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Subject: Haversine formulae for Great Circles
From: Lu Abel (lunav@XXX.XXX)
Date: Sat Nov 17 2001 - 14:08:22 EST
Almost every celestial text provides the formulae for calculating great
circle distance and direction between two points. This is not surprising
since the trigonometry is identical to that of sight reduction. If one
takes any standard celestial reduction method and substitutes the starting
point for the AP and the ending point for the body GP, Zn gives the
direction and 60*(90-Hc) gives the distance.
A couple of years ago this august group was of great help in answering for
me an obvious question I've never seen answered in any standard navigation
text, namely how to calculate the L/Lo of intermediate points.
Recently I've done a bit of further digging on great circles. I learned
that while the law of cosines formula typically used for Hc calculation is
trigonometrically correct, it can produce incorrect answers on a computer
or calculator when the starting and ending points are close because
computers and calculators express numbers with a limited number of
significant digits. An alternate is the "Haversine" formula, called by
that name because of its haversine-like terms [recall that hav(x) =
(1-cos(x))/2 = sin^2(x/2) where "^2" means squared ]
The haversine distance formula goes as follows:
L1, Lo1 are L/Lo of the starting point, L2/Lo2 are the L/Lo of the destination
DLat = L2 - L1
DLo = Lo2 - Lo1
A = sin^2(DLat/2) + cos(L2) * cos(L1) * sin^2(DLo/2)
Distance = 120 * arcsin (sqrt (A))
For a great writeup on the haversine distance formula, see
http://mathforum.org/dr.math/problems/neff.04.21.99.html (note no "www" at
beginning)
Having learned about this formula, I'm going to guess it's used in the
majority of GPS sets, since they're often calculating small distances (like
distance-to-go when approaching a waypoint)
The standard reference on calculating great circles via the haversine
formula seems to be R.W. Sinnott, "Virtues of the Haversine", Sky and
Telescope, vol. 68, no. 2, 1984, p. 159. I've seen it mentioned in several
writeups of the haversine distance formula.
I can't access a copy to see if Sinnott provided formulae beyond the
distance formula, so here's my question for this group:
Are there equivalent "haversine" formulae for initial direction and the
L/Lo of intermediate points?
Thanks
Lu Abel
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