void dsphdr_c ( SpiceDouble x,
SpiceDouble jacobi )
This routine computes the Jacobian of the transformation from
rectangular to spherical coordinates.
Variable I/O Description
-------- --- --------------------------------------------------
x I x-coordinate of point.
y I y-coordinate of point.
z I z-coordinate of point.
jacobi O Matrix of partial derivatives.
z are the rectangular coordinates of the point at
which the Jacobian of the map from rectangular
to spherical coordinates is desired.
jacobi is the matrix of partial derivatives of the conversion
between rectangular and spherical coordinates. It
has the form
| dr/dx dr/dy dr/dz |
| dcolat/dx dcolat/dy dcolat/dz |
| dlon/dx dlon/dy dlon/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.
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 spherical coordinates to gain insights about phenomena
in this coordinate frame.
To transform rectangular velocities to derivatives of coordinates
in a spherical 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 corresponding spherical coordinate derivatives are given by
the matrix equation:
t | t
(dr, dcolat, dlon) = 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 spherical coordinates, one simply multiplies the
Jacobian of the transformation from rectangular to spherical
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 ( t );
recv = dy ( t );
recv = dz ( t );
Determine the Jacobian of the transformation from rectangular to
spherical coordinates at the rectangular coordinates at time t.
dsphdr_c ( x(t), y(t), z(t), jacobi );
Multiply the Jacobian on the right by the rectangular
velocity to obtain the spherical coordinate derivatives
mxv_c ( jacobi, recv, sphv );
W.L. Taber (JPL)
N.J. Bachman (JPL)
-CSPICE Version 1.0.0, 20-JUL-2001 (WLT) (NJB)
Jacobian of spherical w.r.t. rectangular coordinates