Procedure

   q2m_c ( Quaternion to matrix ) 

   void q2m_c ( ConstSpiceDouble  q[4],
                SpiceDouble       r[3][3] )

Abstract

   Find the rotation matrix corresponding to a specified unit
   quaternion.

Required_Reading

   ROTATION

Keywords

   MATH
   MATRIX
   ROTATION

Brief_I/O

   VARIABLE  I/O  DESCRIPTION
   --------  ---  --------------------------------------------------
   q          I   A unit quaternion.
   r          O   A rotation matrix corresponding to `q'.

Detailed_Input

   q           is a unit-length SPICE-style quaternion representing
               a rotation. `q' has the property that

                  || q ||  =  1

               See the discussion of quaternion styles in
               -Particulars below.

Detailed_Output

   r           is a 3 by 3 rotation matrix representing the same
               rotation as does `q'. See the discussion titled
               "Associating SPICE Quaternions with Rotation
               Matrices" in -Particulars below.

Parameters

   None.

Exceptions

   Error free.

   1)  If `q' is not a unit quaternion, the output matrix `r' is
       the rotation matrix that is the result of converting
       normalized `q' to a rotation matrix.

   2)  If `q' is the zero quaternion, the output matrix `r' is
       the identity matrix.

Files

   None.

Particulars

   If a 4-dimensional vector `q' satisfies the equality

      || q ||   =  1

   or equivalently

          2          2          2          2
      q(0)   +   q(1)   +   q(2)   +   q(3)   =  1,

   then we can always find a unit vector `q' and a scalar `theta' such
   that

      q =

      ( cos(theta/2), sin(theta/2)a(1), sin(theta/2)a(2), sin(theta/2)a(3) )

   We can interpret `a' and `theta' as the axis and rotation angle of a
   rotation in 3-space. If we restrict `theta' to the range [0, pi],
   then `theta' and `a' are uniquely determined, except if theta = pi.
   In this special case, `a' and -a are both valid rotation axes.

   Every rotation is represented by a unique orthogonal matrix; this
   routine returns that unique rotation matrix corresponding to `q'.

   The CSPICE routine m2q_c is a one-sided inverse of this routine:
   given any rotation matrix `r', the calls

      m2q_c ( r, q )
      q2m_c ( q, r )

   leave `r' unchanged, except for round-off error. However, the
   calls

      q2m_c ( q, r )
      m2q_c ( r, q )

   might preserve `q' or convert `q' to -q.


   Quaternion Styles
   -----------------

   There are different "styles" of quaternions used in
   science and engineering applications. Quaternion styles
   are characterized by

   -  The order of quaternion elements

   -  The quaternion multiplication formula

   -  The convention for associating quaternions
      with rotation matrices

   Two of the commonly used styles are

      - "SPICE"

         > Invented by Sir William Rowan Hamilton
         > Frequently used in mathematics and physics textbooks

      - "Engineering"

         > Widely used in aerospace engineering applications


   CSPICE function interfaces ALWAYS use SPICE quaternions.
   Quaternions of any other style must be converted to SPICE
   quaternions before they are passed to CSPICE functions.


   Relationship between SPICE and Engineering Quaternions
   ------------------------------------------------------

   Let M be a rotation matrix such that for any vector V,

      M*V

   is the result of rotating V by theta radians in the
   counterclockwise direction about unit rotation axis vector A.
   Then the SPICE quaternions representing M are

      (+/-) (  cos(theta/2),
               sin(theta/2) A(1),
               sin(theta/2) A(2),
               sin(theta/2) A(3)  )

   while the engineering quaternions representing M are

      (+/-) ( -sin(theta/2) A(1),
              -sin(theta/2) A(2),
              -sin(theta/2) A(3),
               cos(theta/2)       )

   For both styles of quaternions, if a quaternion q represents
   a rotation matrix M, then -q represents M as well.

   Given an engineering quaternion

      QENG   = ( q0,  q1,  q2,  q3 )

   the equivalent SPICE quaternion is

      QSPICE = ( q3, -q0, -q1, -q2 )


   Associating SPICE Quaternions with Rotation Matrices
   ----------------------------------------------------

   Let FROM and TO be two right-handed reference frames, for
   example, an inertial frame and a spacecraft-fixed frame. Let the
   symbols

      V    ,   V
       FROM     TO

   denote, respectively, an arbitrary vector expressed relative to
   the FROM and TO frames. Let M denote the transformation matrix
   that transforms vectors from frame FROM to frame TO; then

      V   =  M * V
       TO         FROM

   where the expression on the right hand side represents left
   multiplication of the vector by the matrix.

   Then if the unit-length SPICE quaternion q represents M, where

      q = (q0, q1, q2, q3)

   the elements of M are derived from the elements of q as follows:

        +-                                                         -+
        |           2    2                                          |
        | 1 - 2*( q2 + q3 )   2*(q1*q2 - q0*q3)   2*(q1*q3 + q0*q2) |
        |                                                           |
        |                                                           |
        |                               2    2                      |
    M = | 2*(q1*q2 + q0*q3)   1 - 2*( q1 + q3 )   2*(q2*q3 - q0*q1) |
        |                                                           |
        |                                                           |
        |                                                   2    2  |
        | 2*(q1*q3 - q0*q2)   2*(q2*q3 + q0*q1)   1 - 2*( q1 + q2 ) |
        |                                                           |
        +-                                                         -+

   Note that substituting the elements of -q for those of q in the
   right hand side leaves each element of M unchanged; this shows
   that if a quaternion q represents a matrix M, then so does the
   quaternion -q.

   To map the rotation matrix M to a unit quaternion, we start by
   decomposing the rotation matrix as a sum of symmetric
   and skew-symmetric parts:

                                      2
      M = [ I  +  (1-cos(theta)) OMEGA  ] + [ sin(theta) OMEGA ]

                   symmetric                   skew-symmetric


   OMEGA is a skew-symmetric matrix of the form

                 +-             -+
                 |  0   -n3   n2 |
                 |               |
       OMEGA  =  |  n3   0   -n1 |
                 |               |
                 | -n2   n1   0  |
                 +-             -+

   The vector N of matrix entries (n1, n2, n3) is the rotation axis
   of M and theta is M's rotation angle. Note that N and theta
   are not unique.

   Let

      C = cos(theta/2)
      S = sin(theta/2)

   Then the unit quaternions Q corresponding to M are

      Q = +/- ( C, S*n1, S*n2, S*n3 )

   The mappings between quaternions and the corresponding rotations
   are carried out by the CSPICE routines

      q2m_c {quaternion to matrix}
      m2q_c {matrix to quaternion}

   m2q_c always returns a quaternion with scalar part greater than
   or equal to zero.


   SPICE Quaternion Multiplication Formula
   ---------------------------------------

   Given a SPICE quaternion

      Q = ( q0, q1, q2, q3 )

   corresponding to rotation axis A and angle theta as above, we can
   represent Q using "scalar + vector" notation as follows:

      s =   q0           = cos(theta/2)

      v = ( q1, q2, q3 ) = sin(theta/2) * A

      Q = s + v

   Let Q1 and Q2 be SPICE quaternions with respective scalar
   and vector parts s1, s2 and v1, v2:

      Q1 = s1 + v1
      Q2 = s2 + v2

   We represent the dot product of v1 and v2 by

      <v1, v2>

   and the cross product of v1 and v2 by

      v1 x v2

   Then the SPICE quaternion product is

      Q1*Q2 = s1*s2 - <v1,v2>  + s1*v2 + s2*v1 + (v1 x v2)

   If Q1 and Q2 represent the rotation matrices M1 and M2
   respectively, then the quaternion product

      Q1*Q2

   represents the matrix product

      M1*M2

Examples

   1)  A case amenable to checking by hand calculation:

          To convert the rotation matrix

                   +-              -+
                   |  0     1    0  |
                   |                |
             r  =  | -1     0    0  |
                   |                |
                   |  0     0    1  |
                   +-              -+

          also represented as

             [ pi/2 ]
                     3

          to a quaternion, we can use the code fragment

             rotate_c (  halfpi_c(),  3,  r );
             m2q_c    (  r,               q );

          m2q_c will return `q' as

             ( sqrt(2)/2, 0, 0, -sqrt(2)/2 )

          Why?  Well, `r' is a reference frame transformation that
          rotates vectors by -pi/2 radians about the axis vector

              a = ( 0, 0, 1 )

          Equivalently, `r' rotates vectors by pi/2 radians in
          the counterclockwise sense about the axis vector

             -a = ( 0, 0, -1 )

          so our definition of `q',

             h = theta/2

             q = ( cos(h), sin(h)a , sin(h)a , sin(h)a  )
                                  1         2         3

          implies that in this case,

             q =  ( cos(pi/4),  0,  0,  -sin(pi/4) )

               =  ( sqrt(2)/2,  0,  0,  -sqrt(2)/2 )


   2)  Finding a set of Euler angles that represent a rotation
       specified by a quaternion:

          Suppose our rotation `r' is represented by the quaternion
          `q'. To find angles `tau', `alpha', `delta' such that


             r  =  [ tau ]  [ pi/2 - delta ]  [ alpha ]
                          3                 2          3

          we can use the code fragment


             q2m_c   ( q, r );
             m2eul_c ( r, 3, 2, 3, tau, delta, alpha );

             delta = halfpi_c() - delta;

Restrictions

   None.

Literature_References

   None.

Author_and_Institution

   N.J. Bachman        (JPL)
   J. Diaz del Rio     (ODC Space)
   W.L. Taber          (JPL)
   E.D. Wright         (JPL)

Version

   -CSPICE Version 1.3.3, 10-AUG-2021 (JDR)

       Edited the header to comply with NAIF standard. Moved ROTATIONS required
       reading from -Literature_References to -Required_Reading section.

       Updated entry #1 and added entry #2 to -Exceptions section.

   -CSPICE Version 1.3.2, 27-FEB-2008 (NJB)

       Updated header; added information about SPICE quaternion
       conventions. Made miscellaneous edits throughout header.

   -CSPICE Version 1.3.1, 06-FEB-2003 (EDW)

       Corrected typo error in -Examples section.

   -CSPICE Version 1.3.0, 24-JUL-2001 (NJB)

       Changed prototype: input q is now type (ConstSpiceDouble [4]).
       Implemented interface macro for casting input q to const.

   -CSPICE Version 1.2.0, 08-FEB-1998 (NJB)

       Removed local variables used for temporary capture of outputs.
       Removed tracing calls, since the underlying Fortran routine
       is error-free.

   -CSPICE Version 1.0.0, 25-OCT-1997 (NJB) (WLT)

       Based on SPICELIB Version 1.0.1, 10-MAR-1992 (WLT)

Index_Entries

   quaternion to matrix