eulerangles

Euler angles are often used to represent rigid body rotations in 3D.

These transformations can be defined in many different ways. The world of transformations is filled with ambiguities which can make it harder than necessary to interface softwares which define their transformations differently.

eulerangles is designed to simplify the handling of large sets of Euler angles in Python.

Features

• convert Euler angles into rotation matrices
• convert rotation matrices into Euler angles
• convert between differently defined Euler angles
• simple API
• vectorised implementation

Documentation

Complete documentation is provided on readthedocs.io.

• Quick Start
• Installation
• API Reference

Installation

If you're already at ease with package management in Python, go ahead and

pip install eulerangles

Otherwise, please see the installation page.

Overview

To keep usage simple, the package provides only five functions

• euler2matrix
• matrix2euler
• euler2euler
• convert_eulers
• invert_rotation_matrices

euler2matrix converts sets of Euler angles into rotation matrices.

matrix2euler converts sets of rotation matrices into Euler angles.

euler2euler converts between sets of Euler angles defined in different ways.

convert_eulers provides a simpler interface to euler2euler.

invert_rotation_matrices inverts sets of rotation matrices, yielding the inverse transformation.

Examples

euler2matrix

import numpy as np
from eulerangles import euler2matrix

eulers = np.array([30, 60, 75])
rotation_matrix = euler2matrix(eulers,
                               axes='zyz',
                               intrinsic=True,
                               right_handed_rotation=True)

>>> rotation_matrix
array([[-0.37089098, -0.54766767,  0.75      ],
       [ 0.90122107, -0.01733759,  0.4330127 ],
       [-0.22414387,  0.8365163 ,  0.5       ]])

multiple sets of Euler angles can be passed as an (n, 3) array-like object, in which case an (n, 3, 3) array of rotation matrices will be returned.

matrix2euler

import numpy as np
from eulerangles import matrix2euler

rotation_matrix = np.array([[-0.37089098, -0.54766767,  0.75      ],
                            [ 0.90122107, -0.01733759,  0.4330127 ],
                            [-0.22414387,  0.8365163 ,  0.5       ]])
eulers = matrix2euler(rotation_matrix,
                      axes='zyz',
                      intrinsic=True,
                      right_handed_rotation=True)

>>> eulers
array([29.99999989, 60.        , 74.99999981])

multiple rotation matrices can be passed as an (n, 3, 3) array-like object, in which case an (n, 3) array of Euler angles will be returned.

euler2euler

euler2euler is a verbose function for converting between sets of Euler angles defined differently.

• source_ parameters relate to the input Euler angles
• target_ parameters relate to the output Euler angles
• invert_matrix inverts the rotation matrix or matrices before generating output Euler angles

invert_matrix is useful when one set of Euler angles describe an active transformation, the other a passive transformation.

import numpy as np
from eulerangles import euler2euler

input_eulers = np.array([-47.2730, 1.1777, -132.3000])
output_eulers = euler2euler(input_eulers,
                            source_axes='zxz',
                            source_intrinsic=True,
                            source_right_handed_rotation=True,
                            target_axes='zyz',
                            target_intrinsic=False,
                            target_right_handed_rotation=False,
                            invert_matrix=True)

>>> output_eulers
array([ 42.727 ,  -1.1777, 137.7   ])

multiple sets of Euler angles can be passed as an (n, 3) array-like object, in which case an (n, 3) array of Euler angles will be returned.

convert_eulers

convert_eulers provides a simpler interface to the verbose euler2euler function.

Necessary metadata for conversion between input Euler angles and output Euler angles are stored in a ConversionMeta objects, defined as follows

import numpy as np
from eulerangles import ConversionMeta, convert_eulers

source_metadata = ConversionMeta(name='input', 
                                axes='zyz', 
                                intrinsic=True, 
                                right_handed_rotation=True, 
                                active=True)

target_metadata = ConversionMeta(name='output', 
                                axes='zxz', 
                                intrinsic=False, 
                                right_handed_rotation=False, 
                                active=False)

these objects are used for conversion in convert_eulers

input_eulers = np.array([10, 20, 30])
output_eulers = convert_eulers(input_eulers, 
                               source_meta=source_metadata, 
                               target_meta=target_metadata)

>>> output_eulers
array([-80., -20., 120.])

For a select few software packages for 3D reconstruction from transmission electron microscopy images, source_meta and target_meta can be passed as a string corresponding to the name of the software package.

invert_rotation_matrices

invert_rotation_matrices inverts rotation matrices such that they represent the inverse transform of the input rotation matrix.

Rotation matrices are orthogonal, therefore their inverse is simply the transpose.

import numpy as np
from eulerangles import invert_rotation_matrices
rotation_matrix = np.array([[-0.37089098, -0.54766767,  0.75      ],
                            [ 0.90122107, -0.01733759,  0.4330127 ],
                            [-0.22414387,  0.8365163 ,  0.5       ]])

inverse_matrix = invert_rotation_matrices(rotation_matrix)

>>> inverse_matrix
array([[-0.37089098,  0.90122107, -0.22414387],
       [-0.54766767, -0.01733759,  0.8365163 ],
       [ 0.75      ,  0.4330127 ,  0.5       ]])

This function works equally well on multiple rotation matrices passed as an (n, 3, 3) array.