Return a unit normal vector and constant that define a specified 
   plane. 

   PLANES 

   Variable  I/O  Description 
   --------  ---  -------------------------------------------------- 
   plane      I   A CSPICE plane. 
   normal, 
   constant   O   A normal vector and constant defining the 
                  geometric plane represented by plane. 

   plane          is a CSPICE plane. 

   normal, 
   constant       are, respectively, a unit normal vector and 
                  constant that define the geometric plane 
                  represented by plane.  Let the symbol < a, b > 
                  indicate the inner product of vectors a and b; 
                  then the geometric plane is the set of vectors x 
                  in three-dimensional space that satisfy 

                     < x,  normal >  =  constant. 

                  normal is a unit vector.  constant is the distance of 
                  the plane from the origin; 

                     constant * normal 

                  is the closest point in the plane to the origin. 

   None. 

   CSPICE geometry routines that deal with planes use the `plane' 
   data type to represent input and output planes.  This data type 
   makes the subroutine interfaces simpler and more uniform. 

   The CSPICE routines that produce CSPICE planes from data that 
   define a plane are: 

      nvc2pl_c ( Normal vector and constant to plane ) 
      nvp2pl_c ( Normal vector and point to plane    ) 
      psv2pl_c ( Point and spanning vectors to plane ) 

   The CSPICE routines that convert CSPICE planes to data that 
   define a plane are: 

      pl2nvc_c ( Plane to normal vector and constant ) 
      pl2nvp_c ( Plane to normal vector and point    ) 
      pl2psv_c ( Plane to point and spanning vectors ) 

   1)  Given a point in a plane and a normal vector, find the distance 
       of the plane from the origin.  We make a `plane' from the point
       and normal, then convert the plane to a unit normal and constant.
       The constant is the distance of the plane from the origin. 

          nvp2pl_c ( normal, point,  &plane    ); 
          pl2nvc_c ( &plane, normal, &constant ); 


   2)  Apply a linear transformation represented by the matrix m to 
       a plane represented by the normal vector n and the constant c. 
       Find a normal vector and constant for the transformed plane. 

          /. 
          Make a CSPICE plane from n and c, and then find a 
          point in the plane and spanning vectors for the 
          plane.  n need not be a unit vector. 
          ./ 
          nvc2pl_c ( n,       c,     &plane         ); 
          pl2psv_c ( &plane,  point,  span1,  span2 );


          /.
          Apply the linear transformation to the point and 
          spanning vectors.  All we need to do is multiply 
          these vectors by m, since for any linear 
          transformation T, 

                T ( point  +  t1 * span1     +  t2 * span2 ) 

             =  T (point)  +  t1 * T(span1)  +  t2 * T(span2), 

          which means that T(point), T(span1), and T(span2) 
          are a point and spanning vectors for the transformed 
          plane. 
          ./

          mxv_c ( m, point, tpoint ); 
          mxv_c ( m, span1, tspan1 ); 
          mxv_c ( m, span2, tspan2 ); 

          /.
          Make a new CSPICE plane tplane from the 
          transformed point and spanning vectors, and find a 
          unit normal and constant for this new plane. 
          ./

          psv2pl_c ( tpoint,   tspan1,  tspan2,   &tplane ); 
          pl2nvc_c ( &tplane,  tn,      &tc               ); 

   None. 

   Error free. 

   1)  The input plane MUST have been created by one of the CSPICE 
       routines 

          nvc2pl_c ( Normal vector and constant to plane ) 
          nvp2pl_c ( Normal vector and point to plane    ) 
          psv2pl_c ( Point and spanning vectors to plane ) 

       Otherwise, the results of this routine are unpredictable. 

   None. 

   N.J. Bachman   (JPL) 

   [1] `Calculus and Analytic Geometry', Thomas and Finney. 

   -CSPICE Version 1.0.1, 06-FEB-2003 (EDW)

       Trivial correction to header docs.

   -CSPICE Version 1.0.0, 05-MAR-1999 (NJB)

   plane to normal vector and constant