Re: Another little puzzle

From	"Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de>
Newsgroups	comp.programming
Subject	Re: Another little puzzle
Date	2022-12-31 17:04 +0100
Organization	Aioe.org NNTP Server
Message-ID	<topmiu$4ps$1@gioia.aioe.org> (permalink)
References	(3 earlier) <87tu1diu2s.fsf@bsb.me.uk> <864jtdtkt5.fsf@linuxsc.com> <87o7rlhtsv.fsf@bsb.me.uk> <878rioifnh.fsf@bsb.me.uk> <868rinskhk.fsf@linuxsc.com>

Show all headers | View raw

On 2022-12-31 15:42, Tim Rentsch wrote:
> Ben Bacarisse <ben.usenet@bsb.me.uk> 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:

    CoM = Sum Mi * Ri / Sum Mi
          i = 1..n      i = 1..n

Mi = masses, Ri = vectors. If all Mi are same you get

    CoM = Sum Ri / n
          i = 1..n

>> 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:

    L = Rα, R is radius, α is angle in radians.

Averaging arcs is equivalent to averaging angles.

> 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.

-- 
Regards,
Dmitry A. Kazakov
http://www.dmitry-kazakov.de

Back to comp.programming | Previous | Next — Previous in thread | Next in thread | Find similar

Thread

Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-12-30 01:00 +0000
  Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2022-12-29 23:25 -0800
    Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-12-30 14:04 +0000
      Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-12-31 00:24 +0000
        Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2022-12-31 06:42 -0800
          Re: Another little puzzle "Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de> - 2022-12-31 17:04 +0100
            Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-01 01:24 +0000
            Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2023-01-08 07:45 -0800
              Re: Another little puzzle "Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de> - 2023-01-08 17:17 +0100
                Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-08 20:41 +0000
                Re: Another little puzzle Richard Heathfield <rjh@cpax.org.uk> - 2023-01-08 21:14 +0000
                Re: Another little puzzle "Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de> - 2023-01-08 22:31 +0100
                Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-09 03:25 +0000
                Re: Another little puzzle "Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de> - 2023-01-09 11:22 +0100
                Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-09 20:37 +0000
                Re: Another little puzzle "Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de> - 2023-01-10 09:06 +0100
                Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-11 02:41 +0000
                Re: Another little puzzle "Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de> - 2023-01-11 10:01 +0100
                Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-12 01:00 +0000
                Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2023-01-10 23:36 -0800
                Re: Another little puzzle Y A <ya00000100000@yahoo.com> - 2023-01-11 02:39 -0800
          Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-01 01:10 +0000
            Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2023-01-08 07:17 -0800
              Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-08 19:43 +0000
                Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-08 19:59 +0000
                Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2023-01-10 23:25 -0800
      Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2022-12-31 06:20 -0800
        Re: Another little puzzle Augǝl <angel0000000001000000000000@mail.ee> - 2022-12-31 10:23 -0800

csiph-web