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<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.
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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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