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

```   void  m2eul_c ( ConstSpiceDouble    r,
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 },

then the error SPICE(INPUTOUTOFRANGE) is signalled.

2)   An arbitrary rotation matrix cannot be expressed using
a sequence of Euler angles unless the second rotation axis
differs from the other two.  If axis2 is equal to axis3 or
axis1, then then error SPICE(BADAXISNUMBERS) is signalled.

3)   If the input matrix r is not a rotation matrix, the error
SPICE(NOTAROTATION) is signalled.

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, which
produces a rotation matrix, given a sequence of Euler angles.
This routine is a `right inverse' of EUL2M:  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)  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

ticam =

+-                                                               -+
|  0.49127379678135830  0.50872620321864170  0.70699908539882417  |
|                                                                 |
| -0.50872620321864193 -0.49127379678135802  0.70699908539882428  |
|                                                                 |
|  0.70699908539882406 -0.70699908539882439  0.01745240643728360  |
+-                                                               -+

We want to find angles alpha, delta, kappa such that

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

The code fragment

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

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

calculates the desired angles.  If we wish to make sure that
alpha, delta, and kappa are in the ranges [0, 2pi),
[-pi/2, pi/2], and [0, 2pi) respectively, we may add the code

if ( alpha < 0. )
{
alpha = alpha + twopi_c();
}

if ( kappa < 0. )
{
kappa = kappa + twopi_c();
}

Note that delta is already in the correct range, since ang2
is in the range [0, pi] when the first and third input axes
are equal.

If we wish to print out the results in degrees, we might
use the code

printf ( "Alpha = %25.17f\n"
"Delta = %25.17f\n"
"Kappa = %25.17f\n",
dpr_c() * alpha,
dpr_c() * delta,
dpr_c() * kappa     );

We should see something like

Alpha =     315.00000000000000000
Delta =       1.00000000000000000
Kappa =      45.00000000000000000

possibly formatted a little differently, or degraded slightly
by round-off.

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 pointing for some instrument expressed as

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

with respect to some coordinate system S.  For example, S
could be a spacecraft-fixed system.

We will suppose that the transformation from J2000
coordinates to system S coordinates is given by the rotation
matrix j2s.

The rows of j2s are the unit basis vectors of system S, given
in J2000 coordinates.

We want to express the pointing with respect to the J2000
system as the sequence of rotations

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

First, we use subroutine eul2m_c to form the transformation
from system S to instrument coordinates s2inst.

eul2m_c ( gamma, beta, alpha, 3, 2, 3, s2inst );

Next, we form the transformation from J2000 to instrument
coordinates j2inst.

mxm_c ( s2inst, j2s, j2inst );

Finally, we express j2inst using the desired Euler angles, as
in the first example:

m2eul_c ( j2inst, 3, 1, 3, &twist, &ang2, &ang3 );

ra   =  ang3       - halfpi_c();
dec  =  halfpi_c() - ang2;

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. )
{
ra = ra + twopi_c();
}

if ( twist < 0. )
{
twist = twist + twopi_c();
}

Note that dec is already in the correct range, since ang2
is in the range [0, pi] when the first and third input axes
are equal.

Now ra, dec, and twist express the instrument pointing
as RA, Dec, and Twist, relative to the J2000 system.

A warning 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
```

#### Restrictions

```   None.
```

#### Literature_References

```   None.
```

#### Author_and_Institution

```   N.J. Bachman   (JPL)
```

#### Version

```   -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
```
`Wed Apr  5 17:54:38 2017`