# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Haversine formulae for Great Circles**

**From:**Lu Abel

**Date:**2001 Nov 17, 12:08 PM

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