composing Euler rotation matrices
Hello, I'm trying to compose Euler rotation matrices shown in https://en.wikipedia.org/wiki/Euler_angles#Rotation_matrix. For example, The Z1Y2X3 Tait-Bryan rotation shown in the table can be represented in Numpy using the function: def z1y2x3(alpha, beta, gamma): """Rotation matrix given Euler angles""" return np.array([[np.cos(alpha) * np.cos(beta), np.cos(alpha) * np.sin(beta) * np.sin(gamma) - np.cos(gamma) * np.sin(alpha), np.sin(alpha) * np.sin(gamma) + np.cos(alpha) * np.cos(gamma) * np.sin(beta)], [np.cos(beta) * np.sin(alpha), np.cos(alpha) * np.cos(gamma) + np.sin(alpha) * np.sin(beta) * np.sin(gamma), np.cos(gamma) * np.sin(alpha) * np.sin(beta) - np.cos(alpha) * np.sin(gamma)], [-np.sin(beta), np.cos(beta) * np.sin(gamma), np.cos(beta) * np.cos(gamma)]]) which given alpha, beta, gamma as: angles = np.radians(np.array([30, 20, 10])) returns the following matrix: In [31]: z1y2x3(angles[0], angles[1], angles[2]) Out[31]: array([[ 0.81379768, -0.44096961, 0.37852231], [ 0.46984631, 0.88256412, 0.01802831], [-0.34202014, 0.16317591, 0.92541658]]) If I understand correctly, one should be able to compose this matrix by multiplying the rotation matrices that it is made of. However, I cannot reproduce this matrix via composition; i.e. by multiplying the underlying rotation matrices. Any tips would be appreciated. -- Seb
Could you show what you are doing to get the statement "However, I cannot reproduce this matrix via composition; i.e. by multiplying the underlying rotation matrices.". I would guess something involving the `*` operator instead of `@`, but guessing probably won't help you solve your issue. -Joe On Tue, Jan 31, 2017 at 7:56 PM, Seb <spluque@gmail.com> wrote:
On Tue, 31 Jan 2017 21:23:55 -0500, Joseph Fox-Rabinovitz <jfoxrabinovitz@gmail.com> wrote:
Sure, although composition is not something I can take credit for, as it's a well-described operation for generating linear transformations. It is the matrix multiplication of two or more transformation matrices. In the case of Euler transformations, it's matrices specifying rotations around 3 orthogonal axes by 3 given angles. I'm using `numpy.dot' to perform matrix multiplication on 2D arrays representing matrices. However, it's not obvious from the link I provided what particular rotation matrices are multiplied and in what order (i.e. what composition) is used to arrive at the Z1Y2X3 rotation matrix shown. Perhaps I'm not understanding the conventions used therein. This is one of my attempts at reproducing that rotation matrix via composition: ---<--------------------cut here---------------start------------------->--- import numpy as np angles = np.radians(np.array([30, 20, 10])) def z1y2x3(alpha, beta, gamma): """Z1Y2X3 rotation matrix given Euler angles""" return np.array([[np.cos(alpha) * np.cos(beta), np.cos(alpha) * np.sin(beta) * np.sin(gamma) - np.cos(gamma) * np.sin(alpha), np.sin(alpha) * np.sin(gamma) + np.cos(alpha) * np.cos(gamma) * np.sin(beta)], [np.cos(beta) * np.sin(alpha), np.cos(alpha) * np.cos(gamma) + np.sin(alpha) * np.sin(beta) * np.sin(gamma), np.cos(gamma) * np.sin(alpha) * np.sin(beta) - np.cos(alpha) * np.sin(gamma)], [-np.sin(beta), np.cos(beta) * np.sin(gamma), np.cos(beta) * np.cos(gamma)]]) euler_mat = z1y2x3(angles[0], angles[1], angles[2]) ## Now via composition def rotation_matrix(theta, axis, active=False): """Generate rotation matrix for a given axis Parameters ---------- theta: numeric, optional The angle (degrees) by which to perform the rotation. Default is 0, which means return the coordinates of the vector in the rotated coordinate system, when rotate_vectors=False. axis: int, optional Axis around which to perform the rotation (x=0; y=1; z=2) active: bool, optional Whether to return active transformation matrix. Returns ------- numpy.ndarray 3x3 rotation matrix """ theta = np.radians(theta) if axis == 0: R_theta = np.array([[1, 0, 0], [0, np.cos(theta), -np.sin(theta)], [0, np.sin(theta), np.cos(theta)]]) elif axis == 1: R_theta = np.array([[np.cos(theta), 0, np.sin(theta)], [0, 1, 0], [-np.sin(theta), 0, np.cos(theta)]]) else: R_theta = np.array([[np.cos(theta), -np.sin(theta), 0], [np.sin(theta), np.cos(theta), 0], [0, 0, 1]]) if active: R_theta = np.transpose(R_theta) return R_theta ## The rotations are given as active xmat = rotation_matrix(angles[2], 0, active=True) ymat = rotation_matrix(angles[1], 1, active=True) zmat = rotation_matrix(angles[0], 2, active=True) ## The operation seems to imply this composition euler_comp_mat = np.dot(xmat, np.dot(ymat, zmat)) ---<--------------------cut here---------------end--------------------->--- I believe the matrices `euler_mat' and `euler_comp_mat' should be the same, but they aren't, so it's unclear to me what particular composition is meant to produce the matrix specified by this Z1Y2X3 transformation. What am I missing? -- Seb
Instead of trying to decipher what someone wrote on a Wikipedia, why don't you look at a working piece of source code? e.g. https://github.com/3dem/relion/blob/master/src/euler.cpp Robert On Wed, Feb 1, 2017 at 4:27 AM, Seb <spluque@gmail.com> wrote:
-- Robert McLeod, Ph.D. Center for Cellular Imaging and Nano Analytics (C-CINA) Biozentrum der Universität Basel Mattenstrasse 26, 4058 Basel Work: +41.061.387.3225 robert.mcleod@unibas.ch robert.mcleod@bsse.ethz.ch <robert.mcleod@ethz.ch> robbmcleod@gmail.com
Hi, On Wed, Feb 1, 2017 at 8:28 AM, Robert McLeod <robbmcleod@gmail.com> wrote:
Also - have a look at https://pypi.python.org/pypi/transforms3d - and in particular you might get some use from symbolic versions of the transformations, e.g. here : https://github.com/matthew-brett/transforms3d/blob/master/transforms3d/deriv... It's really easy to mix up the conventions, as I'm sure you know - see http://matthew-brett.github.io/transforms3d/reference/transforms3d.euler.htm... Cheers, Matthew
[off topic] Nothing good ever comes from using Euler matrices. All the cool kids a using quaternions these days. They're (in some ways) simpler, can be interpolated easily, don't suffer from gimbal lock (discontinuity), and are not confused about which axis rotation is applied first (for Euler you much decide whether you want to apply x.y.z or z.y.x). They'd be a good addition to numpy. On Wed, Feb 1, 2017 at 1:42 AM, Matthew Brett <matthew.brett@gmail.com> wrote:
On Wed, 1 Feb 2017 09:42:15 +0000, Matthew Brett <matthew.brett@gmail.com> wrote:
e.g.
It's really easy to mix up the conventions, as I'm sure you know - see http://matthew-brett.github.io/transforms3d/reference/transforms3d.euler.htm...
Thank you very much for providing this package. It looks like this is exactly what I was trying to do (learn). The symbolic versions really help show what is going on in the derivations sub-package by showing how each of the 9 matrix elements are found. I'll try to hack it to use active rotations. -- Seb
Could you show what you are doing to get the statement "However, I cannot reproduce this matrix via composition; i.e. by multiplying the underlying rotation matrices.". I would guess something involving the `*` operator instead of `@`, but guessing probably won't help you solve your issue. -Joe On Tue, Jan 31, 2017 at 7:56 PM, Seb <spluque@gmail.com> wrote:
On Tue, 31 Jan 2017 21:23:55 -0500, Joseph Fox-Rabinovitz <jfoxrabinovitz@gmail.com> wrote:
Sure, although composition is not something I can take credit for, as it's a well-described operation for generating linear transformations. It is the matrix multiplication of two or more transformation matrices. In the case of Euler transformations, it's matrices specifying rotations around 3 orthogonal axes by 3 given angles. I'm using `numpy.dot' to perform matrix multiplication on 2D arrays representing matrices. However, it's not obvious from the link I provided what particular rotation matrices are multiplied and in what order (i.e. what composition) is used to arrive at the Z1Y2X3 rotation matrix shown. Perhaps I'm not understanding the conventions used therein. This is one of my attempts at reproducing that rotation matrix via composition: ---<--------------------cut here---------------start------------------->--- import numpy as np angles = np.radians(np.array([30, 20, 10])) def z1y2x3(alpha, beta, gamma): """Z1Y2X3 rotation matrix given Euler angles""" return np.array([[np.cos(alpha) * np.cos(beta), np.cos(alpha) * np.sin(beta) * np.sin(gamma) - np.cos(gamma) * np.sin(alpha), np.sin(alpha) * np.sin(gamma) + np.cos(alpha) * np.cos(gamma) * np.sin(beta)], [np.cos(beta) * np.sin(alpha), np.cos(alpha) * np.cos(gamma) + np.sin(alpha) * np.sin(beta) * np.sin(gamma), np.cos(gamma) * np.sin(alpha) * np.sin(beta) - np.cos(alpha) * np.sin(gamma)], [-np.sin(beta), np.cos(beta) * np.sin(gamma), np.cos(beta) * np.cos(gamma)]]) euler_mat = z1y2x3(angles[0], angles[1], angles[2]) ## Now via composition def rotation_matrix(theta, axis, active=False): """Generate rotation matrix for a given axis Parameters ---------- theta: numeric, optional The angle (degrees) by which to perform the rotation. Default is 0, which means return the coordinates of the vector in the rotated coordinate system, when rotate_vectors=False. axis: int, optional Axis around which to perform the rotation (x=0; y=1; z=2) active: bool, optional Whether to return active transformation matrix. Returns ------- numpy.ndarray 3x3 rotation matrix """ theta = np.radians(theta) if axis == 0: R_theta = np.array([[1, 0, 0], [0, np.cos(theta), -np.sin(theta)], [0, np.sin(theta), np.cos(theta)]]) elif axis == 1: R_theta = np.array([[np.cos(theta), 0, np.sin(theta)], [0, 1, 0], [-np.sin(theta), 0, np.cos(theta)]]) else: R_theta = np.array([[np.cos(theta), -np.sin(theta), 0], [np.sin(theta), np.cos(theta), 0], [0, 0, 1]]) if active: R_theta = np.transpose(R_theta) return R_theta ## The rotations are given as active xmat = rotation_matrix(angles[2], 0, active=True) ymat = rotation_matrix(angles[1], 1, active=True) zmat = rotation_matrix(angles[0], 2, active=True) ## The operation seems to imply this composition euler_comp_mat = np.dot(xmat, np.dot(ymat, zmat)) ---<--------------------cut here---------------end--------------------->--- I believe the matrices `euler_mat' and `euler_comp_mat' should be the same, but they aren't, so it's unclear to me what particular composition is meant to produce the matrix specified by this Z1Y2X3 transformation. What am I missing? -- Seb
Instead of trying to decipher what someone wrote on a Wikipedia, why don't you look at a working piece of source code? e.g. https://github.com/3dem/relion/blob/master/src/euler.cpp Robert On Wed, Feb 1, 2017 at 4:27 AM, Seb <spluque@gmail.com> wrote:
-- Robert McLeod, Ph.D. Center for Cellular Imaging and Nano Analytics (C-CINA) Biozentrum der Universität Basel Mattenstrasse 26, 4058 Basel Work: +41.061.387.3225 robert.mcleod@unibas.ch robert.mcleod@bsse.ethz.ch <robert.mcleod@ethz.ch> robbmcleod@gmail.com
Hi, On Wed, Feb 1, 2017 at 8:28 AM, Robert McLeod <robbmcleod@gmail.com> wrote:
Also - have a look at https://pypi.python.org/pypi/transforms3d - and in particular you might get some use from symbolic versions of the transformations, e.g. here : https://github.com/matthew-brett/transforms3d/blob/master/transforms3d/deriv... It's really easy to mix up the conventions, as I'm sure you know - see http://matthew-brett.github.io/transforms3d/reference/transforms3d.euler.htm... Cheers, Matthew
[off topic] Nothing good ever comes from using Euler matrices. All the cool kids a using quaternions these days. They're (in some ways) simpler, can be interpolated easily, don't suffer from gimbal lock (discontinuity), and are not confused about which axis rotation is applied first (for Euler you much decide whether you want to apply x.y.z or z.y.x). They'd be a good addition to numpy. On Wed, Feb 1, 2017 at 1:42 AM, Matthew Brett <matthew.brett@gmail.com> wrote:
On Wed, 1 Feb 2017 09:42:15 +0000, Matthew Brett <matthew.brett@gmail.com> wrote:
e.g.
It's really easy to mix up the conventions, as I'm sure you know - see http://matthew-brett.github.io/transforms3d/reference/transforms3d.euler.htm...
Thank you very much for providing this package. It looks like this is exactly what I was trying to do (learn). The symbolic versions really help show what is going on in the derivations sub-package by showing how each of the 9 matrix elements are found. I'll try to hack it to use active rotations. -- Seb
participants (5)
-
Joseph Fox-Rabinovitz -
Matthew Brett -
Robert McLeod -
Seb -
Stuart Reynolds