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| author | Joseph Hunkeler <jhunkeler@gmail.com> | 2015-03-04 21:21:30 -0500 |
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| committer | Joseph Hunkeler <jhunkeler@gmail.com> | 2015-03-04 21:21:30 -0500 |
| commit | d54fe7c1f704a63824c5bfa0ece65245572e9b27 (patch) | |
| tree | afc52015ffc2c74e0266653eecef1c8ef8ba5d91 /src/slalib/sun67.htx/node200.html | |
| download | calfuse-d54fe7c1f704a63824c5bfa0ece65245572e9b27.tar.gz | |
Initial commit
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| -rw-r--r-- | src/slalib/sun67.htx/node200.html | 168 |
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diff --git a/src/slalib/sun67.htx/node200.html b/src/slalib/sun67.htx/node200.html new file mode 100644 index 0000000..cbbad9d --- /dev/null +++ b/src/slalib/sun67.htx/node200.html @@ -0,0 +1,168 @@ +<!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>Vectors and Matrices</TITLE> +<META NAME="description" CONTENT="Vectors and Matrices"> +<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="node202.html"> +<LINK REL="previous" HREF="node198.html"> +<LINK REL="up" HREF="node197.html"> +<LINK REL="next" HREF="node201.html"> +</HEAD> +<BODY > +<BR> <HR> +<A NAME="tex2html2453" HREF="node201.html"> +<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next_motif.gif"></A> +<A NAME="tex2html2451" HREF="node197.html"> +<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up_motif.gif"></A> +<A NAME="tex2html2445" HREF="node199.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="tex2html2454" HREF="node201.html">Using vectors</A> +<BR> +<B>Up:</B> <A NAME="tex2html2452" HREF="node197.html">EXPLANATION AND EXAMPLES</A> +<BR> +<B> Previous:</B> <A NAME="tex2html2446" HREF="node199.html">Formatting angles</A> +<BR> <HR> <P> +<P><!--End of Navigation Panel--> +<H2><A NAME="SECTION00052000000000000000"> +Vectors and Matrices</A> +</H2> +As an alternative to employing a spherical polar coordinate system, +the direction of an object can be defined in terms of the sum of any +three vectors as long as they are different and +not coplanar. In practice, three vectors at right angles are +usually chosen, forming a system +of <I>Cartesian coordinates</I>. The <I>x</I>- and <I>y</I>-axes +lie in the fundamental plane (<I>e.g.</I> the equator in the +case of <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" + SRC="img3.gif" + ALT="$[\,\alpha,\delta\,]$">), with the <I>x</I>-axis pointing to zero longitude. +The <I>z</I>-axis is normal to the fundamental plane and points +towards positive latitudes. The <I>y</I>-axis can lie in either +of the two possible directions, depending on whether the +coordinate system is right-handed or left-handed. +The three axes are sometimes called +a <I>triad</I>. For most applications involving arbitrarily +distant objects such as stars, the vector which defines +the direction concerned is constrained to have unit length. +The <I>x</I>-, <I>y-</I> and <I>z-</I>components +can be regarded as the scalar (dot) product of this vector +onto the three axes of the triad in turn. Because the vector +is a unit vector, +each of the three dot-products is simply the cosine of the angle +between the unit vector and the axis concerned, and the +<I>x</I>-, <I>y-</I> and <I>z-</I>components are sometimes +called <I>direction cosines</I>. +<P> +For some applications involving objects +with the Solar System, unit vectors are inappropriate, and +it is necessary to use vectors scaled in length-units such as +AU, km <I>etc.</I> +In these cases the origin of the coordinate system may not be +the observer, but instead might be the Sun, the Solar-System +barycentre, the centre of the Earth <I>etc.</I> But whatever the application, +the final direction in which the observer sees the object can be +expressed as direction cosines. +<P> +But where has this got us? Instead of two numbers - a longitude and +a latitude - we now have three numbers to look after +- the <I>x</I>-, <I>y-</I> and +<I>z-</I>components - whose quadratic sum we have somehow to contrive to +be unity. And, in addition to this apparent redundancy, +most people find it harder to visualize +problems in terms of <IMG WIDTH="58" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" + SRC="img50.gif" + ALT="$[\,x,y,z\,]$"> than in <IMG WIDTH="45" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" + SRC="img238.gif" + ALT="$[\,\theta,\phi~]$">.Despite these objections, the vector approach turns out to have +significant advantages over the spherical trigonometry approach: +<UL> +<LI> Vector formulae tend to be much more succinct; one vector + operation is the equivalent of strings of sines and cosines. +<LI> The formulae are as a rule rigorous, even at the poles. +<LI> Accuracy is maintained all over the celestial sphere. + When one Cartesian component is nearly unity and + therefore insensitive to direction, the others become small + and therefore more precise. +<LI> Formulations usually deliver the quadrant of the result + without the need for any inspection (except within the + library function ATAN2). +</UL> +A number of important transformations in positional +astronomy turn out to be nothing more than changes of coordinate +system, something which is especially convenient if +the vector approach is used. A direction with respect +to one triad can be expressed relative to another triad simply +by multiplying the <IMG WIDTH="58" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" + SRC="img50.gif" + ALT="$[\,x,y,z\,]$"> column vector by the appropriate +<IMG WIDTH="39" HEIGHT="25" ALIGN="MIDDLE" BORDER="0" + SRC="img18.gif" + ALT="$3\times3$"> orthogonal matrix +(a tensor of Rank 2, or <I>dyadic</I>). The three rows of this +<I>rotation matrix</I> +are the vectors in the old coordinate system of the three +new axes, and the transformation amounts to obtaining the +dot-product of the direction-vector with each of the three +new axes. Precession, nutation, <IMG WIDTH="41" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" + SRC="img29.gif" + ALT="$[\,h,\delta\,]$"> to <IMG WIDTH="66" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" + SRC="img28.gif" + ALT="$[\,Az,El~]$">,<IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" + SRC="img3.gif" + ALT="$[\,\alpha,\delta\,]$"> to <IMG WIDTH="59" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" + SRC="img98.gif" + ALT="$[\,l^{I\!I},b^{I\!I}\,]$"> and so on are typical examples of the +technique. A useful property of the rotation matrices +is that they can be inverted simply by taking the transpose. +<P> +The elements of these vectors and matrices are assorted combinations of +the sines and cosines of the various angles involved (hour angle, +declination and so on, depending on which transformation is +being applied). If you write out the matrix multiplications +in full you get expressions which are essentially the same as the +equivalent spherical trigonometry formulae. Indeed, many of the +standard formulae of spherical trigonometry are most easily +derived by expressing the problem initially in +terms of vectors. +<P> +<BR><HR> +<!--Table of Child-Links--> +<A NAME="CHILD_LINKS"> </A> +<UL> +<LI><A NAME="tex2html2455" HREF="node201.html#SECTION00052100000000000000"> +Using vectors</A> +</UL> +<!--End of Table of Child-Links--> +<BR> <HR> +<A NAME="tex2html2453" HREF="node201.html"> +<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next_motif.gif"></A> +<A NAME="tex2html2451" HREF="node197.html"> +<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up_motif.gif"></A> +<A NAME="tex2html2445" HREF="node199.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="tex2html2454" HREF="node201.html">Using vectors</A> +<BR> +<B>Up:</B> <A NAME="tex2html2452" HREF="node197.html">EXPLANATION AND EXAMPLES</A> +<BR> +<B> Previous:</B> <A NAME="tex2html2446" HREF="node199.html">Formatting angles</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> |