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Re: Haversine- how to derive it?
From: Hanno Ix
Date: 2015 May 17, 09:45 -0700

John, Ed, Samuel,

as I outlined the virtues of the haversine to Frank yesterday,

hav (x) is

(i) always positive,
(ii) no matter the sign of the angles x
(iii) never goes to infinity                   which I forgot to mention.

Contrast this to sin(x), cos(x), tan(x) etc. and their logarithms!

However, using the haversine G. Rudzinski and I can show you another,
very easy calculation of Hc. Only one table, 2 pages, is used and only one
multiplication is needed. There are no complicated sign rules
and special cases - it will work for all permutations of  L, d and LHA.

Given: L, d, LHA, find Hc. Execute these 6 elementary steps:

1. calculate                 n = hav ( L - d )
2. calculate                 p = hav (L + d )
3. calculate                 q = p + n

4. calculate                 n + ( 1 - q ) * hav ( LHA ); this yields hav ( ZD )
5. find in table             ZD  by looking up the table backwards

6. Finally                    Hc = 90 deg  - ZD

As can you see the steps are basic arithmetic - executable by hand in minutes.

We discussed all this on the list under the topic Longhand Sight Reduction.
For the azimuth we suggest using the azimuth table I published there, too.

H

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