This routine computes the Jacobian of the transformation from rectangular to cylindrical 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.
x, y, z are the rectangular coordinates of the point at which the Jacobian of the map from rectangular to cylindrical coordinates is desired.
jacobi is the matrix of partial derivatives of the conversion between rectangular and cylindrical coordinates. It has the form .- -. | dr /dx dr /dy dr /dz | | dlon/dx dlon/dy dlon/dz | | dz /dx dz /dy dz /dz | `- -' evaluated at the input values of x, y, and z.
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 cylindrical coordinates to gain insights about phenomena in this coordinate frame. To transform rectangular velocities to derivatives of coordinates in a cylindrical 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 cylindrical coordinates is given by the matrix equation: t | t (dr, dlon, dz) = jacobi| * (dx, dy, dz) |(x,y,z) This routine computes the matrix | jacobi| |(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 cylindrical coordinates, one simply multiplies the Jacobian of the transformation from rectangular to cylindrical 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: #include "SpiceUsr.h" . . . /. 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 cylindrical coordinates at the given rectangular coordinates at time T. ./ dcyldr_c ( x(t), y(t), z(t), jacobi ); /. Multiply the Jacobian on the right by the rectangular velocity to obtain the cylindrical coordinate derivatives cylv. ./ mxv_c ( jacobi, recv, cylv );
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.
W.L. Taber (JPL) N.J. Bachman (JPL)
-CSPICE Version 1.0.0, 19-JUL-2001 (WLT) (NJB)
Jacobian of cylindrical w.r.t. rectangular coordinates