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diff --git a/src/slalib/sun67.htx/node201.html b/src/slalib/sun67.htx/node201.html new file mode 100644 index 0000000..4613125 --- /dev/null +++ b/src/slalib/sun67.htx/node201.html @@ -0,0 +1,215 @@ +<!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>Using vectors</TITLE> +<META NAME="description" CONTENT="Using vectors"> +<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="previous" HREF="node200.html"> +<LINK REL="up" HREF="node200.html"> +<LINK REL="next" HREF="node202.html"> +</HEAD> +<BODY > +<BR> <HR> +<A NAME="tex2html2462" HREF="node202.html"> +<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next_motif.gif"></A> +<A NAME="tex2html2460" HREF="node200.html"> +<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up_motif.gif"></A> +<A NAME="tex2html2456" HREF="node200.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="tex2html2463" HREF="node202.html">Celestial Coordinate Systems</A> +<BR> +<B>Up:</B> <A NAME="tex2html2461" HREF="node200.html">Vectors and Matrices</A> +<BR> +<B> Previous:</B> <A NAME="tex2html2457" HREF="node200.html">Vectors and Matrices</A> +<BR> <HR> <P> +<P><!--End of Navigation Panel--> +<H3><A NAME="SECTION00052100000000000000"> +Using vectors</A> +</H3> +SLALIB provides conversions between spherical and vector +form +(sla_CS2C, +sla_CC2S +<I>etc.</I>), plus an assortment +of standard vector and matrix operations +(sla_VDV, +sla_MXV +<I>etc.</I>). +There are also routines +(sla_EULER +<I>etc.</I>) for creating a rotation matrix +from three <I>Euler angles</I> (successive rotations about +specified Cartesian axes). Instead of Euler angles, a rotation +matrix can be expressed as an <I>axial vector</I> (the pole of the rotation, +and the amount of rotation), and routines are provided for this +(sla_AV2M, +sla_M2AV +<I>etc.</I>). +<P> +Here is an example where spherical coordinates <TT>P1</TT> and <TT>Q1</TT> +undergo a coordinate transformation and become <TT>P2</TT> and <TT>Q2</TT>; +the transformation consists of a rotation of the coordinate system +through angles <TT>A</TT>, <TT>B</TT> and <TT>C</TT> about the +<I>z</I>, new <I>y</I> and new <I>z</I> axes respectively: +<P><PRE> + REAL A,B,C,R(3,3),P1,Q1,V1(3),V2(3),P2,Q2 + : + * Create rotation matrix + CALL sla_EULER('ZYZ',A,B,C,R) + + * Transform position (P,Q) from spherical to Cartesian + CALL sla_CS2C(P1,Q1,V1) + + * Multiply by rotation matrix + CALL sla_MXV(R,V1,V2) + + * Back to spherical + CALL sla_CC2S(V2,P2,Q2) +</PRE> +<P> +Small adjustments to the direction of a position +vector are often most conveniently described in terms of +<IMG WIDTH="99" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" + SRC="img239.gif" + ALT="$[\,\Delta x,\Delta y, \Delta z\,]$">. Adding the correction +vector needs careful handling if the position +vector is to remain of length unity, an advisable precaution which +ensures that +the <IMG WIDTH="58" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" + SRC="img50.gif" + ALT="$[\,x,y,z\,]$"> components are always available to mean the cosines of +the angles between the vector and the axis concerned. Two types +of shifts are commonly used, +the first where a small vector of arbitrary direction is +added to the unit vector, and the second where there is a displacement +in the latitude coordinate (declination, elevation <I>etc.</I>) alone. +<P> +For a shift produced by adding a small <IMG WIDTH="58" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" + SRC="img50.gif" + ALT="$[\,x,y,z\,]$"> vector <IMG WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" + SRC="img240.gif" + ALT="${\bf D}$"> to a +unit vector <IMG WIDTH="26" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" + SRC="img241.gif" + ALT="${\bf V1}$">, the resulting vector <IMG WIDTH="26" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" + SRC="img242.gif" + ALT="${\bf V2}$"> has direction +<IMG WIDTH="95" HEIGHT="27" ALIGN="MIDDLE" BORDER="0" + SRC="img243.gif" + ALT="$<{\bf V1}+{\bf D}\gt$"> but is no longer of unit length. A better approximation +is available if the result is multiplied by a scaling factor of +<IMG WIDTH="93" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" + SRC="img244.gif" + ALT="$(1-{\bf D}\cdot{\bf V1})$">, where the dot +means scalar product. In Fortran: +<P><PRE> + F = (1D0-(DX*V1X+DY*V1Y+DZ*V1Z)) + V2X = F*(V1X+DX) + V2Y = F*(V1Y+DY) + V2Z = F*(V1Z+DZ) +</PRE> +<P> +The correction for diurnal aberration (discussed later) is +an example of this form of shift. +<P> +As an example of the second kind of displacement +we will apply a small change in elevation <IMG WIDTH="23" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" + SRC="img245.gif" + ALT="$\delta E$"> to an +<IMG WIDTH="66" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" + SRC="img28.gif" + ALT="$[\,Az,El~]$"> direction vector. The direction of the +result can be obtained by making the allowable approximation +<IMG WIDTH="92" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" + SRC="img246.gif" + ALT="${\tan \delta E\approx\delta E}$"> and adding a adjustment +vector of length <IMG WIDTH="23" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" + SRC="img245.gif" + ALT="$\delta E$"> normal +to the direction vector in the vertical plane containing the direction +vector. The <I>z</I>-component of the adjustment vector is +<IMG WIDTH="64" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" + SRC="img247.gif" + ALT="$\delta E \cos E$">,and the horizontal component is +<IMG WIDTH="62" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" + SRC="img248.gif" + ALT="$\delta E \sin E$"> which has then to be +resolved into <I>x</I> and <I>y</I> in proportion to their current sizes. To +approximate a unit vector more closely, a correction factor of +<IMG WIDTH="48" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" + SRC="img249.gif" + ALT="$\cos \delta E$"> can then be applied, which is nearly +<IMG WIDTH="88" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" + SRC="img250.gif" + ALT="$(1-\delta E^2 /2)$"> for +small <IMG WIDTH="23" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" + SRC="img245.gif" + ALT="$\delta E$">. Expressed in Fortran, for initial vector +<TT>V1X,V1Y,V1Z</TT>, change in elevation <TT>DEL</TT> +(+ve <IMG WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" + SRC="img251.gif" + ALT="$\equiv$"> upwards), and result +vector <TT>V2X,V2Y,V2Z</TT>: +<P><PRE> + COSDEL = 1D0-DEL*DEL/2D0 + R1 = SQRT(V1X*V1X+V1Y*V1Y) + F = COSDEL*(R1-DEL*V1Z)/R1 + V2X = F*V1X + V2Y = F*V1Y + V2Z = COSDEL*(V1Z+DEL*R1) +</PRE> +<P> +An example of this type of shift is the correction for atmospheric +refraction (see later). +Depending on the relationship between <IMG WIDTH="23" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" + SRC="img245.gif" + ALT="$\delta E$"> and <I>E</I>, special +handling at the pole (the zenith for our example) may be required. +<P> +SLALIB includes routines for the case where both a position +and a velocity are involved. The routines +sla_CS2C6 +and +sla_CC62S +convert from <IMG WIDTH="69" HEIGHT="35" ALIGN="MIDDLE" BORDER="0" + SRC="img252.gif" + ALT="$[\theta,\phi,\dot{\theta},\dot{\phi}]$">to <IMG WIDTH="106" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" + SRC="img51.gif" + ALT="$[\,x,y,z,\dot{x},\dot{y},\dot{z}\,]$"> and back; +sla_DCS26 +and +sla_DC62S +are double precision equivalents. +<P> +<BR> <HR> +<A NAME="tex2html2462" HREF="node202.html"> +<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next_motif.gif"></A> +<A NAME="tex2html2460" HREF="node200.html"> +<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up_motif.gif"></A> +<A NAME="tex2html2456" HREF="node200.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="tex2html2463" HREF="node202.html">Celestial Coordinate Systems</A> +<BR> +<B>Up:</B> <A NAME="tex2html2461" HREF="node200.html">Vectors and Matrices</A> +<BR> +<B> Previous:</B> <A NAME="tex2html2457" HREF="node200.html">Vectors and Matrices</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> |