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Subject: Spherical Law of Cosines
From: Dan Allen (danallen46@XXX.XXX)
Date: Fri Oct 18 2002 - 13:05:29 EDT
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
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