This routine computes the Jacobian of the transformation from spherical to rectangular coordinates.
Variable I/O Description -------- --- -------------------------------------------------- r I Distance of a point from the origin. colat I Angle of the point from the positive z-axis. lon I Angle of the point from the xy plane. jacobi O Matrix of partial derivatives.
r Distance of a point from the origin. colat Angle between the point and the positive z-axis, in radians. lon Angle of the point from the xz plane in radians. The angle increases in the counterclockwise sense about the +z axis.
jacobi is the matrix of partial derivatives of the conversion between spherical and rectangular coordinates, evaluated at the input coordinates. This matrix has the form .- -. | dx/dr dx/dcolat dx/dlon | | | | dy/dr dy/dcolat dy/dlon | | | | dz/dr dz/dcolat dz/dlon | `- -' evaluated at the input values of r, lon and lat. Here x, y, and z are given by the familiar formulae x = r*cos(lon)*sin(colat) y = r*sin(lon)*sin(colat) z = r*cos(colat)
It is often convenient to describe the motion of an object in the spherical coordinate system. However, when performing vector computations its hard to beat rectangular coordinates. To transform states given with respect to spherical coordinates to states with respect to rectangular coordinates, one uses the Jacobian of the transformation between the two systems. Given a state in spherical coordinates ( r, colat, lon, dr, dcolat, dlon ) the velocity in rectangular coordinates is given by the matrix equation: t | t (dx, dy, dz) = jacobi| * (dr, dcolat, dlon ) |(r,colat,lon) This routine computes the matrix | jacobi| |(r,colat,lon)
Suppose that one has a model that gives the radius, colatitude and longitude as a function of time (r(t), colat(t), lon(t)), for which the derivatives ( dr/dt, dcolat/dt, dlon/dt ) are computable. To find the velocity of the object in bodyfixed rectangular coordinates, one simply multiplies the Jacobian of the transformation from spherical to rectangular coordinates (evaluated at r(t), colat(t), lon(t) ) by the vector of derivatives of the spherical coordinates. In code this looks like: #include "SpiceUsr.h" . . . /. Load the derivatives of r, colat, and lon into the spherical velocity vector sphv. ./ sphv = dr_dt ( t ); sphv = dcolat_dt ( t ); sphv = dlon_dt ( t ); /. Determine the Jacobian of the transformation from cylindrical to rectangular at the coordinates at the given cylindrical coordinates at time t. ./ drdsph_c ( r(t), colat(t), lon(t), jacobi ); /. Multiply the Jacobian on the left by the spherical velocity to obtain the rectangular velocity recv. ./ mxv_c ( jacobi, sphv, 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. spherical coordinates