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Perform quaternion operations using NumPy arrays

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Welcome to the documentation for rowan, a package for working with quaternions! Quaternions, which form a number system with various interesting properties, were originally developed for classical mechanics. Although they have since been largely displaced from this application by vector mathematics, they have become a standard method of representing rotations in three dimensions. Quaternions are now commonly used for this purpose in various fields, including computer graphics and attitude control.

The package is built entirely on top of NumPy and represents quaternions using NumPy arrays, meaning that all functions support arbitrarily high-dimensional arrays of quaternions. Quaternions are encoded as arrays of shape (…, 4), with the convention that the final dimension of an array (a, b, c, d) represents the quaternion a + bi + cj + dk. This package provides tools for standard algebraic operations on quaternions as well as a number of additional tools for e.g. measuring distances between quaternions, interpolating between them, and performing basic point-cloud mapping. A particular focus of the rowan package is working with unit quaternions, which are a popular means of representing rotations in 3D. In order to provide a unified framework for working with the various rotation formalisms in 3D, rowan allows easy interconversion between these formalisms.

Core features of rowan include (but are not limited to):

  • Algebra (multiplication, exponentiation, etc).
  • Derivatives and integrals of quaternions.
  • Rotation and reflection operations, with conversions to and from matrices, axis angles, etc.
  • Various distance metrics for quaternions.
  • Basic point set registration, including solutions of the Procrustes problem and the Iterative Closest Point algorithm.
  • Quaternion interpolation (slerp, squad).

Getting Started


The recommended methods for installing rowan are using pip or conda. To install the package from PyPI, execute:

$ pip install rowan --user

To install the package from conda, first add the conda-forge channel and then install rowan:

$ conda config --add channels conda-forge
$ conda install rowan

If you wish, you may also install rowan by cloning the repository and running the setup script:

$ git clone
$ cd rowan
$ python install --user

The minimum requirements for using rowan are:

  • Python >= 3.6
  • NumPy >= 1.15


This library can be used to work with quaternions by simply instantiating the appropriate NumPy arrays and passing them to the required functions. For example:

import rowan
import numpy as np
one = np.array([10, 0, 0, 0])
one_unit = rowan.normalize(one)
assert(np.all(one_unit == np.array([1, 0, 0, 0])))
if not np.all(one_unit == rowan.multiply(one_unit, one_unit)):
    raise RuntimeError("Multiplication failed!")

one_vec = np.array([1, 0, 0])
rotated_vector = rowan.rotate(one_unit, one_vec)

mat = np.eye(3)
quat_rotate = rowan.from_matrix(mat)
alpha, beta, gamma = rowan.to_euler(quat_rotate)
quat_rotate_returned = rowan.from_euler(alpha, beta, gamma)
identity = rowan.to_matrix(quat_rotate_returned)

Running Tests

The package is currently tested for Python >= 3.6 on Unix-like systems. Continuous integrated testing is performed using CircleCI on these Python versions with NumPy versions 1.15 and above.

To run the packaged unit tests, execute the following line from the root of the repository:

python -m unittest discover tests

To check test coverage, make sure the coverage module is installed:

pip install coverage

and then run the packaged unit tests with the coverage module:

coverage run -m unittest discover tests

Running Benchmarks

Benchmarks for the package are contained in a Jupyter notebook in the benchmarks folder in the root of the repository. If you do not have or do not wish to use the notebook format, an equivalent script is also included. The benchmarks compare rowan to two alternative packages, so you will need to install pyquaternion and numpy_quaternion if you wish to see those comparisons.

Building Documentation

You can also build this documentation from source if you clone the repository. The documentation is written in reStructuredText and compiled using Sphinx. To build from source, first install Sphinx:

pip install sphinx sphinx_rtd_theme

You can then use Sphinx to create the actual documentation in either PDF or HTML form by running the following commands in the rowan root directory:

cd doc
make html # For html output
make latexpdf # For a LaTeX compiled PDF file
open build/html/index.html

Support and Contribution

This package is hosted on GitHub. Please report any bugs or problems that you find on the issue tracker.

All contributions to rowan are welcomed via pull requests! Please see the development guide for more information on requirements for new code.

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