/usr/lib/python2.7/dist-packages/nibabel/eulerangles.py is in python-nibabel 2.0.2-2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 | # emacs: -*- mode: python-mode; py-indent-offset: 4; indent-tabs-mode: nil -*-
# vi: set ft=python sts=4 ts=4 sw=4 et:
### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ##
#
# See COPYING file distributed along with the NiBabel package for the
# copyright and license terms.
#
### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ##
''' Module implementing Euler angle rotations and their conversions
See:
* https://en.wikipedia.org/wiki/Rotation_matrix
* https://en.wikipedia.org/wiki/Euler_angles
* http://mathworld.wolfram.com/EulerAngles.html
See also: *Representing Attitude with Euler Angles and Quaternions: A
Reference* (2006) by James Diebel. A cached PDF link last found here:
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.110.5134
Euler's rotation theorem tells us that any rotation in 3D can be
described by 3 angles. Let's call the 3 angles the *Euler angle vector*
and call the angles in the vector :math:`alpha`, :math:`beta` and
:math:`gamma`. The vector is [ :math:`alpha`,
:math:`beta`. :math:`gamma` ] and, in this description, the order of the
parameters specifies the order in which the rotations occur (so the
rotation corresponding to :math:`alpha` is applied first).
In order to specify the meaning of an *Euler angle vector* we need to
specify the axes around which each of the rotations corresponding to
:math:`alpha`, :math:`beta` and :math:`gamma` will occur.
There are therefore three axes for the rotations :math:`alpha`,
:math:`beta` and :math:`gamma`; let's call them :math:`i` :math:`j`,
:math:`k`.
Let us express the rotation :math:`alpha` around axis `i` as a 3 by 3
rotation matrix `A`. Similarly :math:`beta` around `j` becomes 3 x 3
matrix `B` and :math:`gamma` around `k` becomes matrix `G`. Then the
whole rotation expressed by the Euler angle vector [ :math:`alpha`,
:math:`beta`. :math:`gamma` ], `R` is given by::
R = np.dot(G, np.dot(B, A))
See http://mathworld.wolfram.com/EulerAngles.html
The order :math:`G B A` expresses the fact that the rotations are
performed in the order of the vector (:math:`alpha` around axis `i` =
`A` first).
To convert a given Euler angle vector to a meaningful rotation, and a
rotation matrix, we need to define:
* the axes `i`, `j`, `k`
* whether a rotation matrix should be applied on the left of a vector to
be transformed (vectors are column vectors) or on the right (vectors
are row vectors).
* whether the rotations move the axes as they are applied (intrinsic
rotations) - compared the situation where the axes stay fixed and the
vectors move within the axis frame (extrinsic)
* the handedness of the coordinate system
See: https://en.wikipedia.org/wiki/Rotation_matrix#Ambiguities
We are using the following conventions:
* axes `i`, `j`, `k` are the `z`, `y`, and `x` axes respectively. Thus
an Euler angle vector [ :math:`alpha`, :math:`beta`. :math:`gamma` ]
in our convention implies a :math:`alpha` radian rotation around the
`z` axis, followed by a :math:`beta` rotation around the `y` axis,
followed by a :math:`gamma` rotation around the `x` axis.
* the rotation matrix applies on the left, to column vectors on the
right, so if `R` is the rotation matrix, and `v` is a 3 x N matrix
with N column vectors, the transformed vector set `vdash` is given by
``vdash = np.dot(R, v)``.
* extrinsic rotations - the axes are fixed, and do not move with the
rotations.
* a right-handed coordinate system
The convention of rotation around ``z``, followed by rotation around
``y``, followed by rotation around ``x``, is known (confusingly) as
"xyz", pitch-roll-yaw, Cardan angles, or Tait-Bryan angles.
'''
import math
from .externals.six.moves import reduce
import numpy as np
_FLOAT_EPS_4 = np.finfo(float).eps * 4.0
def euler2mat(z=0, y=0, x=0):
''' Return matrix for rotations around z, y and x axes
Uses the z, then y, then x convention above
Parameters
----------
z : scalar
Rotation angle in radians around z-axis (performed first)
y : scalar
Rotation angle in radians around y-axis
x : scalar
Rotation angle in radians around x-axis (performed last)
Returns
-------
M : array shape (3,3)
Rotation matrix giving same rotation as for given angles
Examples
--------
>>> zrot = 1.3 # radians
>>> yrot = -0.1
>>> xrot = 0.2
>>> M = euler2mat(zrot, yrot, xrot)
>>> M.shape == (3, 3)
True
The output rotation matrix is equal to the composition of the
individual rotations
>>> M1 = euler2mat(zrot)
>>> M2 = euler2mat(0, yrot)
>>> M3 = euler2mat(0, 0, xrot)
>>> composed_M = np.dot(M3, np.dot(M2, M1))
>>> np.allclose(M, composed_M)
True
You can specify rotations by named arguments
>>> np.all(M3 == euler2mat(x=xrot))
True
When applying M to a vector, the vector should column vector to the
right of M. If the right hand side is a 2D array rather than a
vector, then each column of the 2D array represents a vector.
>>> vec = np.array([1, 0, 0]).reshape((3,1))
>>> v2 = np.dot(M, vec)
>>> vecs = np.array([[1, 0, 0],[0, 1, 0]]).T # giving 3x2 array
>>> vecs2 = np.dot(M, vecs)
Rotations are counter-clockwise.
>>> zred = np.dot(euler2mat(z=np.pi/2), np.eye(3))
>>> np.allclose(zred, [[0, -1, 0],[1, 0, 0], [0, 0, 1]])
True
>>> yred = np.dot(euler2mat(y=np.pi/2), np.eye(3))
>>> np.allclose(yred, [[0, 0, 1],[0, 1, 0], [-1, 0, 0]])
True
>>> xred = np.dot(euler2mat(x=np.pi/2), np.eye(3))
>>> np.allclose(xred, [[1, 0, 0],[0, 0, -1], [0, 1, 0]])
True
Notes
-----
The direction of rotation is given by the right-hand rule (orient
the thumb of the right hand along the axis around which the rotation
occurs, with the end of the thumb at the positive end of the axis;
curl your fingers; the direction your fingers curl is the direction
of rotation). Therefore, the rotations are counterclockwise if
looking along the axis of rotation from positive to negative.
'''
Ms = []
if z:
cosz = math.cos(z)
sinz = math.sin(z)
Ms.append(np.array(
[[cosz, -sinz, 0],
[sinz, cosz, 0],
[0, 0, 1]]))
if y:
cosy = math.cos(y)
siny = math.sin(y)
Ms.append(np.array(
[[cosy, 0, siny],
[0, 1, 0],
[-siny, 0, cosy]]))
if x:
cosx = math.cos(x)
sinx = math.sin(x)
Ms.append(np.array(
[[1, 0, 0],
[0, cosx, -sinx],
[0, sinx, cosx]]))
if Ms:
return reduce(np.dot, Ms[::-1])
return np.eye(3)
def mat2euler(M, cy_thresh=None):
''' Discover Euler angle vector from 3x3 matrix
Uses the conventions above.
Parameters
----------
M : array-like, shape (3,3)
cy_thresh : None or scalar, optional
threshold below which to give up on straightforward arctan for
estimating x rotation. If None (default), estimate from
precision of input.
Returns
-------
z : scalar
y : scalar
x : scalar
Rotations in radians around z, y, x axes, respectively
Notes
-----
If there was no numerical error, the routine could be derived using
Sympy expression for z then y then x rotation matrix, which is::
[ cos(y)*cos(z), -cos(y)*sin(z), sin(y)],
[cos(x)*sin(z) + cos(z)*sin(x)*sin(y), cos(x)*cos(z) - sin(x)*sin(y)*sin(z), -cos(y)*sin(x)],
[sin(x)*sin(z) - cos(x)*cos(z)*sin(y), cos(z)*sin(x) + cos(x)*sin(y)*sin(z), cos(x)*cos(y)]
with the obvious derivations for z, y, and x
z = atan2(-r12, r11)
y = asin(r13)
x = atan2(-r23, r33)
Problems arise when cos(y) is close to zero, because both of::
z = atan2(cos(y)*sin(z), cos(y)*cos(z))
x = atan2(cos(y)*sin(x), cos(x)*cos(y))
will be close to atan2(0, 0), and highly unstable.
The ``cy`` fix for numerical instability below is from: *Graphics
Gems IV*, Paul Heckbert (editor), Academic Press, 1994, ISBN:
0123361559. Specifically it comes from EulerAngles.c by Ken
Shoemake, and deals with the case where cos(y) is close to zero:
See: http://www.graphicsgems.org/
The code appears to be licensed (from the website) as "can be used
without restrictions".
'''
M = np.asarray(M)
if cy_thresh is None:
try:
cy_thresh = np.finfo(M.dtype).eps * 4
except ValueError:
cy_thresh = _FLOAT_EPS_4
r11, r12, r13, r21, r22, r23, r31, r32, r33 = M.flat
# cy: sqrt((cos(y)*cos(z))**2 + (cos(x)*cos(y))**2)
cy = math.sqrt(r33*r33 + r23*r23)
if cy > cy_thresh: # cos(y) not close to zero, standard form
z = math.atan2(-r12, r11) # atan2(cos(y)*sin(z), cos(y)*cos(z))
y = math.atan2(r13, cy) # atan2(sin(y), cy)
x = math.atan2(-r23, r33) # atan2(cos(y)*sin(x), cos(x)*cos(y))
else: # cos(y) (close to) zero, so x -> 0.0 (see above)
# so r21 -> sin(z), r22 -> cos(z) and
z = math.atan2(r21, r22)
y = math.atan2(r13, cy) # atan2(sin(y), cy)
x = 0.0
return z, y, x
def euler2quat(z=0, y=0, x=0):
''' Return quaternion corresponding to these Euler angles
Uses the z, then y, then x convention above
Parameters
----------
z : scalar
Rotation angle in radians around z-axis (performed first)
y : scalar
Rotation angle in radians around y-axis
x : scalar
Rotation angle in radians around x-axis (performed last)
Returns
-------
quat : array shape (4,)
Quaternion in w, x, y z (real, then vector) format
Notes
-----
We can derive this formula in Sympy using:
1. Formula giving quaternion corresponding to rotation of theta radians
about arbitrary axis:
http://mathworld.wolfram.com/EulerParameters.html
2. Generated formulae from 1.) for quaternions corresponding to
theta radians rotations about ``x, y, z`` axes
3. Apply quaternion multiplication formula -
https://en.wikipedia.org/wiki/Quaternions#Hamilton_product - to
formulae from 2.) to give formula for combined rotations.
'''
z = z/2.0
y = y/2.0
x = x/2.0
cz = math.cos(z)
sz = math.sin(z)
cy = math.cos(y)
sy = math.sin(y)
cx = math.cos(x)
sx = math.sin(x)
return np.array([
cx*cy*cz - sx*sy*sz,
cx*sy*sz + cy*cz*sx,
cx*cz*sy - sx*cy*sz,
cx*cy*sz + sx*cz*sy])
def quat2euler(q):
''' Return Euler angles corresponding to quaternion `q`
Parameters
----------
q : 4 element sequence
w, x, y, z of quaternion
Returns
-------
z : scalar
Rotation angle in radians around z-axis (performed first)
y : scalar
Rotation angle in radians around y-axis
x : scalar
Rotation angle in radians around x-axis (performed last)
Notes
-----
It's possible to reduce the amount of calculation a little, by
combining parts of the ``quat2mat`` and ``mat2euler`` functions, but
the reduction in computation is small, and the code repetition is
large.
'''
# delayed import to avoid cyclic dependencies
from . import quaternions as nq
return mat2euler(nq.quat2mat(q))
def euler2angle_axis(z=0, y=0, x=0):
''' Return angle, axis corresponding to these Euler angles
Uses the z, then y, then x convention above
Parameters
----------
z : scalar
Rotation angle in radians around z-axis (performed first)
y : scalar
Rotation angle in radians around y-axis
x : scalar
Rotation angle in radians around x-axis (performed last)
Returns
-------
theta : scalar
angle of rotation
vector : array shape (3,)
axis around which rotation occurs
Examples
--------
>>> theta, vec = euler2angle_axis(0, 1.5, 0)
>>> print(theta)
1.5
>>> np.allclose(vec, [0, 1, 0])
True
'''
# delayed import to avoid cyclic dependencies
from . import quaternions as nq
return nq.quat2angle_axis(euler2quat(z, y, x))
def angle_axis2euler(theta, vector, is_normalized=False):
''' Convert angle, axis pair to Euler angles
Parameters
----------
theta : scalar
angle of rotation
vector : 3 element sequence
vector specifying axis for rotation.
is_normalized : bool, optional
True if vector is already normalized (has norm of 1). Default
False
Returns
-------
z : scalar
y : scalar
x : scalar
Rotations in radians around z, y, x axes, respectively
Examples
--------
>>> z, y, x = angle_axis2euler(0, [1, 0, 0])
>>> np.allclose((z, y, x), 0)
True
Notes
-----
It's possible to reduce the amount of calculation a little, by
combining parts of the ``angle_axis2mat`` and ``mat2euler``
functions, but the reduction in computation is small, and the code
repetition is large.
'''
# delayed import to avoid cyclic dependencies
from . import quaternions as nq
M = nq.angle_axis2mat(theta, vector, is_normalized)
return mat2euler(M)
|