[Numpy-discussion] ANN: xarray v0.9 released

[Stephan Hoyer]

I'm pleased to announce the release of the latest major version of xarray,
v0.9.

xarray is an open source project and Python package that provides a toolkit
and data structures for N-dimensional labeled arrays. Its approach combines
an API inspired by pandas with the Common Data Model for self-described
scientific data.

This release includes five months worth of enhancements and bug fixes from
24 contributors, including some significant enhancements to the data model
that are not fully backwards compatible.

Highlights include:
- Coordinates are now optional in the xarray data model, even for
dimensions.
- Changes to caching, lazy loading and pickling to improve xarray’s
experience for parallel computing.
- Improvements for accessing and manipulating pandas.MultiIndex levels.
- Many new methods and functions, including quantile(), cumsum(),
cumprod(), combine_firstset_index(), reset_index(), reorder_levels(),
full_like(), zeros_like(), ones_like(), open_dataarray(), compute(),
Dataset.info(), testing.assert_equal(), testing.assert_identical(), and
testing.assert_allclose().

For more details, read the full release notes:
http://xarray.pydata.org/en/latest/whats-new.html

You can install xarray with pip or conda:
pip install xarray
conda install -c conda-forge xarray

Best,
Stephan
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Re: [Numpy-discussion] composing Euler rotation matrices

[Seb]

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
> `*` operator instead of `@`, but guessing probably won't help you
> solve your issue.

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
"""
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))
------

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

[Joseph Fox-Rabinovitz]

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  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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[Numpy-discussion] composing Euler rotation matrices

[Seb]

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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