Table of contents
CSPICE_DPGRDR computes the Jacobian matrix of the transformation
from rectangular to planetographic coordinates.
Given:
body the name of the body with which the planetographic coordinate
system is associated.
[1,c1] = size(body); char = class(body)
or
[1,1] = size(body); cell = class(body)
`body' is used by this routine to look up from the kernel
pool the prime meridian rate coefficient giving the body's
spin sense.
x,
y,
z the rectangular coordinates of the point(s) at which the
Jacobian of the map from rectangular to planetographic
coordinates is desired.
[1,n] = size(x); double = class(x)
[1,n] = size(y); double = class(y)
[1,n] = size(z); double = class(z)
re the equatorial radius of the reference spheroid.
[1,1] = size(re); double = class(re)
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 the flattening coefficient
f = (re-rp) / re
where `rp' is the polar radius of the spheroid.
[1,1] = size(f); double = class(f)
The units of `rp' match those of `re'. (More importantly
rp = re*(1-f) )
the call:
[jacobi] = cspice_dpgrdr( body, x, y, z, re, f )
returns:
jacobi the matrix(es) of partial derivatives of the conversion from
rectangular to planetographic coordinates.
If [1,1] = size(x) then [3,3] = size(jacobi).
If [1,n] = size(x) then [3,3,n] = size(jacobi).
double = class(jacobi)
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'.
`jacobi' returns with the same vectorization measure (N)
as `x', `y' and `z'.
None.
Any numerical results shown for this example may differ between
platforms as the results depend on the SPICE kernels used as input
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
--------- --------
de405.bsp Planetary ephemeris
pck00008.tpc Planet orientation and
radii
naif0007.tls Leapseconds
\begindata
KERNELS_TO_LOAD = ( 'de405.bsp',
'pck00008.tpc',
'naif0007.tls' )
\begintext
End of meta-kernel
Example code begins here.
function dpgrdr_ex1()
%
% Load SPK, PCK, and LSK kernels, use a meta kernel for
% convenience.
%
cspice_furnsh( '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.
%
[radii] = cspice_bodvrd( 'MARS', 'RADII', 3 );
%
% Compute flattening coefficient.
%
re = radii(1);
rp = radii(3);
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.
%
[et] = cspice_str2et( 'January 1, 2005 TDB' );
[state, lt] = cspice_spkezr( 'Earth', et, 'J2000', ...
'LT+S', 'Mars' );
%
% Convert position to planetographic coordinates.
%
[lon, lat, alt] = cspice_recpgr( 'mars', state(1:3), re, f );
%
% Convert velocity to planetographic coordinates.
%
[jacobi] = cspice_dpgrdr( 'MARS', state(1), state(2), ...
state(3), re, f );
pgrvel = jacobi * state(4:6);
%
% As a check, convert the planetographic state back to
% rectangular coordinates.
%
[rectan] = cspice_pgrrec( 'mars', lon, lat, alt, re, f );
[jacobi] = cspice_drdpgr( 'mars', lon, lat, alt, re, f );
drectn = jacobi * pgrvel;
fprintf( '\n' )
fprintf( 'Rectangular coordinates:\n' )
fprintf( '\n' )
fprintf( ' X (km) = %17.8e\n', state (1) )
fprintf( ' Y (km) = %17.8e\n', state (2) )
fprintf( ' Z (km) = %17.8e\n', state (3) )
fprintf( '\n' )
fprintf( 'Rectangular velocity:\n' )
fprintf( '\n' )
fprintf( ' dX/dt (km/s) = %17.8e\n', state (4) )
fprintf( ' dY/dt (km/s) = %17.8e\n', state (5) )
fprintf( ' dZ/dt (km/s) = %17.8e\n', state (6) )
fprintf( '\n' )
fprintf( 'Ellipsoid shape parameters:\n' )
fprintf( '\n' )
fprintf( ' Equatorial radius (km) = %17.8e\n', re )
fprintf( ' Polar radius (km) = %17.8e\n', rp )
fprintf( ' Flattening coefficient = %17.8e\n', f )
fprintf( '\n' )
fprintf( 'Planetographic coordinates:\n' )
fprintf( '\n' )
fprintf( ' Longitude (deg) = %17.8e\n', ...
lon / cspice_rpd )
fprintf( ' Latitude (deg) = %17.8e\n', ...
lat / cspice_rpd )
fprintf( ' Altitude (km) = %17.8e\n', alt )
fprintf( '\n' )
fprintf( 'Planetographic velocity:\n' )
fprintf( '\n' )
fprintf( ' d Longitude/dt (deg/s) = %17.8e\n', ...
pgrvel(1)/cspice_rpd )
fprintf( ' d Latitude/dt (deg/s) = %17.8e\n', ...
pgrvel(2)/cspice_rpd )
fprintf( ' d Altitude/dt (km/s) = %17.8e\n', pgrvel(3) )
fprintf( '\n' )
fprintf( 'Rectangular coordinates from inverse mapping:\n' )
fprintf( '\n' )
fprintf( ' X (km) = %17.8e\n', rectan (1) )
fprintf( ' Y (km) = %17.8e\n', rectan (2) )
fprintf( ' Z (km) = %17.8e\n', rectan (3) )
fprintf( '\n' )
fprintf( 'Rectangular velocity from inverse mapping:\n' )
fprintf( '\n' )
fprintf( ' dX/dt (km/s) = %17.8e\n', drectn (1) )
fprintf( ' dY/dt (km/s) = %17.8e\n', drectn (2) )
fprintf( ' dZ/dt (km/s) = %17.8e\n', drectn (3) )
fprintf( '\n' )
%
% It's always good form to unload kernels after use,
% particularly in Matlab due to data persistence.
%
cspice_kclear
When this program was executed on a Mac/Intel/Octave6.x/64-bit
platform, the output was:
Rectangular coordinates:
X (km) = 1.46039732e+08
Y (km) = 2.78546607e+08
Z (km) = 1.19750315e+08
Rectangular velocity:
dX/dt (km/s) = -4.70432720e+01
dY/dt (km/s) = 9.07326134e+00
dZ/dt (km/s) = 4.75791694e+00
Ellipsoid shape parameters:
Equatorial radius (km) = 3.39619000e+03
Polar radius (km) = 3.37620000e+03
Flattening coefficient = 5.88600756e-03
Planetographic coordinates:
Longitude (deg) = 2.97667659e+02
Latitude (deg) = 2.08445040e+01
Altitude (km) = 3.36531825e+08
Planetographic velocity:
d Longitude/dt (deg/s) = -8.35770664e-06
d Latitude/dt (deg/s) = 1.59355667e-06
d Altitude/dt (km/s) = -1.12116008e+01
Rectangular coordinates from inverse mapping:
X (km) = 1.46039732e+08
Y (km) = 2.78546607e+08
Z (km) = 1.19750315e+08
Rectangular velocity from inverse mapping:
dX/dt (km/s) = -4.70432720e+01
dY/dt (km/s) = 9.07326134e+00
dZ/dt (km/s) = 4.75791694e+00
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 cspice_furnsh.
The definition of this kernel variable controls the behavior of
the CSPICE planetographic routines
cspice_pgrrec
cspice_recpgr
cspice_dpgrdr
cspice_drdpgr
It does not affect the other CSPICE coordinate conversion
routines.
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 any of the input arguments, `body', `x', `y', `z', `re' or
`f', is undefined, an error is signaled by the Matlab error
handling system.
8) If any of the input arguments, `body', `x', `y', `z', `re' or
`f', is not of the expected type, or it does not have the
expected dimensions and size, an error is signaled by the Mice
interface.
9) If the input vectorizable arguments `x', `y' and `z' do not
have the same measure of vectorization (N), an error is
signaled by the Mice interface.
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 for details.
None.
MICE.REQ
None.
J. Diaz del Rio (ODC Space)
S.C. Krening (JPL)
E.D. Wright (JPL)
-Mice Version 1.1.0, 23-AUG-2021 (EDW) (JDR)
Edited the header to comply with NAIF standard. Added complete
example to the -Examples section.
Added -Parameters, -Exceptions, -Files, -Restrictions,
-Literature_References and -Author_and_Institution sections.
Eliminated use of "lasterror" in rethrow.
Removed reference to the function's corresponding CSPICE header from
-Required_Reading section.
-Mice Version 1.0.0, 11-NOV-2013 (EDW) (SCK)
Jacobian of planetographic w.r.t. rectangular coordinates
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