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Index Page dsphdr_c

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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

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

       void dsphdr_c ( SpiceDouble   x,
                       SpiceDouble   y,
                       SpiceDouble   z,
                       SpiceDouble   jacobi[3][3] )

#### Abstract

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

#### Required_Reading

       None.

#### Keywords

       COORDINATES
       DERIVATIVES
       MATRIX

#### Brief_I/O

       VARIABLE  I/O  DESCRIPTION
       --------  ---  --------------------------------------------------
       x          I   x-coordinate of point.
       y          I   y-coordinate of point.
       z          I   z-coordinate of point.
       jacobi     O   Matrix of partial derivatives.

#### Detailed_Input

       x,
       y,
       z           are the rectangular coordinates of the point at
                   which the Jacobian of the map from rectangular
                   to spherical coordinates is desired.

#### Detailed_Output

       jacobi      is the matrix of partial derivatives of the conversion
                   between rectangular and spherical coordinates. It
                   has the form

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

                 evaluated at the input values of x, y, and z.

#### Parameters

       None.

#### Exceptions

       1)  If the input point is on the z-axis (x and y = 0), the Jacobian
           is undefined, the error SPICE(POINTONZAXIS) is signaled by
           a routine in the call tree of this routine.

#### Files

       None.

#### Particulars

       When performing vector calculations with velocities it is
       usually most convenient to work in rectangular coordinates.
       However, once the vector manipulations have been performed
       it is often desirable to convert the rectangular representations
       into spherical coordinates to gain insights about phenomena
       in this coordinate frame.

       To transform rectangular velocities to derivatives of coordinates
       in a spherical system, one uses the Jacobian of the transformation
       between the two systems.

       Given a state in rectangular coordinates

            ( x, y, z, dx, dy, dz )

       the corresponding spherical coordinate derivatives are given by
       the matrix equation:

                            t          |                    t
          (dr, dcolat, dlon)   = jacobi|      * (dx, dy, dz)
                                       |(x,y,z)

       This routine computes the matrix

                |
          jacobi|
                |(x, y, z)

#### 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: dsphdr_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 dsphdr_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 ( "dsphdr_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)

#### Version

       -CSPICE Version 1.0.1, 01-NOV-2021 (JDR)

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

       -CSPICE Version 1.0.0, 20-JUL-2001 (WLT) (NJB)

#### Index_Entries

       Jacobian of spherical w.r.t. rectangular coordinates

#### Link to routine dsphdr_c source file dsphdr_c.c

    Fri Dec 31 18:41:05 2021