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Re: Spherical Law of Cosines
From: Dave Weilacher
Date: 2002 Oct 18, 10:12 -0700

```Thanks.  Very pleased with you.

Dave

On Fri, 18 Oct 2002 10:05:29 -0700 Dan Allen  wrote:

> On Friday, October 18, 2002, at 07:53 AM, David
> Weilacher wrote:
>
> > Please define little (a), (b), (c), and (ab),
> for me.
>
> With pleasure!
>
> cos(c) = cos(a) * cos(b) + sin(a) * sin(b) *
> cos(ab)
>
> a is the length of the first side of the
> spherical triangle.
> b is the length of the second side of the
> spherical triangle.
> ab is the angle between the two sides a and b.
> c is the length of the side opposite the angle
> ab.
>
> Here is an example of a great circle distance
> between
> San Francisco (SF) and Salt Lake City (SLC).
> All angles
> and lengths are expressed in degrees for this
> example,
> and North and West are positive.
>
> SF:
> lat1 = 37 degrees
> lon1 = 122 degrees
>
> SLC:
> lat2 = 40 degrees
> lon2 = 112 degrees
>
> So,
>
> a is the co-latitude of lat1, or 90-lat1 or 53
> degrees.
> b is the co-latitude of lat2, or 90-lat2 or 50
> degrees.
> ab is the difference of the longitudes, or
> lon1-lon2 or 10 degrees.
>
> Solving for c one learns the distance in
> degrees between SF and SLC,
> which is about 8.375 degrees.  To get a
> distance we understand,
> multiply degrees times 60 nmi per deg and you
> get 502.5 nmi.
>
> One then can re-substitute back in and use the
> formula again to
> determine the initial great circle course.  In
> this case we
> use a as co-latitude of lat1, b is the distance
> we just
> computed (8.375 degrees -- make sure to use
> degrees), c is
> the co-latitude of lat2, and the angle ab is
> the initial
> course.  We have rotated the whole spherical
> triangle around
> to put the sides a,b, and c where we know their
> values,
> leaving ab to be solved for.  Substituting and
> solving one
> determines that the initial course is 65.9588
> degrees.
>
> ---
>
> In sight reduction:
>
> a is the co-latitude of the assumed position
> b is the declination of the body (say the sun)
> ab is the hour angle of the body
> c is the altitude of the body
>
> ---
>
> Note too that there are equivalent ways of
> writing the spherical
> law of cosines.  One can write it as
>
> cos(c) = sin(a)*sin(b) + cos(a)*cos(b)*cos(ab)
>
> where the pairs of sin/cos of a/b are switched.
>  This allows one to do
> great circle calculations using latitudes
> directly,
> without using co-latitudes.  The origin is
> moved from the pole to
> the equator, so to speak.  This form is often
> handier but the first
> version is the canonical version.
>
> Hope this helps.
>
> Dan
>

Dave Weilacher
.    #889968
.ASA certified sailing and celestial
.IBM AS400 RPG contract programmer

```
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