Path: csiph.com!news.mixmin.net!eternal-september.org!reader01.eternal-september.org!.POSTED!not-for-mail From: Tim Rentsch Newsgroups: comp.programming Subject: Re: Another little puzzle Date: Sun, 08 Jan 2023 07:45:23 -0800 Organization: A noiseless patient Spider Lines: 54 Message-ID: <868ridni7g.fsf@linuxsc.com> References: <87tu1diu2s.fsf@bsb.me.uk> <864jtdtkt5.fsf@linuxsc.com> <87o7rlhtsv.fsf@bsb.me.uk> <878rioifnh.fsf@bsb.me.uk> <868rinskhk.fsf@linuxsc.com> MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Injection-Info: reader01.eternal-september.org; posting-host="74dc1f9657693bf61783dca6338fedc2"; logging-data="4074897"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1/5GKp4bsR1RLo+ABMGm6HPy0cuLwetyA4=" User-Agent: Gnus/5.11 (Gnus v5.11) Emacs/22.4 (gnu/linux) Cancel-Lock: sha1:+xi4zWOIoKcHPJHihsxAgUj3SxA= sha1:hqjIQUaZ31W5fuVQAh0zaBifls8= Xref: csiph.com comp.programming:16266 "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?