void dgeodr_c ( SpiceDouble x,
SpiceDouble jacobi )
This routine computes the Jacobian of the transformation from
rectangular to geodetic coordinates.
Variable I/O Description
-------- --- --------------------------------------------------
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.
z are the rectangular coordinates of the point at
which the Jacobian of the map from rectangular
to geodetic coordinates is desired.
re Equatorial radius of the reference spheroid.
f Flattening coefficient = (re-rp) / re, where rp is
the polar radius of the spheroid. (More importantly
rp = re*(1-f).)
jacobi is the matrix of partial derivatives of the conversion
between rectangular and geodetic 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.
1) If the input point is on the z-axis (x and y = 0), the
Jacobian is undefined. The error SPICE(POINTONZAXIS)
will be signaled.
2) If the flattening coefficient is greater than or equal to
one, the error SPICE(VALUEOUTOFRANGE) is signaled.
3) If the equatorial radius is not positive, the error
SPICE(BADRADIUS) is signaled.
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 geodetic coordinates to gain insights about phenomena
in this coordinate frame.
To transform rectangular velocities to derivatives of coordinates
in a geodetic 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 geodetic coordinates is given by the matrix
t | t
(dlon, dlat, dalt) = jacobi| * (dx, dy, dz)
This routine computes the matrix
|(x, y, z)
Suppose one is given the bodyfixed rectangular state of an object
(x(t), y(t), z(t), dx(t), dy(t), dz(t)) as a function of time t.
To find the derivatives of the coordinates of the object in
bodyfixed geodetic coordinates, one simply multiplies the
Jacobian of the transformation from rectangular to geodetic
coordinates (evaluated at x(t), y(t), z(t)) by the rectangular
velocity vector of the object at time t.
In code this looks like:
Load the rectangular velocity vector vector recv.
recv = dx_dt ( t );
recv = dy_dt ( t );
recv = dz_dt ( t );
Determine the Jacobian of the transformation from
rectangular to geodetic coordinates at the rectangular
coordinates at time t.
dgeodr_c ( x(t), y(t), z(t), re, f, jacobi );
Multiply the Jacobian on the right by the rectangular
velocity to obtain the geodetic coordinate derivatives
mxv_c ( jacobi, recv, geov );
W.L. Taber (JPL)
N.J. Bachman (JPL)
-CSPICE Version 1.0.0, 18-JUL-2001 (WLT) (NJB)
Jacobian of geodetic w.r.t. rectangular coordinates