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From: Tim Rentsch <tr.17687@z991.linuxsc.com>
Newsgroups: comp.programming
Subject: Re: Another little puzzle
Date: Tue, 10 Jan 2023 23:25:10 -0800
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Ben Bacarisse <ben.usenet@bsb.me.uk> writes:

> Ben Bacarisse <ben.usenet@bsb.me.uk> writes:
>
>> Tim Rentsch <tr.17687@z991.linuxsc.com> writes:
>>
>>> Ben Bacarisse <ben.usenet@bsb.me.uk> writes:
>>
>> <cut>
>>
>>>> The "other" average, minimises the sum of the of the angular
>>>> distances.
>>>
>>> I think you mean it minimizes the sum of the squares of the
>>> angular distances.
>>
>> Yes thanks.  I originally thought I'd write an abbreviation but then I
>> convinced myself I would not forget any of the "squares of"s.  Proof
>> reading might have been useful too.
>>
>>>>> 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.
>>>>
>>>> It's more interesting because it really is about chords!
>>>
>>> I don't buy the "really" here.  The center of mass is fundamental
>>> and always well-defined.  Furthermore it works for collections of
>>> points that are not all on the same circle or sphere.  Looking at
>>> the problem in terms of chords seems somewhat artificial, or at
>>> least less fundamental.
>>
>> There are other ways for something to be interesting (at least to me).
>> When you plot the two averages, and look at the chords vs the arcs, it
>> becomes clear why one average "looks" more average than the other.
>
> I can see a possible misunderstanding here.  You might have taken my "it
> really is about chords" to mean "it is fundamentally about chords", but
> I said "it really is about chords" because you thought I'd used the word
> chord incorrectly to refer to the lines between the centre of mass and
> the data points.  I was saying "I really did mean chords".

Now that I see what you meant it all makes sense.

Also I agree that comparing the chords view and the arcs view
gives a useful perspective.  I am starting to appreciate the
value of that perspective more as time goes on.