| drdgeo_c |
|
Table of contents
Procedure
drdgeo_c ( Derivative of rectangular w.r.t. geodetic )
void drdgeo_c ( SpiceDouble lon,
SpiceDouble lat,
SpiceDouble alt,
SpiceDouble re,
SpiceDouble f,
SpiceDouble jacobi[3][3] )
AbstractCompute the Jacobian matrix of the transformation from geodetic to rectangular coordinates. Required_ReadingNone. KeywordsCOORDINATES DERIVATIVES MATRIX Brief_I/OVARIABLE I/O DESCRIPTION -------- --- -------------------------------------------------- lon I Geodetic longitude of point (radians). lat I Geodetic latitude of point (radians). alt I Altitude of point above the reference spheroid. re I Equatorial radius of the reference spheroid. f I Flattening coefficient. jacobi O Matrix of partial derivatives. Detailed_Input
lon is the geodetic longitude of point (radians).
lat is the geodetic latitude of point (radians).
alt is the altitude of point above the reference spheroid.
re is the equatorial radius of the reference spheroid.
f is the flattening coefficient = (re-rp) / re, where `rp' is
the polar radius of the spheroid. (More importantly
rp = re*(1-f).)
Detailed_Output
jacobi is the matrix of partial derivatives of the conversion
between geodetic and rectangular coordinates. It
has the form
.- -.
| dx/dlon dx/dlat dx/dalt |
| dy/dlon dy/dlat dy/dalt |
| dz/dlon dz/dlat dz/dalt |
`- -'
evaluated at the input values of `lon', `lat' and `alt'.
The formulae for computing `x', `y', and `z' from
geodetic coordinates are given below.
x = [alt + re/g(lat,f)]*cos(lon)*cos(lat)
y = [alt + re/g(lat,f)]*sin(lon)*cos(lat)
2
z = [alt + re*(1-f) /g(lat,f)]* sin(lat)
where
re is the polar radius of the reference spheroid.
f is the flattening factor (the polar radius is
obtained by multiplying the equatorial radius by 1-f).
g( lat, f ) is given by
2 2 2
sqrt ( cos (lat) + (1-f) * sin (lat) )
ParametersNone. Exceptions
1) 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.
2) If the equatorial radius is non-positive, the error
SPICE(BADRADIUS) is signaled by a routine in the call tree of
this routine.
FilesNone. Particulars
It is often convenient to describe the motion of an object in
the geodetic coordinate system. However, when performing
vector computations its hard to beat rectangular coordinates.
To transform states given with respect to geodetic coordinates
to states with respect to rectangular coordinates, one makes use
of the Jacobian of the transformation between the two systems.
Given a state in geodetic coordinates
( lon, lat, alt, dlon, dlat, dalt )
the velocity in rectangular coordinates is given by the matrix
equation:
t | t
(dx, dy, dz) = jacobi| * (dlon, dlat, dalt)
|(lon,lat,alt)
This routine computes the matrix
|
jacobi|
|(lon,lat,alt)
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 geodetic 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: drdgeo_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 drdgeo_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 geovel [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 ( "drdgeo_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 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, < );
/.
Convert position to geodetic coordinates.
./
recgeo_c ( state, re, f, &lon, &lat, &alt );
/.
Convert velocity to geodetic coordinates.
./
dgeodr_c ( state[0], state[1], state[2], re, f, jacobi );
mxv_c ( jacobi, state+3, geovel );
/.
As a check, convert the geodetic state back to
rectangular coordinates.
./
georec_c ( lon, lat, alt, re, f, rectan );
drdgeo_c ( lon, lat, alt, re, f, jacobi );
mxv_c ( jacobi, geovel, 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( "Ellipsoid shape parameters: \n" );
printf( " \n" );
printf( " Equatorial radius (km) = %17.8e\n", re );
printf( " Polar radius (km) = %17.8e\n", rp );
printf( " Flattening coefficient = %17.8e\n", f );
printf( " \n" );
printf( "Geodetic coordinates:\n" );
printf( " \n" );
printf( " Longitude (deg) = %17.8e\n", lon / rpd_c() );
printf( " Latitude (deg) = %17.8e\n", lat / rpd_c() );
printf( " Altitude (km) = %17.8e\n", alt );
printf( " \n" );
printf( "Geodetic velocity:\n" );
printf( " \n" );
printf( " d Longitude/dt (deg/s) = %17.8e\n", geovel[0]/rpd_c() );
printf( " d Latitude/dt (deg/s) = %17.8e\n", geovel[1]/rpd_c() );
printf( " d Altitude/dt (km/s) = %17.8e\n", geovel[2] );
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
Ellipsoid shape parameters:
Equatorial radius (km) = 3.39619000e+03
Polar radius (km) = 3.37620000e+03
Flattening coefficient = 5.88600756e-03
Geodetic coordinates:
Longitude (deg) = 1.03202903e+02
Latitude (deg) = 8.10898757e+00
Altitude (km) = 3.36531823e+08
Geodetic velocity:
d Longitude/dt (deg/s) = -4.05392876e-03
d Latitude/dt (deg/s) = -3.31899337e-06
d Altitude/dt (km/s) = -1.12116015e+01
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
RestrictionsNone. Literature_ReferencesNone. Author_and_InstitutionN.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 -Examples section to comply with NAIF standard.
Added complete code example.
-CSPICE Version 1.0.0, 20-JUL-2001 (WLT) (NJB)
Index_EntriesJacobian of rectangular w.r.t. geodetic coordinates |
Fri Dec 31 18:41:04 2021