diff options
Diffstat (limited to 'src/slalib/sun67.htx/node198.html')
| -rw-r--r-- | src/slalib/sun67.htx/node198.html | 167 |
1 files changed, 167 insertions, 0 deletions
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 @@ +<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2//EN"> +<!--Converted with LaTeX2HTML 97.1 (release) (July 13th, 1997) + by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds +* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan +* with significant contributions from: + Jens Lippman, Marek Rouchal, Martin Wilck and others --> +<HTML> +<HEAD> +<TITLE>Spherical Trigonometry</TITLE> +<META NAME="description" CONTENT="Spherical Trigonometry"> +<META NAME="keywords" CONTENT="sun67"> +<META NAME="resource-type" CONTENT="document"> +<META NAME="distribution" CONTENT="global"> +<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso_8859_1"> +<LINK REL="STYLESHEET" HREF="sun67.css"> +<LINK REL="next" HREF="node200.html"> +<LINK REL="previous" HREF="node197.html"> +<LINK REL="up" HREF="node197.html"> +<LINK REL="next" HREF="node199.html"> +</HEAD> +<BODY > +<BR> <HR> +<A NAME="tex2html2434" HREF="node199.html"> +<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next_motif.gif"></A> +<A NAME="tex2html2432" HREF="node197.html"> +<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up_motif.gif"></A> +<A NAME="tex2html2426" HREF="node197.html"> +<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="previous_motif.gif"></A> <A HREF="sun67.html#stardoccontents"><IMG ALIGN="BOTTOM" BORDER="0" + SRC="contents_motif.gif"></A> +<BR> +<B> Next:</B> <A NAME="tex2html2435" HREF="node199.html">Formatting angles</A> +<BR> +<B>Up:</B> <A NAME="tex2html2433" HREF="node197.html">EXPLANATION AND EXAMPLES</A> +<BR> +<B> Previous:</B> <A NAME="tex2html2427" HREF="node197.html">EXPLANATION AND EXAMPLES</A> +<BR> <HR> <P> +<P><!--End of Navigation Panel--> +<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> +<BR><HR> +<!--Table of Child-Links--> +<A NAME="CHILD_LINKS"> </A> +<UL> +<LI><A NAME="tex2html2436" HREF="node199.html#SECTION00051100000000000000"> +Formatting angles</A> +</UL> +<!--End of Table of Child-Links--> +<BR> <HR> +<A NAME="tex2html2434" HREF="node199.html"> +<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next_motif.gif"></A> +<A NAME="tex2html2432" HREF="node197.html"> +<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up_motif.gif"></A> +<A NAME="tex2html2426" HREF="node197.html"> +<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="previous_motif.gif"></A> <A HREF="sun67.html#stardoccontents"><IMG ALIGN="BOTTOM" BORDER="0" + SRC="contents_motif.gif"></A> +<BR> +<B> Next:</B> <A NAME="tex2html2435" HREF="node199.html">Formatting angles</A> +<BR> +<B>Up:</B> <A NAME="tex2html2433" HREF="node197.html">EXPLANATION AND EXAMPLES</A> +<BR> +<B> Previous:</B> <A NAME="tex2html2427" HREF="node197.html">EXPLANATION AND EXAMPLES</A> +<BR> <HR> <P> +<P><!--End of Navigation Panel--> +<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> +</ADDRESS> +</BODY> +</HTML> |