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drdazl_c

Table of contents
Procedure
Abstract
Required_Reading
Keywords
Brief_I/O
Detailed_Input
Detailed_Output
Parameters
Exceptions
Files
Particulars
Examples
Restrictions
Literature_References
Author_and_Institution
Version
Index_Entries

Procedure

   drdazl_c ( Derivative of rectangular w.r.t. AZ/EL ) 

   void drdazl_c ( SpiceDouble         range,
                   SpiceDouble         az,
                   SpiceDouble         el,
                   SpiceBoolean        azccw,
                   SpiceBoolean        elplsz,
                   SpiceDouble         jacobi [3][3] )

Abstract

   Compute the Jacobian matrix of the transformation from
   azimuth/elevation to rectangular coordinates.

Required_Reading

   None.

Keywords

   COORDINATES
   DERIVATIVES
   MATRIX


Brief_I/O

   VARIABLE  I/O  DESCRIPTION
   --------  ---  --------------------------------------------------
   range      I   Distance of a point from the origin.
   az         I   Azimuth of input point in radians.
   el         I   Elevation of input point in radians.
   azccw      I   Flag indicating how azimuth is measured.
   elplsz     I   Flag indicating how elevation is measured.
   jacobi     O   Matrix of partial derivatives.

Detailed_Input

   range       is the distance from the origin of the input point
               specified by `range', `az', and `el'.

               Negative values for `range' are not allowed.

               Units are arbitrary and are considered to match those
               of the rectangular coordinate system associated with the
               output matrix `jacobi'.

   az          is the azimuth of the point. This is the angle between
               the projection onto the XY plane of the vector from
               the origin to the point and the +X axis of the
               reference frame. `az' is zero at the +X axis.

               The way azimuth is measured depends on the value of
               the logical flag `azccw'. See the description of the
               argument `azccw' for details.

               The range (i.e., the set of allowed values) of `az' is
               unrestricted. See the -Exceptions section for a
               discussion on the `az' range.

               Units are radians.

   el          is the elevation of the point. This is the angle
               between the vector from the origin to the point and
               the XY plane. `el' is zero at the XY plane.

               The way elevation is measured depends on the value of
               the logical flag `elplsz'. See the description of the
               argument `elplsz' for details.

               The range (i.e., the set of allowed values) of `el' is
               [-pi/2, pi/2], but no error checking is done to ensure
               that `el' is within this range. See the -Exceptions
               section for a discussion on the `el' range.

               Units are radians.

   azccw       is a flag indicating how the azimuth is measured.

               If `azccw' is SPICETRUE, the azimuth increases in the
               counterclockwise direction; otherwise `az' increases
               in the clockwise direction.

   elplsz      if a flag indicating how the elevation is measured.

               If `elplsz' is SPICETRUE, the elevation increases from
               the XY plane toward +Z; otherwise toward -Z.

Detailed_Output

   jacobi      is the matrix of partial derivatives of the
               transformation from azimuth/elevation to rectangular
               coordinates. It has the form

                  .-                                  -.
                  |  dx/drange     dx/daz     dx/del   |
                  |                                    |
                  |  dy/drange     dy/daz     dy/del   |
                  |                                    |
                  |  dz/drange     dz/daz     dz/del   |
                  `-                                  -'

               evaluated at the input values of `range', `az' and `el'.

               `x', `y', and `z' are given by the familiar formulae

                  x = range * cos( az )          * cos( el )
                  y = range * sin( azsnse * az ) * cos( el )
                  z = range * sin( eldir  * el )

               where `azsnse' is +1 when `azccw' is SPICETRUE and -1
               otherwise, and `eldir' is +1 when `elplsz' is SPICETRUE
               and -1 otherwise.

Parameters

   None.

Exceptions

   1)  If the value of the input parameter `range' is negative, the
       error SPICE(VALUEOUTOFRANGE) is signaled by a routine in the
       call tree of this routine.

   2)  If the value of the input argument `el' is outside the
       range [-pi/2, pi/2], the results may not be as
       expected.

   3)  If the value of the input argument `az' is outside the
       range [0, 2*pi], the value will be mapped to a value
       inside the range that differs from the input value by an
       integer multiple of 2*pi.

Files

   None.

Particulars

   It is often convenient to describe the motion of an object
   in azimuth/elevation coordinates. It is also convenient to
   manipulate vectors associated with the object in rectangular
   coordinates.

   The transformation of an azimuth/elevation state into an
   equivalent rectangular state makes use of the Jacobian matrix
   of the transformation between the two systems.

   Given a state in latitudinal coordinates,

      ( r, az, el, dr, daz, del )

   the velocity in rectangular coordinates is given by the matrix
   equation
                  t          |                             t
      (dx, dy, dz)   = jacobi|             * (dr, daz, del)
                             |(r,az,el)

   This routine computes the matrix

            |
      jacobi|
            |(r,az,el)

   In the azimuth/elevation coordinate system, several conventions
   exist on how azimuth and elevation are measured. Using the `azccw'
   and `elplsz' flags, users indicate which conventions shall be used.
   See the descriptions of these input arguments for details.

Examples

   The numerical results shown for this example may differ across
   platforms. The results depend on the SPICE kernels used as
   input, the compiler and supporting libraries, and the machine
   specific arithmetic implementation.

   1) Find the azimuth/elevation state of Venus as seen from the
      DSS-14 station at a given epoch. Map this state back to
      rectangular coordinates as a check.

      Task description
      ================

      In this example, we will obtain the apparent state of Venus as
      seen from the DSS-14 station in the DSS-14 topocentric
      reference frame. We will use a station frames kernel and
      transform the resulting rectangular coordinates to azimuth,
      elevation and range and its derivatives using recazl_c and
      dazldr_c.

      We will map this state back to rectangular coordinates using
      azlrec_c and drdazl_c.

      In order to introduce the usage of the logical flags `azccw'
      and `elplsz', we will request the azimuth to be measured
      clockwise and the elevation positive towards +Z
      axis of the DSS-14_TOPO reference frame.

      Kernels
      =======

      Use the meta-kernel shown below to load the required SPICE
      kernels.


         KPL/MK

         File name: drdazl_ex1.tm

         This meta-kernel is intended to support operation of SPICE
         example programs. The kernels shown here should not be
         assumed to contain adequate or correct versions of data
         required by SPICE-based user applications.

         In order for an application to use this meta-kernel, the
         kernels referenced here must be present in the user's
         current working directory.

         The names and contents of the kernels referenced
         by this meta-kernel are as follows:

            File name                        Contents
            ---------                        --------
            de430.bsp                        Planetary ephemeris
            naif0011.tls                     Leapseconds
            earth_720101_070426.bpc          Earth historical
                                                binary PCK
            earthstns_itrf93_050714.bsp      DSN station SPK
            earth_topo_050714.tf             DSN station FK

         \begindata

         KERNELS_TO_LOAD = ( 'de430.bsp',
                             'naif0011.tls',
                             'earth_720101_070426.bpc',
                             'earthstns_itrf93_050714.bsp',
                             'earth_topo_050714.tf'         )

         \begintext

         End of meta-kernel.


      Example code begins here.


      /.
         Program drdazl_ex1
      ./
      #include <stdio.h>
      #include "SpiceUsr.h"

      int main( )
      {

         /.
         Local parameters
         ./
         #define META         "drdazl_ex1.tm"

         /.
         Local variables
         ./
         SpiceChar          * abcorr;
         SpiceChar          * obs;
         SpiceChar          * obstim;
         SpiceChar          * ref;
         SpiceChar          * target;

         SpiceDouble          az;
         SpiceDouble          azlvel [3];
         SpiceDouble          drectn [3];
         SpiceDouble          el;
         SpiceDouble          et;
         SpiceDouble          jacobi [3][3];
         SpiceDouble          lt;
         SpiceDouble          state  [6];
         SpiceDouble          r;
         SpiceDouble          rectan [3];

         SpiceBoolean         azccw;
         SpiceBoolean         elplsz;

         /.
         Load SPICE kernels.
         ./
         furnsh_c ( META );

         /.
         Convert the observation time to seconds past J2000 TDB.
         ./
         obstim = "2003 OCT 13 06:00:00.000000 UTC";

         str2et_c ( obstim, &et );

         /.
         Set the target, observer, observer frame, and
         aberration corrections.
         ./
         target = "VENUS";
         obs    = "DSS-14";
         ref    = "DSS-14_TOPO";
         abcorr = "CN+S";

         /.
         Compute the observer-target state.
         ./
         spkezr_c ( target, et, ref, abcorr, obs, state, &lt );

         /.
         Convert position to azimuth/elevation coordinates,
         with azimuth increasing clockwise and elevation
         positive towards +Z axis of the DSS-14_TOPO
         reference frame
         ./
         azccw  = SPICEFALSE;
         elplsz = SPICETRUE;

         recazl_c ( state, azccw, elplsz, &r, &az, &el );

         /.
         Convert velocity to azimuth/elevation coordinates.
         ./
         dazldr_c ( state[0], state[1], state[2], azccw, elplsz, jacobi );

         mxv_c ( jacobi, state+3, azlvel );

         /.
         As a check, convert the azimuth/elevation state back to
         rectangular coordinates.
         ./
         azlrec_c ( r, az, el, azccw, elplsz, rectan );

         drdazl_c ( r, az, el, azccw, elplsz, jacobi );

         mxv_c ( jacobi, azlvel, drectn );

         printf( "\n" );
         printf( "AZ/EL coordinates:\n" );
         printf( "\n" );
         printf( "   Range      (km)        =  %19.8f\n", r );
         printf( "   Azimuth    (deg)       =  %19.8f\n", az * dpr_c() );
         printf( "   Elevation  (deg)       =  %19.8f\n", el * dpr_c() );
         printf( "\n" );
         printf( "AZ/EL velocity:\n" );
         printf( "\n" );
         printf( "   d Range/dt     (km/s)  =  %19.8f\n", azlvel[0] );
         printf( "   d Azimuth/dt   (deg/s) =  %19.8f\n",
                                      azlvel[1] * dpr_c() );
         printf( "   d Elevation/dt (deg/s) =  %19.8f\n",
                                      azlvel[2] * dpr_c() );
         printf( "\n" );
         printf( "Rectangular coordinates:\n" );
         printf( "\n" );
         printf( "   X (km)                 =  %19.8f\n", state[0] );
         printf( "   Y (km)                 =  %19.8f\n", state[1] );
         printf( "   Z (km)                 =  %19.8f\n", state[2] );
         printf( "\n" );
         printf( "Rectangular velocity:\n" );
         printf( "\n" );
         printf( "   dX/dt (km/s)           =  %19.8f\n", state[3] );
         printf( "   dY/dt (km/s)           =  %19.8f\n", state[4] );
         printf( "   dZ/dt (km/s)           =  %19.8f\n", state[5] );
         printf( "\n" );
         printf( "Rectangular coordinates from inverse mapping:\n" );
         printf( "\n" );
         printf( "   X (km)                 =  %19.8f\n", rectan[0] );
         printf( "   Y (km)                 =  %19.8f\n", rectan[1] );
         printf( "   Z (km)                 =  %19.8f\n", rectan[2] );
         printf( "\n" );
         printf( "Rectangular velocity from inverse mapping:\n" );
         printf( "\n" );
         printf( "   dX/dt (km/s)           =  %19.8f\n", drectn[0] );
         printf( "   dY/dt (km/s)           =  %19.8f\n", drectn[1] );
         printf( "   dZ/dt (km/s)           =  %19.8f\n", drectn[2] );
         printf( "\n" );

         return ( 0 );
      }


      When this program was executed on a Mac/Intel/cc/64-bit
      platform, the output was:


      AZ/EL coordinates:

         Range      (km)        =   245721478.99272084
         Azimuth    (deg)       =         294.48543372
         Elevation  (deg)       =         -48.94609726

      AZ/EL velocity:

         d Range/dt     (km/s)  =          -4.68189834
         d Azimuth/dt   (deg/s) =           0.00402256
         d Elevation/dt (deg/s) =          -0.00309156

      Rectangular coordinates:

         X (km)                 =    66886767.37916667
         Y (km)                 =   146868551.77222887
         Z (km)                 =  -185296611.10841590

      Rectangular velocity:

         dX/dt (km/s)           =        6166.04150307
         dY/dt (km/s)           =      -13797.77164550
         dZ/dt (km/s)           =       -8704.32385654

      Rectangular coordinates from inverse mapping:

         X (km)                 =    66886767.37916658
         Y (km)                 =   146868551.77222890
         Z (km)                 =  -185296611.10841590

      Rectangular velocity from inverse mapping:

         dX/dt (km/s)           =        6166.04150307
         dY/dt (km/s)           =      -13797.77164550
         dZ/dt (km/s)           =       -8704.32385654

Restrictions

   None.

Literature_References

   None.

Author_and_Institution

   J. Diaz del Rio     (ODC Space)

Version

   -CSPICE Version 1.0.0, 30-DEC-2021 (JDR)

Index_Entries

   Jacobian matrix of rectangular w.r.t. AZ/EL coordinates
   range, azimuth and elevation to rectangular derivative
   Range, AZ and EL to rectangular velocity conversion
Fri Dec 31 18:41:04 2021