void m2eul_c ( ConstSpiceDouble r[3][3],
SpiceInt axis3,
SpiceInt axis2,
SpiceInt axis1,
SpiceDouble * angle3,
SpiceDouble * angle2,
SpiceDouble * angle1 )
Factor a rotation matrix as a product of three rotations about
specified coordinate axes.
ROTATION
ANGLE
MATRIX
ROTATION
TRANSFORMATION
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.
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.
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.
None.
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.
None.
A word about notation: the symbol
[ x ]
i
indicates a coordinate system 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 |
+- -+
The input matrix is assumed to be the product of three
rotation matrices, each one of the form
+- -+
| 1 0 0 |
| |
| 0 cos(r) sin(r) | (rotation of r radians about the
| | x-axis),
| 0 -sin(r) cos(r) |
+- -+
+- -+
| cos(s) 0 -sin(s) |
| |
| 0 1 0 | (rotation of s radians about the
| | y-axis),
| sin(s) 0 cos(s) |
+- -+
or
+- -+
| cos(t) sin(t) 0 |
| |
| -sin(t) cos(t) 0 | (rotation of t radians about the
| | 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.
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
used in your application.
None.
None.
N.J. Bachman (JPL)
-CSPICE Version 1.3.1, 13-OCT-2004 (NJB)
Fixed header typo.
-CSPICE Version 1.3.0, 21-OCT-1998 (NJB)
Made input matrix const.
-CSPICE Version 1.2.0, 13-FEB-1998 (EDW)
Minor corrections to header.
-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)
matrix to euler angles
Link to routine m2eul_c source file m2eul_c.c
|