From d54fe7c1f704a63824c5bfa0ece65245572e9b27 Mon Sep 17 00:00:00 2001 From: Joseph Hunkeler Date: Wed, 4 Mar 2015 21:21:30 -0500 Subject: Initial commit --- src/slalib/sun67.htx/node198.html | 167 ++++++++++++++++++++++++++++++++++++++ 1 file changed, 167 insertions(+) create mode 100644 src/slalib/sun67.htx/node198.html (limited to 'src/slalib/sun67.htx/node198.html') diff --git a/src/slalib/sun67.htx/node198.html b/src/slalib/sun67.htx/node198.html new file mode 100644 index 0000000..30aa3e8 --- /dev/null +++ b/src/slalib/sun67.htx/node198.html @@ -0,0 +1,167 @@ + + + + +Spherical Trigonometry + + + + + + + + + + + + +
+ +next + +up + +previous +
+ Next: Formatting angles +
+Up: EXPLANATION AND EXAMPLES +
+ Previous: EXPLANATION AND EXAMPLES +

+

+

+Spherical Trigonometry +

+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 celestial sphere. Problems involving +positions and orientations in the sky can then be solved by +using the formulae of spherical trigonometry, which +apply to spherical triangles, the sides of which are +great circles. +

+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 +(e.g. right ascension and declination $[\,\alpha,\delta\,]$,Galactic longitude and latitude $[\,l^{I\!I},b^{I\!I}\,]$)or left-handed (e.g. hour angle and +declination $[\,h,\delta\,]$). 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. +

+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. +

+As well as providing many routines which work in terms of specific +spherical coordinates such as $[\,\alpha,\delta\,]$, 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 position angle) +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: +

+            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
+
+

(The result is 17011 km, $61^\circ$.) +

+The routines +sla_PAV and +sla_DPAV +are equivalents of sla_BEAR and sla_DBEAR but starting from +direction-cosines instead of spherical coordinates. +

+

+ +  + + +
+ +next + +up + +previous +
+ Next: Formatting angles +
+Up: EXPLANATION AND EXAMPLES +
+ Previous: EXPLANATION AND EXAMPLES +

+

+

+SLALIB --- Positional Astronomy Library
Starlink User Note 67
P. T. Wallace
12 October 1999
E-mail:ptw@star.rl.ac.uk +
+ + -- cgit