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Index Page diags2_c

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Procedure Abstract Required_Reading Keywords Brief_I/O Detailed_Input | Detailed_Output Parameters Exceptions Files Particulars Examples | Restrictions Literature_References Author_and_Institution Version Index_Entries

#### Procedure

       diags2_c ( Diagonalize symmetric 2x2 matrix )

       void diags2_c ( ConstSpiceDouble    symmat [2][2],
                       SpiceDouble         diag   [2][2],
                       SpiceDouble         rotate [2][2]  )

#### Abstract

       Diagonalize a symmetric 2x2 matrix.

#### Required_Reading

       ROTATION

#### Keywords

       ELLIPSE
       MATRIX
       ROTATION
       TRANSFORMATION

#### Brief_I/O

       VARIABLE  I/O  DESCRIPTION
       --------  ---  --------------------------------------------------
       symmat     I   A symmetric 2x2 matrix.
       diag       O   A diagonal matrix similar to `symmat'.
       rotate     O   A rotation used as the similarity transformation.

#### Detailed_Input

       symmat      is a symmetric 2x2 matrix. That is, `symmat' has the
                   form

                      .-        -.
                      |  a    b  |
                      |          |
                      |  b    C  |
                      `-        -'

                   This routine uses only the upper-triangular
                   elements of `symmat', that is, the elements

                      symmat[0][0]
                      symmat[0][1]
                      symmat[1][1]

                   to determine the outputs `diag' and `rotate'.

#### Detailed_Output

       diag,
       rotate      are, respectively, a diagonal matrix and a 2x2
                   rotation matrix that satisfy the equation

                                       T
                      diag   =   rotate   *  symmat  *  rotate.

                   In other words, `diag' is similar to `symmat', and `rotate'
                   is a change-of-basis matrix that diagonalizes `symmat'.
                   diags2_c chooses `rotate' so that its angle of rotation has
                   the smallest possible magnitude. If there are two rotations
                   that meet these criteria (they will be inverses of one
                   another), either rotation may be chosen.

#### Parameters

       None.

#### Exceptions

       Error free.

       1)  The matrix element symmat[1][0] is not used in this routine's
           computations, so the condition

              symmat[0][1]  !=  symmat[1][0]

           has no effect on this routine's outputs.

#### Files

       None.

#### Particulars

       The capability of diagonalizing a 2x2 symmetric matrix is
       especially useful in a number of geometric applications
       involving quadratic curves such as ellipses. Such curves are
       described by expressions of the form

             2                    2
          A x   +   B xy   +   C y   +   D x    +    E y   +   F   =   0.

       Diagonalization of the matrix

          .-         -.
          | A     B/2 |
          |           |
          | B/2     C |
          `-         -'

       allows us to perform a coordinate transformation (a rotation,
       specifically) such that the equation of the curve becomes

             2         2
          P u   +   Q v   +   R u    +    S v   +   T   =   0

       in the transformed coordinates. This form is much easier to
       handle. If the quadratic curve in question is an ellipse,
       we can easily find its center, semi-major axis, and semi-minor
       axis from the second equation.

       Ellipses turn up frequently in navigation geometry problems;
       for example, the limb and terminator (if we treat the Sun as a
       point source) of a body modeled as a tri-axial ellipsoid are
       ellipses.

       A mathematical note: because `symmat' is symmetric, we can ALWAYS
       find an orthogonal similarity transformation that diagonalizes
       `symmat', and we can choose the similarity transformation to be a
       rotation matrix. By `orthogonal' we mean that if the `rotate' is
       the matrix in question, then

                T                         T
          rotate  rotate  =  rotate rotate  =  I.

       The reasons this routine handles only the 2x2 case are: first,
       the 2x2 case is much simpler than the general case, in which
       iterative diagonalization methods must be used, and second, the
       2x2 case is adequate for solving problems involving ellipses in
       3 dimensional space. Finally, this routine can be used to
       support a routine that solves the general-dimension diagonalization
       problem for symmetric matrices.

       Another feature of the routine that might provoke curiosity is
       its insistence on choosing the diagonalization matrix that
       rotates the original basis vectors by the smallest amount. The
       rotation angle of `rotate' is of no concern for most applications,
       but can be important if this routine is used as part of an
       iterative diagonalization method for higher-dimensional matrices.
       In that case, it is most undesirable to interchange diagonal
       matrix elements willy-nilly; the matrix to be diagonalized could
       get ever closer to being diagonal without converging. Choosing
       the smallest rotation angle precludes this possibility.

#### Examples

       1)  A case that can be verified by hand computation:
           Suppose symmat is

              .-            -.
              |  1.0    4.0  |
              |              |
              |  4.0   -5.0  |
              `-            -'

           Then symmat is similar to the diagonal matrix

              .-            -.
              |  3.0    0.0  |
              |              |
              |  0.0   -7.0  |
              `-            -'

           so

              diag[0][0] =  3.
              diag[1][0] =  0.
              diag[0][1] =  0.
              diag[1][1] = -7.

           and rotate is

              .-                                             -.
              |  0.89442719099991588    -0.44721359549995794  |
              |                                               |
              |  0.44721359549995794     0.89442719099991588  |
              `-                                             -'

          which is an approximation to

              .-                                   -.
              |  .4 * 5**(1/2)     -.2 * 5**(1/2)   |
              |                                     |
              |  .2 * 5**(1/2)      .4 * 5**(1/2)   |
              `-                                   -'

       2)  Suppose we want to find the semi-axes of the ellipse defined
           by
                  2                 2
              27 x  +  10 xy  +  3 y   =  1

           We can write the above equation as the matrix equation

              .-     -.  .-         -.  .- -.
              | x   y |  | 27     5  |  | x |    =   1
              `-     -'  |           |  |   |
                         |  5     3  |  | y |
                         `-         -'  `- -'

           Let symmat be the symmetric matrix on the left. The code
           fragment

              symmat[0][0]  =  27.0;
              symmat[1][0]  =   5.0;
              symmat[0][1]  =   5.0;
              symmat[1][1]  =   3.0;

              diags2_c ( symmat, diag, rotate );

           will return diag, an array containing the eigenvalues of
           symmat, and rotate, the coordinate transformation required
           to diagonalize symmat. In this case,

              diag[0][0]   =  28.
              diag[1][0]   =   0.
              diag[0][1]   =   0.
              diag[1][1]   =   2.

            and

              rotate[0][0] =   0.980580675690920
              rotate[1][0] =   0.196116135138184
              rotate[0][1] =  -0.196116135138184
              rotate[1][1] =   0.980580675690920

           The columns of rotate give the ellipse's axes, after scaling
           them by

                     1                            1
              ----------------     and     ---------------
                ____________                 ____________
              \/  diag[0][0]               \/  diag[1][1]

           respectively.

           If smajor and sminor are semi-major and semi-minor axes,
           we can find them as shown below. For brevity, we omit the
           check for zero or negative eigenvalues.

              for ( i = 0;  i < 2;  i++ )
              {
                 smajor[i] = rotate[i][0]  /  sqrt( diag[0][0] );
                 sminor[i] = rotate[i][1]  /  sqrt( diag[1][1] );
              }

#### Restrictions

       None.

#### Literature_References

       [1]  T. Apostol, "Calculus, Vol. II," chapter 5, "Eigenvalues of
            Operators Acting on Euclidean Spaces," John Wiley & Sons,
            1969.

#### Author_and_Institution

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

#### Version

       -CSPICE Version 1.0.1, 17-JUN-2021 (JDR)

           Edited the header to comply with NAIF standard.

       -CSPICE Version 1.0.0, 13-JUL-1999 (NJB)

#### Index_Entries

       diagonalize symmetric 2x2_matrix

#### Link to routine diags2_c source file diags2_c.c

    Fri Dec 31 18:41:04 2021