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<H2><A NAME="SECTION00051000000000000000">
Spherical Trigonometry</A>
</H2>
Celestial phenomena occur at such vast distances from the
observer that for most practical purposes there is no need to
work in 3D; only the direction
of a source matters, not how far away it is. Things can
therefore be viewed as if they were happening
on the inside of sphere with the observer at the centre -
the <I>celestial sphere</I>. Problems involving
positions and orientations in the sky can then be solved by
using the formulae of <I>spherical trigonometry</I>, which
apply to <I>spherical triangles</I>, the sides of which are
<I>great circles</I>.
<P>
Positions on the celestial sphere may be specified by using
a spherical polar coordinate system, defined in terms of
some fundamental plane and a line in that plane chosen to
represent zero longitude. Mathematicians usually work with the
co-latitude, with zero at the principal pole, whereas most
astronomical coordinate systems use latitude, reckoned plus and
minus from the equator.
Astronomical coordinate systems may be either right-handed
(<I>e.g.</I> right ascension and declination <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$">,Galactic longitude and latitude <IMG WIDTH="59" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img98.gif"
ALT="$[\,l^{I\!I},b^{I\!I}\,]$">)or left-handed (<I>e.g.</I> hour angle and
declination <IMG WIDTH="41" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img29.gif"
ALT="$[\,h,\delta\,]$">). In some cases
different conventions have been used in the past, a fruitful source of
mistakes. Azimuth and geographical longitude are examples; azimuth
is now generally reckoned north through east
(making a left-handed system); geographical longitude is now usually
taken to increase eastwards (a right-handed system) but astronomers
used to employ a west-positive convention. In reports
and program comments it is wise to spell out what convention
is being used, if there is any possibility of confusion.
<P>
When applying spherical trigonometry formulae, attention must be
paid to
rounding errors (for example it is a bad idea to find a
small angle through its cosine) and to the possibility of
problems close to poles.
Also, if a formulation relies on inspection to establish
the quadrant of the result, it is an indication that a vector-related
method might be preferable.
<P>
As well as providing many routines which work in terms of specific
spherical coordinates such as <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$">, SLALIB provides
two routines which operate directly on generic spherical
coordinates:
sla_SEP
computes the separation between
two points (the distance along a great circle) and
sla_BEAR
computes the bearing (or <I>position angle</I>)
of one point seen from the other. The routines
sla_DSEP
and
sla_DBEAR
are double precision equivalents. As a simple demonstration
of SLALIB, we will use these facilities to estimate the distance from
London to Sydney and the initial compass heading:
<P><PRE>
IMPLICIT NONE
* Degrees to radians
REAL D2R
PARAMETER (D2R=0.01745329252)
* Longitudes and latitudes (radians) for London and Sydney
REAL AL,BL,AS,BS
PARAMETER (AL=-0.2*D2R,BL=51.5*D2R,AS=151.2*D2R,BS=-33.9*D2R)
* Earth radius in km (spherical approximation)
REAL RKM
PARAMETER (RKM=6375.0)
REAL sla_SEP,sla_BEAR
* Distance and initial heading (N=0, E=90)
WRITE (*,'(1X,I5,'' km,'',I4,'' deg'')')
: NINT(sla_SEP(AL,BL,AS,BS)*RKM),NINT(sla_BEAR(AL,BL,AS,BS)/D2R)
END
</PRE>
<P>(The result is 17011 km, <IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img235.gif"
ALT="$61^\circ$">.)
<P>
The routines
sla_PAV and
sla_DPAV
are equivalents of sla_BEAR and sla_DBEAR but starting from
direction-cosines instead of spherical coordinates.
<P>
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<ADDRESS>
<I>SLALIB --- Positional Astronomy Library<BR>Starlink User Note 67<BR>P. T. Wallace<BR>12 October 1999<BR>E-mail:ptw@star.rl.ac.uk</I>
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