| dpgrdr_c |
|
Table of contents
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] )
AbstractCompute the Jacobian matrix of the transformation from rectangular to planetographic coordinates. Required_ReadingNone. KeywordsCOORDINATES DERIVATIVES MATRIX Brief_I/OVARIABLE 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'.
ParametersNone. 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, < );
/.
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
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, 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_EntriesJacobian of planetographic w.r.t. rectangular coordinates |
Fri Dec 31 18:41:04 2021