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I've been pondering a bit further about how azimuth tables might be improved
a bit. This was triggered by realising in an earlier posting, that it was
equally valid to tabulate log (cos X sin y)  as it was to tabulate 1000 (cos
X sin Y) in the two sets of tables being compared. Mathematically, this is
because we are solving the equation-
.cos Alt sin Az = cos Dec sin LHA
and it would be equally valid instead to solve the equation-
log ( cos Alt sin Az ) = log (cos Dec sin LHA), which is equally true. That
is what HO171 sets out to do, in tabulating log (cos X sin Y).

But it's just struck me that any other simple single-valued function of
(cos Dec sin LHA) could be used instead, and might offer advantages in
manipulating the tables. The difficulty with (cos X sin Y) is that it's
changing very slowly as it approaches 1, which occurs toward the bottom left
of Bennett's table. What if, instead, we tabulated the angle whose sine had
the value (cos X sin Y)? Or better (if the arc sine was calculated in
degrees) then mutiplying that angle (in the range 0 to 90) by 11.11, and
rounding to the nearest integer, would give a result, up to maximum of 1000,
that increased smoothly and steadily throughout the table. If arc sine is
calculated in radians, as in most computer programs, it would involve
multiplying by 2000/pi, or 636.6, instead.

We would be solving the equation-
11.11 arcsine( cos Alt sin Az ) = 11.11 arcsine (cos Dec sin LHA)
where arcsine is calculated in degrees.

Can anyone see any snags in doing that? I must admit to not having thought
out the matter in any detail.

All we are doing here is improving the precision of the internal
manipulations, over part of the range. It would do nothing to resolve the
difficulty of the extreme senstivity of the method to small changes in
altitude or LHA, when the azimuth is anywhere near East or West, especially
when these have to be rounded to the nearest integer. To overcome that, it's
necessary to work instead in terms of lat, dec, and LHA, avoiding the use of
a calculated altitude.

George.

contact George Huxtable at george@huxtable.u-net.com
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.


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