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Subject: Re: Spherical Law of Cosines
From: Dan Allen (danallen46@XXX.XXX)
Date: Wed Oct 23 2002 - 12:53:21 EDT
I spoke too hastily about my error in alternate renderings of
the spherical law of cosines formula.
Originally I had said that
cos(c) = sin(a)*sin(b) + cos(a)*cos(b)*cos(ab)
was an alternate form, and then Bill Arden pointed out that for
use with Hc it should have read
sin(Hc) = sin(a)*sin(b) + cos(a)*cos(b)*cos(ab)
the difference being the left hand side of the equation.
However, I went back and found support in Smart's book for the
form that I had written, i.e.,
cos(c) = sin(a)*sin(b) + cos(a)*cos(b)*cos(ab)
in determining the length of twilight and other such calculations.
In thinking about things I realized that both versions are
right, but it simply is a matter of origin. Are the angles
measured down from the pole (co-latitudes and such) or are
they measured from the equator up (latitudes)?
They are equivalent.
The mental picture that I work from is the canonical version,
cos(c) = cos(a)*cos(b) + sin(a)*sin(b)*cos(ab)
and then I don't get into problems, because that is the one
that I learned from.
So we were both right.
Dan
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