void drdcyl_c ( SpiceDouble r,
SpiceDouble jacobi )
This routine computes the Jacobian of the transformation from
cylindrical to rectangular coordinates.
Variable I/O Description
-------- --- --------------------------------------------------
r I Distance of a point from the origin.
lon I Angle of the point from the xz plane in radians.
z I Height of the point above the xy plane.
jacobi O Matrix of partial derivatives.
r Distance of the point of interest from z axis.
lon Cylindrical angle (in radians) of the point of
interest from xz plane. The angle increases in the
counterclockwise sense about the +z axis.
z Height of the point above xy plane.
jacobi is the matrix of partial derivatives of the conversion
between cylindrical and rectangular coordinates. It
has the form
| dx/dr dx/dlon dx/dz |
| dy/dr dy/dlon dy/dz |
| dz/dr dz/dlon dz/dz |
evaluated at the input values of r, lon and z.
Here x,y, and z are given by the familiar formulae
x = r*cos(lon)
y = r*sin(lon)
z = z
It is often convenient to describe the motion of an object in
the cylindrical coordinate system. However, when performing
vector computations its hard to beat rectangular coordinates.
To transform states given with respect to cylindrical coordinates
to states with respect to rectangular coordinates, one uses
the Jacobian of the transformation between the two systems.
Given a state in cylindrical coordinates
( r, lon, z, dr, dlon, dz )
the velocity in rectangular coordinates is given by the matrix
t | t
(dx, dy, dz) = jacobi| * (dr, dlon, dz)
This routine computes the matrix
Suppose that one has a model that gives radius, longitude and
height as a function of time (r(t), lon(t), z(t)) for
which the derivatives ( dr/dt, dlon/dt, dz/dt ) are computable.
To find the corresponing velocity in bodyfixed rectangular
coordinates, one simply multiplies the Jacobian of the
transformation from cylindrical to rectangular coordinates
(evaluated at r(t), lon(t), z(t) ) by the vector of derivatives
of the cylindrical coordinates.
In code this looks like:
Load the derivatives of r, lon, and z into the
cylindrical velocity vector sphv.
cylv = dr_dt ( t );
cylv = dlon_dt ( t );
cylv = dz_dt ( t );
Determine the Jacobian of the transformation from
cylindrical to rectangular at the coordinates at the
given cylindrical coordinates at time t.
drdcyl_c ( r(t), lon(t), z(t), jacobi );
Multiply the Jacobian on the left by the cylindrical
velocity to obtain the rectangular velocity recv.
mxv_c ( jacobi, cylv, recv );
W.L. Taber (JPL)
I.M. Underwood (JPL)
N.J. Bachman (JPL)
-CSPICE Version 1.0.0, 19-JUL-2001 (WLT) (IMU) (NJB)
Jacobian of rectangular w.r.t. cylindrical coordinates