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Index Page drdlat_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

       drdlat_c ( Derivative of rectangular w.r.t. latitudinal )

       void drdlat_c ( SpiceDouble   r,
                       SpiceDouble   lon,
                       SpiceDouble   lat,
                       SpiceDouble   jacobi[3][3] )

#### Abstract

       Compute the Jacobian matrix of the transformation from
       latitudinal 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.
       lon        I   Angle of the point from the XZ plane in radians.
       lat        I   Angle of the point from the XY plane in radians.
       jacobi     O   Matrix of partial derivatives.

#### Detailed_Input

       r           is the distance of a point from the origin.

       lon         is the angle of the point from the XZ plane in radians.
                   The angle increases in the counterclockwise sense
                   about the +Z axis.

       lat         is the angle of the point from the XY plane in radians.
                   The angle increases in the direction of the +Z axis.

#### Detailed_Output

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

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

                 evaluated at the input values of r, lon and lat.
                 Here x, y, and z are given by the familiar formulae

                      x = r * cos(lon) * cos(lat)
                      y = r * sin(lon) * cos(lat)
                      z = r *            sin(lat).

#### Parameters

       None.

#### Exceptions

       Error free.

#### Files

       None.

#### Particulars

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

       The transformation of a latitudinal state into an equivalent
       rectangular state makes use of the Jacobian of the
       transformation between the two systems.

       Given a state in latitudinal coordinates,

            ( r, lon, lat, dr, dlon, dlat )

       the velocity in rectangular coordinates is given by the matrix
       equation
                      t          |                               t
          (dx, dy, dz)   = jacobi|             * (dr, dlon, dlat)
                                 |(r,lon,lat)

       This routine computes the matrix

                |
          jacobi|
                |(r,lon,lat)

#### 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 latitudinal 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: drdlat_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 drdlat_ex1
          ./
          #include <stdio.h>
          #include "SpiceUsr.h"

          int main( )
          {

             /.
             Local variables
             ./
             SpiceDouble          drectn [3];
             SpiceDouble          et;
             SpiceDouble          jacobi [3][3];
             SpiceDouble          lat;
             SpiceDouble          lon;
             SpiceDouble          lt;
             SpiceDouble          latvel [3];
             SpiceDouble          rectan [3];
             SpiceDouble          r;
             SpiceDouble          state  [6];

             /.
             Load SPK, PCK and LSK kernels, use a meta kernel for
             convenience.
             ./
             furnsh_c ( "drdlat_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 latitudinal coordinates.
             ./
             reclat_c ( state, &r, &lon, &lat );

             /.
             Convert velocity to latitudinal coordinates.
             ./

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

             mxv_c ( jacobi, state+3, latvel );

             /.
             As a check, convert the latitudinal state back to
             rectangular coordinates.
             ./
             latrec_c ( r, lon, lat, rectan );

             drdlat_c ( r, lon, lat, jacobi );

             mxv_c ( jacobi, latvel, 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( "Latitudinal coordinates:\n" );
             printf( " \n" );
             printf( " Radius    (km)         =  %17.8e\n", r );
             printf( " Longitude (deg)        =  %17.8e\n", lon/rpd_c() );
             printf( " Latitude  (deg)        =  %17.8e\n", lat/rpd_c() );
             printf( " \n" );
             printf( "Latitudinal velocity:\n" );
             printf( " \n" );
             printf( " d Radius/dt    (km/s)  =  %17.8e\n", latvel[0] );
             printf( " d Longitude/dt (deg/s) =  %17.8e\n", latvel[1]/rpd_c() );
             printf( " d Latitude/dt  (deg/s) =  %17.8e\n", latvel[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

          Latitudinal coordinates:

           Radius    (km)         =     3.36535219e+08
           Longitude (deg)        =     1.03202903e+02
           Latitude  (deg)        =     8.10898662e+00

          Latitudinal velocity:

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

          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.

           Updated -Brief_I/O and -Detailed_Input sections to correct `r'
           argument name, which in previous version was `radius'.

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

#### Index_Entries

       Jacobian of rectangular w.r.t. latitudinal coordinates

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

    Fri Dec 31 18:41:04 2021