Groups | Search | Server Info | Keyboard shortcuts | Login | Register [http] [https] [nntp] [nntps]

Groups > comp.programming > #16266

Re: Another little puzzle

Show all headers | View raw

"Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de> writes:

> 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:

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?

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