Find the separation angle in radians between two double 
   precision, 3-dimensional vectors.  This angle is defined as zero 
   if either vector is zero. 

   None. 

   VARIABLE  I/O  DESCRIPTION 
   --------  ---  -------------------------------------------------- 
    v1        I     First vector. 
    v2        I     Second vector. 

   v1      is an arbitrary double precision, 3-dimensional vector. 
   v2      is also an arbitrary double precision, 3-dimensional 
            vector.  v1 or v2 or both may be the zero vector. 

   vsep_c    is the angle between v1 and v2 expressed in radians. 
            vsep_c is strictly non-negative.  If either v1 or v2 is 
            the zero vector, then vsep_c is defined to be 0 radians. 

   None. 

   In the plane, it is a simple matter to calculate the angle 
   between two vectors once the two vectors have been made to be 
   unit length.  Then, since the two vectors form the two equal 
   sides of an isosceles triangle, the length of the third side 
   is given by the expression 

          length = 2.0 * sin ( vsep /2.0 ) 

   The length is given by the magnitude of the difference of the 
   two unit vectors 

          length = norm_c ( u1 - u2 ) 

   Once the length is found, the value of vsep_c may be calculated 
   by inverting the first expression given above as 

          vsep  = 2.0 * arcsin ( length/2.0 ) 

   This expression becomes increasingly unstable when vsep_c gets 
   larger than PI/2 or 90 degrees.  In this situation (which is 
   easily detected by determining the sign of the dot product of 
   v1 and v2) the supplementary angle is calculated first and 
   then vsep_c is given by 

          vsep  = pi - supplementary_angle 

   The following table gives sample values for v1, v2 and vsep_c 
   implied by the inputs. 

   v1                        v2                      vsep_c 
   ----------------------------------------------------------------- 
   (1, 0, 0)                (1, 0, 0)                0.0 
   (1, 0, 0)                (0, 1, 0)                PI/2 (=1.71...) 

   The user is required to insure that the input vectors will not 
   cause floating point overflow upon calculation of the vector 
   dot product since no error detection or correction code is 
   implemented.  In practice, this is not a significant 
   restriction. 

   Error free. 

   None 

   K.R. Gehringer  (JPL) 
   W.M. Owen       (JPL) 
   W.L. Taber      (JPL) 

   None 

   -CSPICE Version 1.1.0, 22-OCT-1998 (NJB)

      Made input vectors const.

   -CSPICE Version 1.0.0, 08-FEB-1998   (EDW)

   angular separation of 3-dimensional vectors