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twovxf_c

 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.
```

```   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

'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
```
`Fri Dec 31 18:41:14 2021`