/*

-Procedure dvsep_c ( Time derivative of separation angle )

-Abstract
 
   Calculate the time derivative of the separation angle between
   two input states, S1 and S2.
 
-Disclaimer
 
   THIS SOFTWARE AND ANY RELATED MATERIALS WERE CREATED BY THE 
   CALIFORNIA INSTITUTE OF TECHNOLOGY (CALTECH) UNDER A U.S. 
   GOVERNMENT CONTRACT WITH THE NATIONAL AERONAUTICS AND SPACE 
   ADMINISTRATION (NASA). THE SOFTWARE IS TECHNOLOGY AND SOFTWARE 
   PUBLICLY AVAILABLE UNDER U.S. EXPORT LAWS AND IS PROVIDED "AS-IS" 
   TO THE RECIPIENT WITHOUT WARRANTY OF ANY KIND, INCLUDING ANY 
   WARRANTIES OF PERFORMANCE OR MERCHANTABILITY OR FITNESS FOR A 
   PARTICULAR USE OR PURPOSE (AS SET FORTH IN UNITED STATES UCC 
   SECTIONS 2312-2313) OR FOR ANY PURPOSE WHATSOEVER, FOR THE 
   SOFTWARE AND RELATED MATERIALS, HOWEVER USED. 
 
   IN NO EVENT SHALL CALTECH, ITS JET PROPULSION LABORATORY, OR NASA 
   BE LIABLE FOR ANY DAMAGES AND/OR COSTS, INCLUDING, BUT NOT 
   LIMITED TO, INCIDENTAL OR CONSEQUENTIAL DAMAGES OF ANY KIND, 
   INCLUDING ECONOMIC DAMAGE OR INJURY TO PROPERTY AND LOST PROFITS, 
   REGARDLESS OF WHETHER CALTECH, JPL, OR NASA BE ADVISED, HAVE 
   REASON TO KNOW, OR, IN FACT, SHALL KNOW OF THE POSSIBILITY. 
 
   RECIPIENT BEARS ALL RISK RELATING TO QUALITY AND PERFORMANCE OF 
   THE SOFTWARE AND ANY RELATED MATERIALS, AND AGREES TO INDEMNIFY 
   CALTECH AND NASA FOR ALL THIRD-PARTY CLAIMS RESULTING FROM THE 
   ACTIONS OF RECIPIENT IN THE USE OF THE SOFTWARE. 
 
-Required_Reading
 
   None. 
 
-Keywords
 
   GEOMETRY
 
*/

   #include "SpiceUsr.h"
   #include "SpiceZfc.h"
   #undef dvsep_c

   SpiceDouble dvsep_c (ConstSpiceDouble s1[6], ConstSpiceDouble s2[6] ) 

/*

-Brief_I/O
 
   VARIABLE  I/O  DESCRIPTION 
   --------  ---  -------------------------------------------------- 
   s1         I   State vector of the first body 
   s2         I   State vector of the second  body 
 
-Detailed_Input
 
   s1         the state vector of the first target body as seen from 
              the observer 
 
   s2         the state vector of the second target body as seen from 
              the observer 
 
   An implicit assumption exists that both states lie in the same 
   refrence frame with the same observer for the same epoch. If this 
   is not the case, the numerical result has no meaning. 
 
-Detailed_Output
 
   The function returns the double precision value of the time derivative 
   of the angular separation between S1 and S2.
 
-Parameters
 
    None. 
 
-Exceptions

   1) The error SPICE(NUMERICOVERFLOW) signals if the inputs S1, S2
      define states with an angular separation rate ~ DPMAX().

   2) If called in RETURN mode, the return has value 0.

   3) Linear dependent position components of S1 and S1 constitutes
      a non-error exception. The function returns 0 for this case.
 
-Files
 
    None. 
 
-Particulars
 

   In this discussion, the notation

      < V1, V2 >

   indicates the dot product of vectors V1 and V2. The notation

      V1 x V2

   indicates the cross product of vectors V1 and V2.

   To start out, note that we need consider only unit vectors,
   since the angular separation of any two non-zero vectors
   equals the angular separation of the corresponding unit vectors.
   Call these vectors U1 and U2; let their velocities be V1 and V2.

   For unit vectors having angular separation

      THETA

   the identity

      || U1 x U1 || = ||U1|| * ||U2|| * sin(THETA)                (1)

   reduces to

      || U1 x U2 || = sin(THETA)                                  (2)

   and the identity

      | < U1, U2 > | = || U1 || * || U2 || * cos(THETA)           (3)

   reduces to

      | < U1, U2 > | = cos(THETA)                                 (4)

   Since THETA is an angular separation, THETA is in the range

      0 : Pi

   Then letting s be +1 if cos(THETA) > 0 and -1 if cos(THETA) < 0, 
     we have for any value of THETA other than 0 or Pi


                                2          1/2
      cos(THETA) = s * ( 1 - sin (THETA)  )                       (5)

   or  

                                2          1/2
      < U1, U2 > = s * ( 1 - sin (THETA)  )                       (6)


   At this point, for any value of THETA other than 0 or Pi,
   we can differentiate both sides with respect to time (T) 
   to obtain

                                                    2        -1/2
      < U1, V2 > + < V1, U2 > =    s * (1/2)(1 - sin (THETA))

                                 * (-2) sin(THETA)*cos(THETA)

                                 * d(THETA)/dT                   (7a)


   Using equation (5), and noting that s = 1/s, we can cancel 
   the cosine terms on the right hand side

                                                    -1
      < U1, V2 > + < V1, U2 > =    (1/2)(cos(THETA))

                                 * (-2) sin(THETA)*cos(THETA)

                                 * d(THETA)/dT                   (7b)

   With (7b) reducing to

      < U1, V2 > + < V1, U2 > = - sin(THETA) * d(THETA)/dT        (8)

   Using equation (2) and switching sides, we obtain

      || U1 x U2 || * d(THETA)/dT  =  - < U1, V2 > - < V1, U2 >   (9)

   or, provided U1 and U2 are linearly independent,
     
      d(THETA)/dT = ( - < U1, V2 > - < V1, U2 > ) / ||U1 x U2||  (10)

   Note for times when U1 and U2 have angular separation 0 or Pi
   radians, the derivative of angular separation with respect to
   time doesn't exist. (Consider the graph of angular separation
   with respect to time; typically the graph is roughly v-shaped at
   the singular points.)
 
-Examples

   None. 

-Restrictions

   None. 

-Literature_References

   None. 

-Author_and_Institution

   E.D. Wright    (JPL) 

-Version

   -CSPICE Version 1.0.0, 09-MAR-2009 (EDW) (NJB)

-Index_Entries
 
   time derivative of angular separation 
 
-&
*/

{ /* Begin dvsep_c */

   /*
   Local variabes.
   */
   SpiceDouble       retval;

   /*
   Participate in error tracing.
   */
   chkin_c ( "dvsep_c" );

   /*
   Call the f2c'd Fortran routine.
   */
   retval = (SpiceDouble) dvsep_( ( doublereal * ) s1,
                                  ( doublereal * ) s2 );

   chkout_c ( "dvsep_c" );

   return(retval);

} /* End dvsep_c */
