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Index Page eul2m_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

       eul2m_c ( Euler angles to matrix )

       void eul2m_c ( SpiceDouble  angle3,
                      SpiceDouble  angle2,
                      SpiceDouble  angle1,
                      SpiceInt     axis3,
                      SpiceInt     axis2,
                      SpiceInt     axis1,
                      SpiceDouble  r [3][3] )

#### Abstract

       Construct a rotation matrix from a set of Euler angles.

#### Required_Reading

       ROTATION

#### Keywords

       MATRIX
       ROTATION
       TRANSFORMATION

#### Brief_I/O

       VARIABLE  I/O  DESCRIPTION
       --------  ---  --------------------------------------------------
       angle3,
       angle2,
       angle1     I   Rotation angles about third, second, and first
                      rotation axes (radians).
       axis3,
       axis2,
       axis1      I   Axis numbers of third, second, and first rotation
                      axes.
       r          O   Product of the 3 rotations.

#### Detailed_Input

       angle3,
       angle2,
       angle1,
       axis3,
       axis2,
       axis1       are, respectively, a set of three angles and three
                   coordinate axis numbers; each pair angleX and axisX
                   specifies a coordinate transformation consisting of a
                   rotation by ANGLEx radians about the coordinate axis
                   indexed by AXISx. These coordinate transformations are
                   typically symbolized by

                      [ angleX ]     .
                                axisX

                   See the -Particulars section below for details concerning
                   this notation.

                   Note that these coordinate transformations rotate vectors
                   by

                      -angleX

                   radians about the axis indexed by axisX.

                   The values of axisX may be 1, 2, or 3, indicating the X,
                   Y, and Z axes respectively.

#### Detailed_Output

       r           is a rotation matrix representing the composition of the
                   rotations defined by the input angle-axis pairs.
                   Together, the three pairs specify a composite
                   transformation that is the result of performing the
                   rotations about the axes indexed by `axis1', `axis2', and
                   `axis3', in that order. So,

                      r = [ angle3 ]      [ angle2 ]      [ angle1 ]
                                    axis3           axis2           axis1

                   See the -Particulars section below for details concerning
                   this notation.

                   The resulting matrix `r' may be thought of as a coordinate
                   transformation; applying it to a vector yields the
                   vector's coordinates in the rotated system.

                   Viewing `r' as a coordinate transformation matrix, the
                   basis that `r' transforms vectors to is created by rotating
                   the original coordinate axes first by `angle1' radians
                   about the coordinate axis indexed by `axis1', next by
                   `angle2' radians about the coordinate axis indexed by
                   `axis2', and finally by `angle3' radians about coordinate
                   axis indexed by `axis3'. At the second and third steps of
                   this process, the coordinate axes about which rotations
                   are performed belong to the bases resulting from the
                   previous rotations.

#### Parameters

       None.

#### Exceptions

       1)  If any of `axis3', `axis2', or `axis1' do not have values in

              { 1, 2, 3 }

           the error SPICE(BADAXISNUMBERS) is signaled by a routine in
           the call tree of this routine.

#### Files

       None.

#### Particulars

       A word about notation: the symbol

          [ x ]
               i

       indicates a rotation of x radians about the ith coordinate axis.
       To be specific, the symbol

          [ x ]
               1

       indicates a coordinate system rotation of x radians about the
       first, or x-, axis; the corresponding matrix is

          .-                    -.
          |  1      0       0    |
          |                      |
          |  0    cos(x)  sin(x) |
          |                      |
          |  0   -sin(x)  cos(x) |
          `-                    -'

       Remember, this is a COORDINATE SYSTEM rotation by x radians; this
       matrix, when applied to a vector, rotates the vector by -x
       radians, not x radians. Applying the matrix to a vector yields
       the vector's representation relative to the rotated coordinate
       system.

       The analogous rotation about the second, or y-, axis is
       represented by

          [ x ]
               2

       which symbolizes the matrix

          .-                    -.
          | cos(x)   0   -sin(x) |
          |                      |
          |  0       1      0    |
          |                      |
          | sin(x)   0    cos(x) |
          `-                    -'

       and the analogous rotation about the third, or z-, axis is
       represented by

          [ x ]
               3

       which symbolizes the matrix

          .-                    -.
          |  cos(x)  sin(x)   0  |
          |                      |
          | -sin(x)  cos(x)   0  |
          |                      |
          |  0        0       1  |
          `-                    -'

       From time to time, (depending on one's line of work, perhaps) one
       may happen upon a pair of coordinate systems related by a
       sequence of rotations. For example, the coordinate system
       defined by an instrument such as a camera is sometime specified
       by RA, DEC, and twist; if alpha, delta, and phi are the rotation
       angles, then the series of rotations

          [ phi ]     [ pi/2  -  delta ]     [ alpha ]
                 3                      2             3

       produces a transformation from inertial to camera coordinates.

       This routine is related to the CSPICE routine m2eul_c, which
       produces a sequence of Euler angles, given a rotation matrix.
       This routine is a "left inverse" of m2eul_c: the sequence of
       calls

          m2eul_c ( r,  axis3,   axis2,   axis1,
                       &angle3, &angle2, &angle1     );

          eul2m_c (     angle3,  angle2,  angle1,
                        axis3,   axis2,   axis1,   r );

       preserves `r', except for round-off error.

       On the other hand, the sequence of calls

          eul2m_c (     angle3,  angle2,  angle1,
                        axis3,   axis2,   axis1,   r );

          m2eul_c ( r,  axis3,   axis2,   axis1,
                       &angle3, &angle2, &angle1     );

       preserve `angle3', `angle2', and `angle1' only if these angles start
       out in the ranges that m2eul_c's outputs are restricted to.

#### Examples

       1)  Create a coordinate transformation matrix by rotating
           the original coordinate axes first by 30 degrees about
           the z axis, next by 60 degrees about the y axis resulting
           from the first rotation, and finally by -50 degrees about
           the z axis resulting from the first two rotations.

                    /.

                    Create the coordinate transformation matrix

                                  o          o          o
                       R  =  [ -50  ]   [  60  ]   [  30  ]
                                     3          2          3

                    All angles in radians, please. The CSPICE
                    function rpd_c (radians per degree) gives the
                    conversion factor.

                    The z axis is `axis 3'; the y axis is `axis 2'.
                    ./

                    angle1 = rpd_c() *  30.;
                    angle2 = rpd_c() *  60.;
                    angle3 = rpd_c() * -50.;

                    axis1  = 3;
                    axis2  = 2;
                    axis3  = 3;

                    eul2m_c (  angle3, angle2, angle1,
                               axis3,  axis2,  axis1,   r  );

       2)  A trivial example using actual numbers.

           The call

              eul2m_c (  0.,     0.,     halfpi_c(),
                         1,        1,            3,      r  );

           sets r equal to the matrix

              +-                  -+
              |  0      1       0  |
              |                    |
              | -1      0       0  |.
              |                    |
              |  0      0       1  |
              +-                  -+

       3)  Finding the rotation matrix specified by a set of `clock,
           cone, and twist' angles, as defined on the Voyager 2 project:

              Voyager 2 narrow angle camera pointing, relative to the
              Sun-Canopus coordinate system, was frequently specified
              by a set of Euler angles called `clock, cone, and twist'.
              These defined a 3-2-3 coordinate transformation matrix
              TSCTV as the product

                 [ twist ]  [ cone ]   [ clock ] .
                          3         2           3

              Given the angles clock, cone, and twist (in units of
              radians), we can compute tsctv with the call

                 eul2m_c (  twist,  cone,  clock,
                            3,      2,     3,      tsctv  );

       4)  Finding the rotation matrix specified by a set of `right
           ascension, declination, and twist' angles, as defined on the
           Galileo project:

              Galileo scan platform pointing, relative to an inertial
              reference frame, (EME50 variety) is frequently specified
              by a set of Euler angles called `right ascension (RA),
              declination (Dec), and twist'. These define a 3-2-3
              coordinate transformation matrix TISP as the product

                 [ Twist ]  [ pi/2 - Dec ]   [ RA ] .
                          3               2        3

              Given the angles ra, dec, and twist (in units of radians),
              we can compute tisp with the code fragment

                 eul2m_c (  twist,   halfpi_c()-dec,   ra,
                            3,       2,                3,   tisp  );

#### Restrictions

       1)  Beware: more than one definition of "RA, DEC and twist"
           exists.

#### Literature_References

       [1]  W. Owen, "Galileo Attitude and Camera Models," JPL
            Interoffice Memorandum 314-323, Nov. 11, 1983. NAIF document
            number 204.0.

#### Author_and_Institution

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

#### Version

       -CSPICE Version 1.0.3, 02-AUG-2021 (JDR)

           Edited the header to comply with NAIF standard.

       -CSPICE Version 1.0.2, 26-DEC-2006 (NJB)

           Fixed header typo.

       -CSPICE Version 1.0.1, 13-OCT-2004 (NJB)

           Fixed header typo.

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

           Based on SPICELIB Version 1.1.1, 10-MAR-1992 (WLT)

#### Index_Entries

       euler angles to matrix

#### Link to routine eul2m_c source file eul2m_c.c

    Fri Dec 31 18:41:06 2021