Procedure

   void dsphdr_c ( SpiceDouble   x,
                   SpiceDouble   y,
                   SpiceDouble   z,
                   SpiceDouble   jacobi[3][3] ) 

Abstract

 
   This routine computes the Jacobian of the transformation from 
   rectangular to spherical coordinates. 
 

Required_Reading

 
   None. 
 

Keywords

 
   COORDINATES 
   DERIVATIVES 
   MATRIX 
 

Brief_I/O

 
   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. 
 

Detailed_Input

 
   x, 
   y, 
   z          are the rectangular coordinates of the point at 
              which the Jacobian of the map from rectangular 
              to spherical coordinates is desired. 
 

Detailed_Output

 
   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. 
 

Parameters

 
   None. 
 

Exceptions

 
   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. 
 

Files

 
   None. 
 

Particulars

 
   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) 
                                   |(x,y,z) 
 
   This routine computes the matrix  
 
            | 
      jacobi| 
            |(x, y, z) 
 

Examples

 
   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: 
 

      #include "SpiceUsr.h"
            .
            .
            .
      /.
      Load the rectangular velocity vector vector recv. 
      ./ 
      recv[0] = dx ( t );
      recv[1] = dy ( t );
      recv[2] = 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  
      sphv. 
      ./ 
      mxv_c ( jacobi, recv, sphv );
 

Restrictions

 
   None. 
 

Literature_References

 
   None. 
 

Author_and_Institution

 
   W.L. Taber     (JPL) 
   N.J. Bachman   (JPL)

Version

 
   -CSPICE Version 1.0.0, 20-JUL-2001 (WLT) (NJB)

Index_Entries

 
   Jacobian of spherical w.r.t. rectangular coordinates 
 

Link to routine dsphdr_c source file dsphdr_c.c