitry A. Kazakov"  writes:   
      
   > On 2022-12-31 15:42, Tim Rentsch wrote:   
   >   
   >> Ben Bacarisse  writes:   
   >>   
   >>> For the "vector average", we convert the t(i) to unit vectors u(i) and   
   >>> we calculate the mean if the u(i) to get a vector m.  The "average", A,   
   >>> is just the direction of this vector -- another point on the unit   
   >>> circle.  In this case we are minimising the sum of squares of the   
   >>> /chord/ lengths between A and the t(i).   
   >>   
   >> I think of this approach differently.  I take the time values   
   >> t(i) as being unit masses on the unit circle, and calculate the   
   >> center of mass.  As long as the center of mass is not the origin   
   >> we can project it from the origin to find a corresponding time   
   >> value on the unit circle (which in my case is done implicitly by   
   >> using atan2()).   
   >   
   > Center of mass of a set of ideal points (particles) and vector   
   > average are same:   
      
   Yes, I thought the equivalence is obvious and not in need of   
   explanation.   
      
   >>> This distinction between arc lengths and chord lengths helps to   
   >>> visualise where these averages differ, and why the conventional   
   >>> average may seem more intuitive.   
   >>   
   >> Interesting perspective.  I wouldn't call them chord lengths   
   >> because I think of a chord as being between two points both on   
   >> the same circle, and the center of mass is never on the unit   
   >> circle (not counting the case when all the time values are the   
   >> same).  Even so it's an interesting way to view the distinction.   
   >   
   > Arc length is proportional to angle:   
      
   A trivial and useless observation.   
      
   > Averaging arcs is equivalent to averaging angles.   
      
   Angles are a one-dimensional measure.  Finding an arc length   
   "average" of points on a sphere needs a two-dimensional result.   
      
   >> Now that I think about it, finding the point that minimizes the   
   >> great circle distances squared would be at least computationally   
   >> unpleasant.   
   >   
   > See above, it is just angles to average.   
      
   Apparently you have not yet understood the problem.  Why don't   
   you try writing a program that inputs a set of points normalized   
   to be on the unit sphere, and then calculates the arc length   
   average point (on the unit sphere) of those input points?   
      
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