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Index Page dpgrdr_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

       dpgrdr_c ( Derivative of planetographic w.r.t. rectangular )

       void dpgrdr_c ( ConstSpiceChar  * body,
                       SpiceDouble       x,
                       SpiceDouble       y,
                       SpiceDouble       z,
                       SpiceDouble       re,
                       SpiceDouble       f,
                       SpiceDouble       jacobi[3][3]  )

#### Abstract

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

#### Required_Reading

       None.

#### Keywords

       COORDINATES
       DERIVATIVES
       MATRIX

#### Brief_I/O

       VARIABLE  I/O  DESCRIPTION
       --------  ---  --------------------------------------------------
       body       I   Body with which coordinate system is associated.
       x          I   X-coordinate of point.
       y          I   Y-coordinate of point.
       z          I   Z-coordinate of point.
       re         I   Equatorial radius of the reference spheroid.
       f          I   Flattening coefficient.
       jacobi     O   Matrix of partial derivatives.

#### Detailed_Input

       body        is the name of the body with which the planetographic
                   coordinate system is associated.

                   `body' is used by this routine to look up from the
                   kernel pool the prime meridian rate coefficient giving
                   the body's spin sense. See the -Files and -Particulars
                   header sections below for details.

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

       re          is the equatorial radius of the reference spheroid. This
                   spheroid is a volume of revolution: its horizontal
                   cross sections are circular. The shape of the
                   spheroid is defined by an equatorial radius `re' and
                   a polar radius `rp'.

       f           is the flattening coefficient =

                      (re-rp) / re

                   where `rp' is the polar radius of the spheroid. The
                   units of `rp' match those of `re'. (More importantly
                   rp = re*(1-f) )

#### Detailed_Output

       jacobi      is the matrix of partial derivatives of the conversion
                   from rectangular to planetographic coordinates. It
                   has the form

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

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

#### Parameters

       None.

#### Exceptions

       1)  If the body name `body' cannot be mapped to a NAIF ID code, and
           if `body' is not a string representation of an integer, the
           error SPICE(IDCODENOTFOUND) is signaled by a routine in the
           call tree of this routine.

       2)  If the kernel variable

              BODY<ID code>_PGR_POSITIVE_LON

           is present in the kernel pool but has a value other
           than one of

               'EAST'
               'WEST'

           the error SPICE(INVALIDOPTION) is signaled by a routine in the
           call tree of this routine. Case and blanks are ignored when
           these values are interpreted.

       3)  If polynomial coefficients for the prime meridian of `body' are
           not available in the kernel pool, and if the kernel variable
           BODY<ID code>_PGR_POSITIVE_LON is not present in the kernel
           pool, the error SPICE(MISSINGDATA) is signaled by a routine in
           the call tree of this routine.

       4)  If the equatorial radius is non-positive, the error
           SPICE(VALUEOUTOFRANGE) is signaled by a routine in the call
           tree of this routine.

       5)  If the flattening coefficient is greater than or equal to one,
           the error SPICE(VALUEOUTOFRANGE) is signaled by a routine in
           the call tree of this routine.

       6)  If the input point is on the Z-axis (X = 0 and Y = 0), the
           Jacobian matrix is undefined, an error is signaled by a
           routine in the call tree of this routine.

       7)  If the `body' input string pointer is null, the error
           SPICE(NULLPOINTER) is signaled.

       8)  If the `body' input string has zero length, the error
           SPICE(EMPTYSTRING) is signaled.

#### Files

       This routine expects a kernel variable giving body's prime
       meridian angle as a function of time to be available in the
       kernel pool. Normally this item is provided by loading a PCK
       file. The required kernel variable is named

          BODY<body ID>_PM

       where <body ID> represents a string containing the NAIF integer
       ID code for `body'. For example, if `body' is "JUPITER", then
       the name of the kernel variable containing the prime meridian
       angle coefficients is

          BODY599_PM

       See the PCK Required Reading for details concerning the prime
       meridian kernel variable.

       The optional kernel variable

          BODY<body ID>_PGR_POSITIVE_LON

       also is normally defined via loading a text kernel. When this
       variable is present in the kernel pool, the prime meridian
       coefficients for `body' are not required by this routine. See the
       -Particulars section below for details.

#### 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
       planetographic coordinates to gain insights about phenomena in
       this coordinate frame.

       To transform rectangular velocities to derivatives of coordinates
       in a planetographic 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 velocity in planetographic coordinates is given by the matrix
       equation:
                            t          |                     t
          (dlon, dlat, dalt)   = jacobi|       * (dx, dy, dz)
                                       |(x,y,z)

       This routine computes the matrix

                |
          jacobi|
                |(x, y, z)

       The planetographic definition of latitude is identical to the
       planetodetic (also called "geodetic" in SPICE documentation)
       definition. In the planetographic coordinate system, latitude is
       defined using a reference spheroid. The spheroid is
       characterized by an equatorial radius and a polar radius. For a
       point P on the spheroid, latitude is defined as the angle between
       the X-Y plane and the outward surface normal at P. For a point P
       off the spheroid, latitude is defined as the latitude of the
       nearest point to P on the spheroid. Note if P is an interior
       point, for example, if P is at the center of the spheroid, there
       may not be a unique nearest point to P.

       In the planetographic coordinate system, longitude is defined
       using the spin sense of the body. Longitude is positive to the
       west if the spin is prograde and positive to the east if the spin
       is retrograde. The spin sense is given by the sign of the first
       degree term of the time-dependent polynomial for the body's prime
       meridian Euler angle "W":  the spin is retrograde if this term is
       negative and prograde otherwise. For the sun, planets, most
       natural satellites, and selected asteroids, the polynomial
       expression for W may be found in a SPICE PCK kernel.

       The earth, moon, and sun are exceptions: planetographic longitude
       is measured positive east for these bodies.

       If you wish to override the default sense of positive longitude
       for a particular body, you can do so by defining the kernel
       variable

          BODY<body ID>_PGR_POSITIVE_LON

       where <body ID> represents the NAIF ID code of the body. This
       variable may be assigned either of the values

          'WEST'
          'EAST'

       For example, you can have this routine treat the longitude
       of the earth as increasing to the west using the kernel
       variable assignment

          BODY399_PGR_POSITIVE_LON = 'WEST'

       Normally such assignments are made by placing them in a text
       kernel and loading that kernel via furnsh_c.

       The definition of this kernel variable controls the behavior of
       the CSPICE planetographic routines

          pgrrec_c
          recpgr_c
          dpgrdr_c
          drdpgr_c

       It does not affect the other CSPICE coordinate conversion
       routines.

#### 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 planetographic state of the earth as seen from
          Mars in the J2000 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: dpgrdr_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
                pck00008.tpc                  Planet orientation and
                                              radii
                naif0009.tls                  Leapseconds

             \begindata

                KERNELS_TO_LOAD = ( 'de421.bsp',
                                    'pck00008.tpc',
                                    'naif0009.tls'  )

             \begintext

             End of meta-kernel

          Example code begins here.

          /.
             Program dpgrdr_ex1
          ./

          #include <stdio.h>
          #include "SpiceUsr.h"

          int main()
          {
             /.
             Local variables
             ./
             SpiceDouble             alt;
             SpiceDouble             drectn [3];
             SpiceDouble             et;
             SpiceDouble             f;
             SpiceDouble             jacobi [3][3];
             SpiceDouble             lat;
             SpiceDouble             lon;
             SpiceDouble             lt;
             SpiceDouble             pgrvel [3];
             SpiceDouble             radii  [3];
             SpiceDouble             re;
             SpiceDouble             rectan [3];
             SpiceDouble             rp;
             SpiceDouble             state  [6];

             SpiceInt                n;

             /.
             Load SPK, PCK, and LSK kernels, use a meta kernel for
             convenience.
             ./
             furnsh_c ( "dpgrdr_ex1.tm" );

             /.
             Look up the radii for Mars.  Although we
             omit it here, we could first call badkpv_c
             to make sure the variable BODY499_RADII
             has three elements and numeric data type.
             If the variable is not present in the kernel
             pool, bodvrd_c will signal an error.
             ./
             bodvrd_c ( "MARS", "RADII", 3, &n, radii );

             /.
             Compute flattening coefficient.
             ./
             re  =  radii[0];
             rp  =  radii[2];
             f   =  ( re - rp ) / re;

             /.
             Look up the geometric state of earth as seen from Mars at
             January 1, 2005 TDB, relative to the J2000 reference
             frame.
             ./
             str2et_c ( "January 1, 2005 TDB", &et);

             spkezr_c ( "Earth", et,    "J2000", "LT+S",
                        "Mars",  state, &lt            );

             /.
             Convert position to planetographic coordinates.
             ./
             recpgr_c ( "mars", state, re, f, &lon, &lat, &alt );

             /.
             Convert velocity to planetographic coordinates.
             ./

             dpgrdr_c ( "MARS",  state[0],  state[1],  state[2],
                        re,      f,         jacobi               );

             mxv_c ( jacobi, state+3, pgrvel );

             /.
             As a check, convert the planetographic state back to
             rectangular coordinates.
             ./
             pgrrec_c ( "mars", lon, lat, alt, re, f, rectan );
             drdpgr_c ( "mars", lon, lat, alt, re, f, jacobi );

             mxv_c ( jacobi, pgrvel, drectn );

             printf ( "\n"
                      "Rectangular coordinates:\n"
                      "\n"
                      "  X (km)                 = %18.9e\n"
                      "  Y (km)                 = %18.9e\n"
                      "  Z (km)                 = %18.9e\n"
                      "\n"
                      "Rectangular velocity:\n"
                      "\n"
                      "  dX/dt (km/s)           = %18.9e\n"
                      "  dY/dt (km/s)           = %18.9e\n"
                      "  dZ/dt (km/s)           = %18.9e\n"
                      "\n",
                      state [0],
                      state [1],
                      state [2],
                      state [3],
                      state [4],
                      state [5]                );

             printf ( "Ellipsoid shape parameters:\n"
                      "\n"
                      "  Equatorial radius (km) = %18.9e\n"
                      "  Polar radius      (km) = %18.9e\n"
                      "  Flattening coefficient = %18.9e\n"
                      "\n",
                      re,
                      rp,
                      f                );

             printf ( "Planetographic coordinates:\n"
                      "\n"
                      "  Longitude (deg)        = %18.9e\n"
                      "  Latitude  (deg)        = %18.9e\n"
                      "  Altitude  (km)         = %18.9e\n"
                      "\n"
                      "Planetographic velocity:\n"
                      "\n"
                      "  d Longitude/dt (deg/s) = %18.9e\n"
                      "  d Latitude/dt  (deg/s) = %18.9e\n"
                      "  d Altitude/dt  (km/s)  = %18.9e\n"
                      "\n",
                      lon / rpd_c(),
                      lat / rpd_c(),
                      alt,
                      pgrvel[0]/rpd_c(),
                      pgrvel[1]/rpd_c(),
                      pgrvel[2]                );

             printf ( "Rectangular coordinates from inverse mapping:\n"
                      "\n"
                      "  X (km)                 = %18.9e\n"
                      "  Y (km)                 = %18.9e\n"
                      "  Z (km)                 = %18.9e\n"
                      "\n"
                      "Rectangular velocity from inverse mapping:\n"
                      "\n"
                      "  dX/dt (km/s)           = %18.9e\n"
                      "  dY/dt (km/s)           = %18.9e\n"
                      "  dZ/dt (km/s)           = %18.9e\n"
                      "\n",
                      rectan [0],
                      rectan [1],
                      rectan [2],
                      drectn [0],
                      drectn [1],
                      drectn [2]                );

             return ( 0 );
          }

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

          Rectangular coordinates:

            X (km)                 =    1.460397337e+08
            Y (km)                 =    2.785466054e+08
            Z (km)                 =    1.197503176e+08

          Rectangular velocity:

            dX/dt (km/s)           =   -4.704327200e+01
            dY/dt (km/s)           =    9.073261550e+00
            dZ/dt (km/s)           =    4.757916901e+00

          Ellipsoid shape parameters:

            Equatorial radius (km) =    3.396190000e+03
            Polar radius      (km) =    3.376200000e+03
            Flattening coefficient =    5.886007556e-03

          Planetographic coordinates:

            Longitude (deg)        =    2.976676594e+02
            Latitude  (deg)        =    2.084450444e+01
            Altitude  (km)         =    3.365318255e+08

          Planetographic velocity:

            d Longitude/dt (deg/s) =   -8.357706663e-06
            d Latitude/dt  (deg/s) =    1.593556685e-06
            d Altitude/dt  (km/s)  =   -1.121160078e+01

          Rectangular coordinates from inverse mapping:

            X (km)                 =    1.460397337e+08
            Y (km)                 =    2.785466054e+08
            Z (km)                 =    1.197503176e+08

          Rectangular velocity from inverse mapping:

            dX/dt (km/s)           =   -4.704327200e+01
            dY/dt (km/s)           =    9.073261550e+00
            dZ/dt (km/s)           =    4.757916901e+00

#### 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, 10-AUG-2021 (JDR)

           Edited the header comments to comply with NAIF standard.
           Modified code example to use meta-kernel to load kernels.

           Updated example code to split printf statement in three in order to
           comply with ANSI-C maximum string literal of length.

       -CSPICE Version 1.0.0, 26-DEC-2004 (NJB) (WLT)

#### Index_Entries

       Jacobian of planetographic  w.r.t. rectangular coordinates

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

    Fri Dec 31 18:41:04 2021