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Index Page twovxf_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
       twovxf_c ( Two states defining a frame transformation )

       void twovxf_c ( ConstSpiceDouble    axdef  [6],
                       SpiceInt            indexa,
                       ConstSpiceDouble    plndef [6],
                       SpiceInt            indexp,
                       SpiceDouble         xform  [6][6] )
#### Abstract
       Find the state transformation from a base frame to the
       right-handed frame defined by two state vectors: one state
       vector defining a specified axis and a second state vector
       defining a specified coordinate plane.
#### Required_Reading
       None.
#### Keywords
       AXES
       FRAMES
       MATRIX
       TRANSFORMATION
#### Brief_I/O
       VARIABLE  I/O  DESCRIPTION
       --------  ---  --------------------------------------------------
       axdef      I   State defining a principal axis.
       indexa     I   Principal axis number of `axdef' (x=1, y=2, z=3).
       plndef     I   State defining (with `axdef') a principal plane.
       indexp     I   Second axis number (with `indexa') of principal
                      plane.
       xform      O   Output state transformation matrix.
#### Detailed_Input
       axdef       is a "generalized" state vector defining one of the
                   principal axes of a reference frame. This vector
                   consists of three components of a vector-valued
                   function of one independent variable t followed by
                   the derivatives of the components with respect to that
                   variable:

                      ( a, b, c, da/dt, db/dt, dc/dt )

                   This routine treats the input states as unitless, but
                   in most applications the input states represent
                   quantities that have associated units. The first three
                   components must have the same units, and the units of
                   the last three components must be compatible with
                   those of the first three: if the first three
                   components of `axdef'

                      ( a, b, c )

                   have units U and t has units T, then the units of
                   `axdef' normally would be

                      ( U, U, U, U/T, U/T, U/T )

                   Note that the direction and angular velocity defined
                   by `axdef' are actually independent of U, so scaling
                   `axdef' doesn't affect the output of this routine.

                   `axdef' could represent position and velocity; it could
                   also represent velocity and acceleration. `axdef' could
                   for example represent the velocity and acceleration of
                   a time-dependent position vector ( x(t), y(t), z(t) ),
                   in which case `axdef' would be defined by

                      a     = dx/dt
                      b     = dy/dt
                      c     = dz/dt

                               2      2
                      da/dt = d x / dt

                               2      2
                      db/dt = d y / dt

                               2      2
                      dc/dt = d z / dt

                   Below, we'll call the normalized (unit length) version
                   of

                      ( a, b, c )

                   the "direction" of `axdef'.

                   We call the frame relative to which `axdef' is specified
                   the "base frame." The input state `plndef' must be
                   specified relative to the same base frame.

       indexa      is the index of the reference frame axis that is
                   parallel to the direction of `axdef'.

                      indexa   Axis
                      ------   ----
                         1       x
                         2       y
                         3       z

       plndef      is a state vector defining (with `axdef') a principal
                   plane of the reference frame. This vector consists
                   of three components followed by their derivatives with
                   respect to the independent variable `t' associated with
                   `axdef', so `plndef' is

                      ( e, f, g, de/dt, df/dt, dg/dt )

                   Below, we'll call the unitized version of

                      ( e, f, g )

                   the "direction" of `plndef'.

                   The second axis of the principal plane containing the
                   direction vectors of `axdef' and `plndef' is perpendicular
                   to the first axis and has positive dot product with
                   the direction vector of `plndef'.

                   The first three components of `plndef' must have the
                   same units, and the units of the last three components
                   must be compatible with those of the first three: if
                   the first three components of `plndef'

                      ( e, f, g )

                   have units U2 and `t' has units T, then the units of
                   `plndef' normally would be

                      ( U2, U2, U2, U2/T, U2/T, U2/T )

                   Note that ***for meaningful results, the angular
                   velocities defined by `axdef' and `plndef' must both have
                   units of 1/T.***

                   As with `axdef', scaling `plndef' doesn't affect the
                   output of this routine.

                   `axdef' and `plndef' must be specified relative to a
                   common reference frame, which we call the "base
                   frame."

       indexp      is the index of  second axis of the principal frame
                   determined by `axdef' and `plndef'. The association of
                   integer values and axes is the same as for `indexa'.
#### Detailed_Output
       xform       is the 6x6 matrix that transforms states from the
                   frame relative to which `axdef' and `plndef' are specified
                   (the "base frame") to the frame whose axes and
                   derivative are determined by `axdef', `plndef', `indexa' and
                   `indexp'.

                   The matrix `xform' has the structure shown below:

                      .-              -.
                      |        :       |
                      |    r   :   0   |
                      |        :       |
                      | .......:.......|
                      |        :       |
                      |  dr/dt :   r   |
                      |        :       |
                      `-              -'

                   where `r' is a rotation matrix that is a function of
                   the independent variable associated with `axdef' and
                   `plndef', and where dr/dt is the derivative of `r'
                   with respect to that independent variable.
#### Parameters
       None.
#### Exceptions
       1)  If `indexa' or `indexp' is not in the set {1,2,3}, the error
           SPICE(BADINDEX) is signaled by a routine in the call tree of
           this routine.

       2)  If `indexa' and `indexp' are the same, the error
           SPICE(UNDEFINEDFRAME) is signaled by a routine in the call
           tree of this routine.

       3)  If the cross product of the vectors `axdef' and `plndef' is zero,
           the error SPICE(DEPENDENTVECTORS) is signaled by a routine in
           the call tree of this routine.
#### Files
       None.
#### Particulars
       Given two linearly independent state vectors `axdef' and `plndef',
       define vectors `dir1' and `dir2' by

          dir1 = ( axdef[0],   axdef[1],   axdef[2]  )
          dir2 = ( plndef[0],  plndef[1],  plndef[2] )

       Then there is a unique right-handed reference frame `f' having:

          `dir1' lying along the `indexa' axis.

          `dir2' lying in the indexa-indexp coordinate plane, such that
          the dot product of `dir2' with the positive `indexp' axis is
          positive.

       This routine determines the 6x6 matrix that transforms states
       from the base frame used to represent the input vectors to the
       the frame `f' determined by `axdef' and `plndef'. Thus a state vector

          s       = ( x, y, z, dx/dt, dy/dt, dz/dt )
           base

       in the input reference frame will be transformed to

          s       = xform * s
           f                 base

       in the frame `f' determined by `axdef' and `plndef'.
#### Examples
       The numerical results shown for this example may differ across
       platforms. The results depend on the SPICE kernels used as input,
       the compiler and supporting libraries, and the machine specific
       arithmetic implementation.

       1) The time-dependent Sun-Canopus reference frame associated with
          a spacecraft uses the spacecraft-sun state to define the Z axis
          and the Canopus direction to define the X-Z plane.

          Find the geometric position of the Earth as seen from the Mars
          Reconnaissance Orbiter spacecraft (MRO) at a specified time,
          relative to the Sun-Canopus reference frame associated with
          MRO.

          Use the meta-kernel shown below to load the required SPICE
          kernels.

             KPL/MK

             File: twovxf_ex1.tm

             This meta-kernel is intended to support operation of SPICE
             example programs. The kernels shown here should not be
             assumed to contain adequate or correct versions of data
             required by SPICE-based user applications.

             In order for an application to use this meta-kernel, the
             kernels referenced here must be present in the user's
             current working directory.

             The names and contents of the kernels referenced
             by this meta-kernel are as follows:

                File name                        Contents
                ---------                        --------
                naif0012.tls                     Leapseconds
                de430.bsp                        Planetary ephemeris
                mro_psp4_ssd_mro95a.bsp          MRO ephemeris

             \begindata

                KERNELS_TO_LOAD = ( 'naif0012.tls',
                                    'de430.bsp',
                                    'mro_psp4_ssd_mro95a.bsp' )

             \begintext

             End of meta-kernel

          Example code begins here.

          /.
             Program twovxf_ex1
          ./
          #include <stdio.h>
          #include "SpiceUsr.h"

          int main( )
          {

             /.
             Local parameters
             ./
             #define META         "twovxf_ex1.tm"

             /.
             Define the Right Ascension and Declination, and the
             proper motion in both coordinates, of Canopus, relative
             to the J2000 frame at J2000 epoch, in degrees and
             arcsecond/yr respectively. Note that the values used here
             may not be suitable for real applications.
             ./
             #define RAJ2K        90.3991968556
             #define DECJ2K       -52.6956610556
             #define PMRA         19.93e-3
             #define PMDEC        23.24e-3

             /.
             Local variables
             ./
             SpiceDouble          dec;
             SpiceDouble          et;
             SpiceDouble          lt;
             SpiceDouble          pcano  [3];
             SpiceDouble          ra;
             SpiceDouble          rpmra;
             SpiceDouble          rpmdec;
             SpiceDouble          state  [6];
             SpiceDouble          stcano [6];
             SpiceDouble          sterth [6];
             SpiceDouble          stsun  [6];
             SpiceDouble          xfisc  [6][6];

             /.
             Load kernel files via the meta-kernel.
             ./
             furnsh_c ( META );

             /.
             Convert the TDB input time string to seconds past
             J2000, TDB.
             ./
             str2et_c ( "2007 SEP 30 00:00:00 TDB", &et );

             /.
             Define an approximate "state vector" for Canopus using
             the J2000-relative, unit direction vector toward Canopus
             at a specified time `et' (time is needed to compute proper
             motion) as position and the zero vector as velocity.
             ./
             convrt_c ( PMRA,  "ARCSECONDS", "RADIANS", &rpmra  );
             convrt_c ( PMDEC, "ARCSECONDS", "RADIANS", &rpmdec );

             ra  = RAJ2K  * rpd_c() + rpmra  * et/jyear_c();
             dec = DECJ2K * rpd_c() + rpmdec * et/jyear_c();

             radrec_c ( 1.0, ra, dec, pcano );

             /.
             Compute MRO geometric velocity w.r.t. the Solar System
             Barycenter, and use it to correct the Canopus direction
             for stellar aberration.
             ./
             spkezr_c ( "MRO", et, "J2000", "NONE", "SSB", state, &lt );

             stelab_c ( pcano, state+3, stcano );

             vpack_c ( 0.0, 0.0, 0.0, stcano+3 );

             /.
             Let `stsun' be the J2000-relative apparent state of the Sun
             relative to the spacecraft at `et'.
             ./
             spkezr_c ( "SUN", et, "J2000", "CN+S", "MRO", stsun, &lt );

             /.
             The matrix `xfisc' transforms states from J2000 frame
             to the Sun-Canopus reference frame at `et'.
             ./
             twovxf_c ( stsun, 3, stcano, 1, xfisc );

             /.
             Compute the apparent state of the Earth as seen from MRO
             in the J2000 frame at `et' and transform that vector into
             the Sun-Canopus reference frame.
             ./
             spkezr_c ( "EARTH", et, "J2000", "CN+S", "MRO", state, &lt );

             mxvg_c ( xfisc, state, 6, 6, sterth );

             /.
             Display the results.
             ./
             printf( "Earth as seen from MRO in Sun-Canopus frame (km and "
                     "km/s):\n" );
             printf( "   position: %15.3f %15.3f %15.3f\n",
                            sterth[0], sterth[1], sterth[2] );
             printf( "   velocity: %15.3f %15.3f %15.3f\n",
                            sterth[3], sterth[4], sterth[5] );

             return ( 0 );
          }

          When this program was executed on a Mac/Intel/cc/64-bit
          platform, the output was:

          Earth as seen from MRO in Sun-Canopus frame (km and km/s):
             position:   -16659764.322    97343706.915   106745539.738
             velocity:           2.691         -10.345          -7.877
#### Restrictions
       None.
#### Literature_References
       None.
#### Author_and_Institution
       J. Diaz del Rio     (ODC Space)
#### Version
       -CSPICE Version 1.0.0, 05-AUG-2021 (JDR)
#### Index_Entries
       define a state transformation matrix from two states
#### Link to routine twovxf_c source file twovxf_c.c

    Fri Dec 31 18:41:14 2021