Procedure

   isrot_c ( Indicate whether a matrix is a rotation matrix ) 

   SpiceBoolean isrot_c ( ConstSpiceDouble    m   [3][3],
                          SpiceDouble         ntol,
                          SpiceDouble         dtol       )

Abstract

   Indicate whether a 3x3 matrix is a rotation matrix.

Required_Reading

   ROTATION

Keywords

   ERROR
   MATRIX
   ROTATION

Brief_I/O

   VARIABLE  I/O  DESCRIPTION
   --------  ---  --------------------------------------------------
   m          I   A matrix to be tested.
   ntol       I   Tolerance for the norms of the columns of m.
   dtol       I   Tolerance for the determinant of a matrix whose
                  columns are the unitized columns of m.

   The function returns the value SPICETRUE if and only if m is
   a rotation matrix.

Detailed_Input

   m           is a 3x3 matrix to be tested.

   ntol        is the tolerance for the norms of the columns
               of m.

   dtol        is the tolerance for the determinant of a matrix
               whose columns are the unitized columns of m.

Detailed_Output

   The function returns the value SPICETRUE if and only if m is found
   to be a rotation matrix. The criteria that m must meet are:


      1) The norm of each column of m must satisfy the relation

            1. - ntol  <   || column ||   <  1. + ntol.
                       -                  -

      2) The determinant of the matrix whose columns are the
         unitized columns of m must satisfy

            1. - dtol  <   determinant   <  1. + dtol.
                       -                 -

Parameters

   None.

Exceptions

   1)  If either of `ntol' or `dtol' is negative, the error
       SPICE(VALUEOUTOFRANGE) is signaled. isrot_c returns the value
       SPICEFALSE in this case.

Files

   None.

Particulars

   This routine is an error checking "filter"; its purpose is to
   detect gross errors, such as uninitialized matrices. Matrices
   that do not pass the tests used by this routine hardly qualify as
   rotation matrices. The test criteria can be adjusted by varying
   the parameters ntol and dtol.

   A property of rotation matrices is that their columns form a
   right-handed, orthonormal basis in 3-dimensional space. The
   converse is true: all 3x3 matrices with this property are
   rotation matrices.

   An ordered set of three vectors V1, V2, V3 forms a right-handed,
   orthonormal basis if and only if

      1)   || V1 ||  =  || V2 ||  =  || V3 ||  =  1

      2)   V3 = V1 x V2. Since V1, V2, and V3 are unit vectors,
           we also have

           < V3, V1 x V2 > = 1.

           This quantity is the determinant of the matrix whose
           columns are V1, V2 and V3.

   When finite precision numbers are used, rotation matrices will
   usually fail to satisfy these criteria exactly. We must use
   criteria that indicate approximate conformance to the criteria
   listed above. We choose

      1)   |   || Vi ||  -  1   |   <   ntol,  i = 1, 2, 3.
                                    -

      2)   Let

                     Vi
              Ui = ------ ,   i = 1, 2, 3.
                   ||Vi||

           Then we require

              | < U3, U1 x U2 > - 1 |  <  dtol;
                                       -

           equivalently, letting U be the matrix whose columns
           are U1, U2, and U3, we insist on

              | det(U) - 1 |  <  dtol.
                              _

Examples

   1)  We have obtained an instrument pointing matrix C from a
       C-kernel, and we wish to test whether it is in fact a
       rotation matrix. We can use isrot_c to check this:

          #include "SpiceUsr.h"
               .
               .
               .
          /.
          Obtain pointing matrix:
          ./
          ckgp_c ( inst, timein, tol, ref, c, &timout, &found );


          /.
          Verify that c is a rotation:
          ./

          if ( !isrot_c( c ) )
          {

             [ perform exception handling ]

          }
          else
          {

             [ code for the normal case goes here ]

          }

Restrictions

   None.

Literature_References

   None.

Author_and_Institution

   N.J. Bachman        (JPL)
   J. Diaz del Rio     (ODC Space)
   H.A. Neilan         (JPL)

Version

   -CSPICE Version 1.0.1, 03-AUG-2021 (JDR)

       Edited the header to comply with NAIF standard.

   -CSPICE Version 1.0.0, 16-AUG-1999 (NJB) (HAN)

Index_Entries

   indicate whether a matrix is a rotation matrix