From: J. G. Locke (lockejg@redshift.com)
Date: Sat Dec 04 1999 - 08:02:47 PST
From: "J. G. Locke" <lockejg@redshift.com> Subject: RE: Pinhole Digest #301 - Displacement on Sphere Date: Sat, 4 Dec 1999 08:02:47 -0800 Message-ID: <000f01bf3e71$01ccf8a0$2a32c8d8@daddysmachine>
||
|| Subject: RE: Pinhole Digest #300 -- angular displacement
|| From: "Marc Afifi" <marc_afifi@yahoo.com>
|| Date: Fri, 3 Dec 1999 08:14:22 -0800 (PST)
||
|| Thanks for the reply John,
||
|| Perhaps my question was not clear enough. If I walk 3m
|| north and 4m west, my distance traveled is 7m but my
|| displacement is only 5m. This is the actual distance
|| away from the starting point that we refer to as
|| displacement. Your reply seems to correlate to the
|| idea of distance traveled but indeed it is called the
|| angular displacement in every physics book I've looked
|| in. My confusion is this: Since there are only 2pi
|| radians in any circle, the maximum displacement you
|| could experience would be some angle smaller than 2pi
|| else you arrive back at the starting point and your
|| displacement is zero. It doesn't matter how many times
|| you go around the circle, the displacement is still
|| less than 2pi from the starting point. The total
|| angular distance is a different story, but no book I
|| have seen discusses an angular distance. Is this just
|| a problem of semantics?
||
|| -Marc
Oh, I get it, now.
Suppose we travel from Point A to Point B. The "distance" would be the
length of the path from A to B measured along the path. The "displacement"
would be the length of the shortest great circle arc measured from A to B.
For example, we start in Britain,
fly NW to Iceland,
fly W to Alaska,
fly W to Finland, and
fly SSW to Morocco.
Then our displacement is measured along the
great circle from Britain to Spain to Morocco. However, choice of the
displacement arc
Britain-Spain-Morocco or the arc
Britain-ArcticOcean-Samoa-NewZealand-Antarctica-Morocco
may depend upon topological considerations. My gut feeling is that the
displacement is the shorter great circle arc. This is based upon a thought
experiment in which the route is changed infinitesimally from the direct,
great circle arc Britain-Spain-Morocco to routes which differ from direct by
increasing deviations. (Compare these different routes:
Britain-Barcelona-Morocco; Britain-Madrid-Morocco; Britain-Lisbon-Morocco.
We would agree, wouldn't we, that all these routes are associated with the
great circle displacement Britain-Spain-Morocco.)
This has become an interesting diversion -- thanks, Marc, for providing this
exercise.
Respectfully, John
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