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drdlat_c

 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 )

Abstract

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

None.

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

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).

None.

Error free.

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
naif0009.tls                  Leapseconds

\begindata

'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 ;
SpiceDouble          et;
SpiceDouble          jacobi ;
SpiceDouble          lat;
SpiceDouble          lon;
SpiceDouble          lt;
SpiceDouble          latvel ;
SpiceDouble          rectan ;
SpiceDouble          r;
SpiceDouble          state  ;

/.
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, state, state, 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 );
printf( " Y (km)                 =  %17.8e\n", state );
printf( " Z (km)                 =  %17.8e\n", state );
printf( " \n" );
printf( "Rectangular velocity:\n" );
printf( " \n" );
printf( " dX/dt (km/s)           =  %17.8e\n", state );
printf( " dY/dt (km/s)           =  %17.8e\n", state );
printf( " dZ/dt (km/s)           =  %17.8e\n", state );
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 );
printf( " d Longitude/dt (deg/s) =  %17.8e\n", latvel/rpd_c() );
printf( " d Latitude/dt  (deg/s) =  %17.8e\n", latvel/rpd_c() );
printf( " \n" );
printf( "Rectangular coordinates from inverse mapping:\n" );
printf( " \n" );
printf( " X (km)                 =  %17.8e\n", rectan );
printf( " Y (km)                 =  %17.8e\n", rectan );
printf( " Z (km)                 =  %17.8e\n", rectan );
printf( " \n" );
printf( "Rectangular velocity from inverse mapping:\n" );
printf( " \n" );
printf( " dX/dt (km/s)           =  %17.8e\n", drectn );
printf( " dY/dt (km/s)           =  %17.8e\n", drectn );
printf( " dZ/dt (km/s)           =  %17.8e\n", drectn );
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:

Longitude (deg)        =     1.03202903e+02
Latitude  (deg)        =     8.10898662e+00

Latitudinal velocity:

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

None.

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.