Next message: Herbert Prinz: "Re: Azimuth and Declination formulae"
Many thanks Lu - you saved me the trouble. There is probably no modern
use to the formul;a posted - just, in may opinion, an interesting
historical note and perhaps part of one's math education.
On Tue, 19 Jul 2005 21:05:42 -0700 Lu Abel <> writes:
> 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?
> >
> >
>
