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drdsph_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

   drdsph_c ( Derivative of rectangular w.r.t. spherical ) 

   void drdsph_c ( SpiceDouble    r,
                   SpiceDouble    colat,
                   SpiceDouble    slon,
                   SpiceDouble    jacobi[3][3] )

Abstract

   Compute the Jacobian matrix of the transformation from spherical
   to rectangular coordinates.

Required_Reading

   None.

Keywords

   COORDINATES
   DERIVATIVES
   MATRIX


Brief_I/O

   VARIABLE  I/O  DESCRIPTION
   --------  ---  --------------------------------------------------
   r          I   Distance of a point from the origin.
   colat      I   Angle of the point from the positive z-axis.
   slon       I   Angle of the point from the xy plane.
   jacobi     O   Matrix of partial derivatives.

Detailed_Input

   r           isn the distance of a point from the origin.

   colat       is the angle between the point and the positive z-axis,
               in radians.

   slon        is the angle of the point from the xz plane in radians.
               The angle increases in the counterclockwise sense
               about the +z axis.

Detailed_Output

   jacobi      is the matrix of partial derivatives of the conversion
               between spherical and rectangular coordinates,
               evaluated at the input coordinates. This matrix has
               the form

                   .-                                  -.
                   |  dx/dr     dx/dcolat     dx/dslon  |
                   |                                    |
                   |  dy/dr     dy/dcolat     dy/dslon  |
                   |                                    |
                   |  dz/dr     dz/dcolat     dz/dslon  |
                   `-                                  -'

             evaluated at the input values of `r', `slon' and `colat'.
             Here `x', `y', and `z' are given by the familiar formulae

                  x = r*cos(slon)*sin(colat)
                  y = r*sin(slon)*sin(colat)
                  z = r*cos(colat)

Parameters

   None.

Exceptions

   Error free.

Files

   None.

Particulars

   It is often convenient to describe the motion of an object in
   the spherical coordinate system. However, when performing
   vector computations its hard to beat rectangular coordinates.

   To transform states given with respect to spherical coordinates
   to states with respect to rectangular coordinates, one uses
   the Jacobian of the transformation between the two systems.

   Given a state in spherical coordinates

      ( r, colat, slon, dr, dcolat, dslon )

   the velocity in rectangular coordinates is given by the matrix
   equation:
                  t          |                                    t
      (dx, dy, dz)   = jacobi|              * (dr, dcolat, dslon )
                             |(r,colat,slon)

   This routine computes the matrix

            |
      jacobi|
            |(r,colat,slon)

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 spherical state of the Earth as seen from
      Mars in the IAU_MARS reference frame at January 1, 2005 TDB.
      Map this state back to rectangular coordinates as a check.

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


         KPL/MK

         File name: drdsph_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
            ---------                     --------
            de421.bsp                     Planetary ephemeris
            pck00010.tpc                  Planet orientation and
                                          radii
            naif0009.tls                  Leapseconds


         \begindata

            KERNELS_TO_LOAD = ( 'de421.bsp',
                                'pck00010.tpc',
                                'naif0009.tls'  )

         \begintext

         End of meta-kernel


      Example code begins here.


      /.
         Program drdsph_ex1
      ./
      #include <stdio.h>
      #include "SpiceUsr.h"

      int main( )
      {

         /.
         Local variables
         ./
         SpiceDouble          colat;
         SpiceDouble          drectn [3];
         SpiceDouble          et;
         SpiceDouble          jacobi [3][3];
         SpiceDouble          lt;
         SpiceDouble          sphvel [3];
         SpiceDouble          rectan [3];
         SpiceDouble          r;
         SpiceDouble          slon;
         SpiceDouble          state  [6];

         /.
         Load SPK, PCK and LSK kernels, use a meta kernel for
         convenience.
         ./
         furnsh_c ( "drdsph_ex1.tm" );

         /.
         Look up the apparent state of earth as seen from Mars
         at January 1, 2005 TDB, relative to the IAU_MARS reference
         frame.
         ./
         str2et_c ( "January 1, 2005 TDB", &et );

         spkezr_c ( "Earth", et, "IAU_MARS", "LT+S", "Mars", state, &lt );

         /.
         Convert position to spherical coordinates.
         ./
         recsph_c ( state, &r, &colat, &slon );

         /.
         Convert velocity to spherical coordinates.
         ./

         dsphdr_c ( state[0], state[1], state[2], jacobi );

         mxv_c ( jacobi, state+3, sphvel );

         /.
         As a check, convert the spherical state back to
         rectangular coordinates.
         ./
         sphrec_c ( r, colat, slon, rectan );

         drdsph_c ( r, colat, slon, jacobi );

         mxv_c ( jacobi, sphvel, drectn );

         printf( " \n" );
         printf( "Rectangular coordinates:\n" );
         printf( " \n" );
         printf( " X (km)                  =  %17.8e\n", state[0] );
         printf( " Y (km)                  =  %17.8e\n", state[1] );
         printf( " Z (km)                  =  %17.8e\n", state[2] );
         printf( " \n" );
         printf( "Rectangular velocity:\n" );
         printf( " \n" );
         printf( " dX/dt (km/s)            =  %17.8e\n", state[3] );
         printf( " dY/dt (km/s)            =  %17.8e\n", state[4] );
         printf( " dZ/dt (km/s)            =  %17.8e\n", state[5] );
         printf( " \n" );
         printf( "Spherical coordinates:\n" );
         printf( " \n" );
         printf( " Radius     (km)         =  %17.8e\n", r );
         printf( " Colatitude (deg)        =  %17.8e\n", colat/rpd_c() );
         printf( " Longitude  (deg)        =  %17.8e\n", slon/rpd_c()  );
         printf( " \n" );
         printf( "Spherical velocity:\n" );
         printf( " \n" );
         printf( " d Radius/dt     (km/s)  =  %17.8e\n", sphvel[0] );
         printf( " d Colatitude/dt (deg/s) =  %17.8e\n",
                                                 sphvel[1]/rpd_c() );
         printf( " d Longitude/dt  (deg/s) =  %17.8e\n",
                                                 sphvel[2]/rpd_c() );
         printf( " \n" );
         printf( "Rectangular coordinates from inverse mapping:\n" );
         printf( " \n" );
         printf( " X (km)                  =  %17.8e\n", rectan[0] );
         printf( " Y (km)                  =  %17.8e\n", rectan[1] );
         printf( " Z (km)                  =  %17.8e\n", rectan[2] );
         printf( " \n" );
         printf( "Rectangular velocity from inverse mapping:\n" );
         printf( " \n" );
         printf( " dX/dt (km/s)            =  %17.8e\n", drectn[0] );
         printf( " dY/dt (km/s)            =  %17.8e\n", drectn[1] );
         printf( " dZ/dt (km/s)            =  %17.8e\n", drectn[2] );
         printf( " \n" );

         return ( 0 );
      }


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


      Rectangular coordinates:

       X (km)                  =    -7.60961826e+07
       Y (km)                  =     3.24363805e+08
       Z (km)                  =     4.74704840e+07

      Rectangular velocity:

       dX/dt (km/s)            =     2.29520749e+04
       dY/dt (km/s)            =     5.37601112e+03
       dZ/dt (km/s)            =    -2.08811490e+01

      Spherical coordinates:

       Radius     (km)         =     3.36535219e+08
       Colatitude (deg)        =     8.18910134e+01
       Longitude  (deg)        =     1.03202903e+02

      Spherical velocity:

       d Radius/dt     (km/s)  =    -1.12116011e+01
       d Colatitude/dt (deg/s) =     3.31899303e-06
       d Longitude/dt  (deg/s) =    -4.05392876e-03

      Rectangular coordinates from inverse mapping:

       X (km)                  =    -7.60961826e+07
       Y (km)                  =     3.24363805e+08
       Z (km)                  =     4.74704840e+07

      Rectangular velocity from inverse mapping:

       dX/dt (km/s)            =     2.29520749e+04
       dY/dt (km/s)            =     5.37601112e+03
       dZ/dt (km/s)            =    -2.08811490e+01

Restrictions

   None.

Literature_References

   None.

Author_and_Institution

   N.J. Bachman        (JPL)
   J. Diaz del Rio     (ODC Space)
   W.L. Taber          (JPL)
   I.M. Underwood      (JPL)

Version

   -CSPICE Version 1.1.0, 01-NOV-2021 (JDR)

       Changed the output argument name "lon" to "slon" for
       consistency with other routines.

       Edited the header to comply with NAIF standard.
       Added complete code example.

   -CSPICE Version 1.0.0, 19-JUL-2001 (WLT) (IMU) (NJB)

Index_Entries

   Jacobian of rectangular w.r.t. spherical coordinates
Fri Dec 31 18:41:04 2021