# NavList:

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

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Re: Azimuth and Declination formulae
From: Lu Abel
Date: 2005 Jul 19, 21:05 -0700

```Whoa on the haversines.  It's not half of a sine, it's half of a versine.

A versine (x) = 1 - cos (x).   Note that vers (x) has a range from 0 to 2.

Haversine (x) = vers (x) / 2.   This just makes hav (x) have a range
from 0 to 1.

The whole reason for versines and haversines was to allow sight
reductions to be done using logarithms (and therefore the requisite
multiplications become additions); but logs are not defined for negative
numbers, hence the need to shift everything to have a positive value.

Versines and haversines can also be expressed in terms of sine squared,
vers (x) = 2 sin^^2 (x/2).

As a side note, the traditional formula for the great circle distance
between two points breaks down into finding the difference between two
nearly equal large quantities for small distances.  This can produce
inaccurate answers because calculators and computers only carry out
calculations with a limited number of digits.  The equivalent haversine
formula is well behaved, subtracting two small numbers.  Therefore all
GPS's actually use the haversine formula for calculating the distance
between two points.

Lu Abel

Peter Fogg wrote:
>>From: Henry C. Halboth
>>A bit more complicated, but generally employed with the Time Sight is ...
>>
>>hav Z = sec ho x sec L x sin 1/2S - ho x sin 1/2S -L, where ...
>>
>>Z = azimith, named according to Latitude + meridian angle, E or W
>>ho = corrected altitude
>>L = Latitude
>>pd = polar distance
>>S = ho + L + pd
>
>
> Interesting. Presumably 'hav' stands for haversine, which I vaguely recall
> is a half sine? And 'sec' is secant? I don't know what that is.
>
> What do I need to be able to use this formula? Scientific calculators I
> have. Do I need tables of havesines and secants?
>
> What is the advantage of this formula?
>
>

```
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