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m2eul_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

```   m2eul_c ( Matrix to Euler angles )

void  m2eul_c ( ConstSpiceDouble    r[3][3],
SpiceInt            axis3,
SpiceInt            axis2,
SpiceInt            axis1,
SpiceDouble       * angle3,
SpiceDouble       * angle2,
SpiceDouble       * angle1  )

```

#### Abstract

```   Factor a rotation matrix as a product of three rotations about
specified coordinate axes.
```

```   ROTATION
```

#### Keywords

```   ANGLE
MATRIX
ROTATION
TRANSFORMATION

```

#### Brief_I/O

```   VARIABLE  I/O  DESCRIPTION
--------  ---  --------------------------------------------------
r          I   A rotation matrix to be factored.
axis3,
axis2,
axis1      I   Numbers of third, second, and first rotation axes.
angle3,
angle2,
angle1     O   Third, second, and first Euler angles, in radians.
```

#### Detailed_Input

```   r           is a 3x3 rotation matrix that is to be factored as
a product of three rotations about a specified
coordinate axes. The angles of these rotations are
called "Euler angles."

axis3,
axis2,
axis1       are the indices of the rotation axes of the
"factor" rotations, whose product is `r'. `r' is
factored as

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

The axis numbers must belong to the set {1, 2, 3}.
The second axis number MUST differ from the first
and third axis numbers.

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

#### Detailed_Output

```   angle3,
angle2,
angle1      are the Euler angles corresponding to the matrix
`r' and the axes specified by `axis3', `axis2', and
`axis1'. These angles satisfy the equality

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

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

The range of `angle3' and `angle1' is (-pi, pi].

The range of `angle2' depends on the exact set of
axes used for the factorization. For
factorizations in which the first and third axes
are the same,

r = [R]  [S]  [T] ,
a    b    a

the range of `angle2' is [0, pi].

For factorizations in which the first and third
axes are different,

r = [R]  [S]  [T] ,
a    b    c

the range of `angle2' is [-pi/2, pi/2].

For rotations such that `angle3' and `angle1' are not
uniquely determined, `angle3' will always be set to
zero; `angle1' is then uniquely determined.
```

#### 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.

2)  If `axis2' is equal to `axis3' or `axis1', the error
SPICE(BADAXISNUMBERS) is signaled by a routine in the call
tree of this routine. An arbitrary rotation matrix cannot be
expressed using a sequence of Euler angles unless the second
rotation axis differs from the other two.

3)  If the input matrix `r' is not a rotation matrix, the error
SPICE(NOTAROTATION) is signaled by a routine in the call tree
of this routine.

4)  If `angle3' and `angle1' are not uniquely determined, `angle3'
is set to zero, and `angle1' is determined.
```

#### Files

```   None.
```

#### Particulars

```   A word about notation: the symbol

[ x ]
i

ith coordinate axis. To be specific, the symbol

[ x ]
1

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
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  |
`-                    -'

The input matrix is assumed to be the product of three
rotation matrices, each one of the form

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

.-                    -.
| cos(s)   0   -sin(s) |
|                      |
|                      |      y-axis),
| sin(s)   0    cos(s) |
`-                    -'

or

.-                    -.
|  cos(t)  sin(t)   0  |
|                      |
|                      |      z-axis),
|  0        0       1  |
`-                    -'

where the second rotation axis is not equal to the first or
third. Any rotation matrix can be factored as a sequence of
three such rotations, provided that this last criterion is met.

This routine is related to the CSPICE routine eul2m_c, which
produces a rotation matrix, given a sequence of Euler angles.
This routine is a `right inverse' of eul2m_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

```   The numerical results shown for these examples 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) Conversion of instrument pointing from a matrix representation
to Euler angles:

Suppose we want to find camera pointing in alpha, delta, and
kappa, given the inertial-to-camera coordinate transformation

.-                                                         -.
|  0.491273796781358  0.508726203218642  0.706999085398824  |
|                                                           |
| -0.508726203218642 -0.491273796781358  0.706999085398824  |
|                                                           |
|  0.706999085398824 -0.706999085398824  0.017452406437284  |
`-                                                         -'

Find angles alpha, delta, kappa such that

TICAM  =  [ kappa ]  [ pi/2 - delta ]  [ pi/2 + alpha ] .
3                 1                 3

Example code begins here.

/.
Program m2eul_ex1
./
#include <stdio.h>
#include "SpiceUsr.h"

int main( )
{

/.
Local variables
./
SpiceDouble          alpha;
SpiceDouble          ang1;
SpiceDouble          ang2;
SpiceDouble          delta;
SpiceDouble          kappa;

SpiceDouble          ticam  [3][3] = {
{ 0.491273796781358,  0.508726203218642,  0.706999085398824},
{-0.508726203218642, -0.491273796781358,  0.706999085398824},
{ 0.706999085398824, -0.706999085398824,  0.017452406437284} };

m2eul_c ( ticam, 3, 1, 3, &kappa, &ang2, &ang1 );

delta = halfpi_c ( ) - ang2;
alpha = ang1     - halfpi_c ( );

if ( kappa < 0.0 )
{
kappa = kappa + twopi_c ( );
}

if ( alpha < 0.0 )
{
alpha = alpha + twopi_c ( );
}

printf( "Alpha (deg):  %23.14f\n", dpr_c ( ) * alpha );
printf( "Delta (deg):  %23.14f\n", dpr_c ( ) * delta );
printf( "Kappa (deg):  %23.14f\n", dpr_c ( ) * kappa );

return ( 0 );
}

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

Alpha (deg):       315.00000000000000
Delta (deg):         1.00000000000003
Kappa (deg):        45.00000000000000

2) Conversion of instrument pointing angles from a non-J2000,
not necessarily inertial frame to J2000-relative RA, Dec,
and Twist.

Suppose that we have orientation for the CASSINI Narrow Angle
Camera (NAC) frame expressed as

[ gamma ]  [ beta ]  [ alpha ]
1         2          3

with respect to the CASSINI spacecraft frame.

We want to express that orientation with respect to the J2000
frame as the sequence of rotations

[ twist ]  [ dec ]  [ ra ] .
3        1       3

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

KPL/MK

File name: m2eul_ex2.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
---------                      --------
naif0010.tls                   Leapseconds
cas00145.tsc                   Cassini SCLK
cas_v40.tf                     Cassini frames
08052_08057ra.bc               Orientation for Cassini

\begindata

'cas00145.tsc'
'cas_v40.tf'
'08052_08057ra.bc')

\begintext

End of meta-kernel

Example code begins here.

/.
Program m2eul_ex2
./
#include <stdio.h>
#include "SpiceUsr.h"

int main( )
{

/.
Local parameters
./
#define META         "m2eul_ex2.tm"
#define ALPHA         89.9148
#define BETA          -0.03300
#define GAMMA        -90.009796

/.
Local variables
./
SpiceDouble          ra;
SpiceDouble          ang1;
SpiceDouble          ang2;
SpiceDouble          dec;
SpiceDouble          et;
SpiceDouble          inst2j [3][3];
SpiceDouble          inst2s [3][3];
SpiceDouble          s2j    [3][3];
SpiceDouble          twist;

/.
./
furnsh_c ( META );

/.
First, we use function eul2m_c to form the
transformation from instrument coordinates to
CASSINI spacecraft frame.
./
eul2m_c ( GAMMA * rpd_c ( ), BETA * rpd_c ( ), ALPHA * rpd_c ( ),
1,                 2,                3,         inst2s );

/.
Now we compute the transformation from CASSINI
spacecraft frame to J2000, at a given time.
./
str2et_c ( "2008 Feb 25", &et );
pxform_c ( "CASSINI_SC_COORD", "J2000", et, s2j );

/.
Next, we form the transformation from instrument
coordinates to J2000 frame.
./
mxm_c ( s2j, inst2s, inst2j );

/.
Finally, we express `inst2j' using the desired Euler
angles.
./
m2eul_c ( inst2j, 3, 1, 3, &twist, &ang1, &ang2 );

ra   =  ang2 - halfpi_c ( );
dec  =  halfpi_c ( ) - ang1;

/.
If we wish to make sure that `ra', `dec', and `twist' are in
the ranges [0, 2pi), [-pi/2, pi/2], and [0, 2pi)
respectively, we may add the code
./
if ( ra < 0.0 )
{
ra = ra + twopi_c ( );
}

if ( twist < 0.0 )
{
twist = twist + twopi_c ( );
}

/.
Now `ra', `dec', and `twist' express the instrument pointing
as RA, Dec, and Twist, relative to the J2000 system.
./
printf( "RA    (deg):  %23.14f\n", dpr_c ( ) * ra );
printf( "Dec   (deg):  %23.14f\n", dpr_c ( ) * dec );
printf( "Twist (deg):  %23.14f\n", dpr_c ( ) * twist );

return ( 0 );
}

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

RA    (deg):        83.77802387778848
Dec   (deg):       -14.92925498590898
Twist (deg):       294.55732942050986

Note:  more than one definition of RA, Dec, and Twist is
extant. Before using this example in an application, check
that the definition given here is consistent with that used
```

#### Restrictions

```   None.
```

#### Literature_References

```   None.
```

#### Author_and_Institution

```   N.J. Bachman        (JPL)
J. Diaz del Rio     (ODC Space)
E.D. Wright         (JPL)
```

#### Version

```   -CSPICE Version 1.3.2, 10-AUG-2021 (JDR)

complete code examples from existing fragments.

-CSPICE Version 1.3.1, 13-OCT-2004 (NJB)

-CSPICE Version 1.3.0, 21-OCT-1998 (NJB)

-CSPICE Version 1.2.0, 13-FEB-1998 (EDW)

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

Removed local variables used for temporary capture of outputs.

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

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

#### Index_Entries

```   matrix to euler angles
```
`Fri Dec 31 18:41:09 2021`