### Re: [Numpy-discussion] composing Euler rotation matrices

```On Wed, 1 Feb 2017 09:42:15 +,
Matthew Brett  wrote:

> Hi,
> On Wed, Feb 1, 2017 at 8:28 AM, Robert McLeod  wrote:
>> 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

> 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/derivations/eulerangles.py

> 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.html

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

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### Re: [Numpy-discussion] composing Euler rotation matrices

```[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
wrote:

> Hi,
>
> On Wed, Feb 1, 2017 at 8:28 AM, Robert McLeod
> wrote:
> > 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
>
> 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/
> derivations/eulerangles.py
>
> 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.html
>
> Cheers,
>
> Matthew
> ___
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>
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### Re: [Numpy-discussion] composing Euler rotation matrices

```Hi,

On Wed, Feb 1, 2017 at 8:28 AM, Robert McLeod  wrote:
> 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

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/derivations/eulerangles.py

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.html

Cheers,

Matthew
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### Re: [Numpy-discussion] composing Euler rotation matrices

```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  wrote:

> On Tue, 31 Jan 2017 21:23:55 -0500,
> Joseph Fox-Rabinovitz  wrote:
>
> > 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
>
> 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:
>
> --- ->---
> 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
> """
> 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))
> --- ->---
>
> 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
>
> ___
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>

--
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.mcl...@unibas.ch
robert.mcl...@bsse.ethz.ch
robbmcl...@gmail.com
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### Re: [Numpy-discussion] composing Euler rotation matrices

```On Tue, 31 Jan 2017 21:23:55 -0500,
Joseph Fox-Rabinovitz  wrote:

> 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

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:

------
import numpy as np

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
"""
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))
------

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

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### Re: [Numpy-discussion] composing Euler rotation matrices

```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

-Joe

On Tue, Jan 31, 2017 at 7:56 PM, Seb  wrote:

> 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
>
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