Quaternions in numpy
This Python module adds a quaternion dtype to NumPy.
The code was originally based on code by Martin Ling (which he wrote with help from Mark Wiebe), but has been rewritten with ideas from rational to work with both python 2.x and 3.x (and to fix a few bugs), and greatly expands the applications of quaternions.
conda install -c conda-forge quaternion
pip install --user numpy numpy-quaternion
The basic requirements for this code are reasonably current versions
numpy. In particular,
python versions 2.7, 3.6,
and 3.7 are routinely tested. Also, any
numpy version greater than
1.13.0 should work,
but the tests are run on the most recent release at the time of the
However, certain advanced functions in this package (including
integrate_angular_velocity, and related functions) require
scipy and can automatically use
Scipy is a standard python
package for scientific computation, and implements interfaces to C and
Fortran codes for optimization (among other things) need for finding
mean and optimal rotors.
Numba uses LLVM to
compile python code to machine code, accelerating many numerical
functions by factors of anywhere from 2 to 2000. It is possible to
run all the code without
numba, but these particular functions are
roughly 4 to 400 times slower without it.
The only drawback of
numba is that it is nontrivial to install on
its own. Fortunately, the best python installer for scientific
anaconda, makes it trivial.
Just install the main
anaconda package, which installs both
scipy. If you prefer the smaller download size of
miniconda (which comes
with minimal extras), you'll also have to run this command:
conda install numpy scipy numba
Assuming you use
conda to manage your python installation (which is
currently the preferred choice for science and engineering with
python), you can install this package simply as
conda install -c conda-forge quaternion
If you prefer to use
pip (whether or not you use
conda), you can
pip install numpy numpy-quaternion
If you refuse to use
conda, you might want to install inside your
home directory without root privileges. (Conda does this by default
anyway.) This is done by adding
--user to the above command:
pip install --user numpy numpy-quaternion
Note that pip will attempt to compile the code — which requires a
Finally, there's also the fully manual option of just downloading the code, changing to the code directory, and running
python setup.py install
This should work regardless of the installation method, as long as you
have a compiler hanging around. However, note that you will need to
have at least
numpy installed before this can compile (because
this package uses a header file provided by
The full documentation can be found on Read the Docs, and most functions have docstrings that should explain the relevant points. The following are mostly for the purposes of example.
>>> import numpy as np >>> import quaternion >>> np.quaternion(1,0,0,0) quaternion(1, 0, 0, 0) >>> q1 = np.quaternion(1,2,3,4) >>> q2 = np.quaternion(5,6,7,8) >>> q1 * q2 quaternion(-60, 12, 30, 24) >>> a = np.array([q1, q2]) >>> a array([quaternion(1, 2, 3, 4), quaternion(5, 6, 7, 8)], dtype=quaternion) >>> exp(a) array([quaternion(1.69392, -0.78956, -1.18434, -1.57912), quaternion(138.909, -25.6861, -29.9671, -34.2481)], dtype=quaternion)
The following ufuncs are implemented (which means they run fast on numpy arrays):
add, subtract, multiply, divide, log, exp, power, negative, conjugate, copysign, equal, not_equal, less, less_equal, isnan, isinf, isfinite, absolute
Quaternion components are stored as doubles. Numpy arrays with
dtype=quaternion can be accessed as arrays of doubles without any
(slow, memory-consuming) copying of data; rather, a
view of the
exact same memory space can be created within a microsecond,
regardless of the shape or size of the quaternion array.
Comparison operations follow the same lexicographic ordering as tuples.
The unary tests isnan and isinf return true if they would return true for any individual component; isfinite returns true if it would return true for all components.
Real types may be cast to quaternions, giving quaternions with zero for all three imaginary components. Complex types may also be cast to quaternions, with their single imaginary component becoming the first imaginary component of the quaternion. Quaternions may not be cast to real or complex types.
Several array-conversion functions are also included. For example, to
convert an Nx4 array of floats to an N-dimensional array of
>>> import numpy as np >>> import quaternion >>> a = np.random.rand(7, 4) >>> a array([[ 0.93138726, 0.46972279, 0.18706385, 0.86605021], [ 0.70633523, 0.69982741, 0.93303559, 0.61440879], [ 0.79334456, 0.65912598, 0.0711557 , 0.46622885], [ 0.88185987, 0.9391296 , 0.73670503, 0.27115149], [ 0.49176628, 0.56688076, 0.13216632, 0.33309146], [ 0.11951624, 0.86804078, 0.77968826, 0.37229404], [ 0.33187593, 0.53391165, 0.8577846 , 0.18336855]]) >>> qs = quaternion.as_quat_array(a) >>> qs array([ quaternion(0.931387262880247, 0.469722787598354, 0.187063852060487, 0.866050210100621), quaternion(0.706335233363319, 0.69982740767353, 0.933035590130247, 0.614408786768725), quaternion(0.793344561317281, 0.659125976566815, 0.0711557025000925, 0.466228847713644), quaternion(0.881859869074069, 0.939129602918467, 0.736705031709562, 0.271151494174001), quaternion(0.491766284854505, 0.566880763189927, 0.132166320200012, 0.333091463422536), quaternion(0.119516238634238, 0.86804077992676, 0.779688263524229, 0.372294043850009), quaternion(0.331875925159073, 0.533911652483908, 0.857784598617977, 0.183368547490701)], dtype=quaternion)
[Note that quaternions are printed with full precision, unlike floats,
which is why you see extra digits above. But the actual data is
identical in the two cases.] To convert an N-dimensional array of
quaternions to an Nx4 array of floats, use
>>> b = quaternion.as_float_array(qs) >>> b array([[ 0.93138726, 0.46972279, 0.18706385, 0.86605021], [ 0.70633523, 0.69982741, 0.93303559, 0.61440879], [ 0.79334456, 0.65912598, 0.0711557 , 0.46622885], [ 0.88185987, 0.9391296 , 0.73670503, 0.27115149], [ 0.49176628, 0.56688076, 0.13216632, 0.33309146], [ 0.11951624, 0.86804078, 0.77968826, 0.37229404], [ 0.33187593, 0.53391165, 0.8577846 , 0.18336855]])
It is also possible to convert a quaternion to or from a 3x3 array of
floats representing a rotation matrix, or an array of N quaternions to
or from an Nx3x3 array of floats representing N rotation matrices,
conversions are possible for rotation vectors using
from_rotation_vector, and for spherical
Finally, it is possible to derive the Euler angles from a quaternion
as_euler_angles, or create a quaternion from Euler angles
from_euler_angles — though be aware that Euler angles are
basically the worst things
ever.1 Before you complain
about those functions using something other than your favorite
conventions, please read this
Bug reports and feature requests
Bug reports and feature requests are entirely welcome (with very few exceptions). The best way to do this is to open an issue on this code's github page. For bug reports, please try to include a minimal working example demonstrating the problem.
Pull requests are also entirely welcome, of course, if you have an idea where the code is going wrong, or have an idea for a new feature that you know how to implement.
This code is routinely tested on recent versions of both python (2.7,
3.6, and 3.7) and numpy (>=1.13). But the test coverage is not
necessarily as complete as it could be, so bugs may certainly be
present, especially in the higher-level functions like
This code is, of course, hosted on github. Because it is an open-source project, the hosting is free, and all the wonderful features of github are available, including free wiki space and web page hosting, pull requests, a nice interface to the git logs, etc. Github user Hannes Ovrén (hovren) pointed out some errors in a previous version of this code and suggested some nice utility functions for rotation matrices, etc. Github user Stijn van Drongelen (rhymoid) contributed some code that makes compilation work with MSVC++. Github user Jon Long (longjon) has provided some elegant contributions to substantially improve several tricky parts of this code. Rebecca Turner (9999years) and Leo Stein (duetosymmetry) did all the work in getting the documentation onto Read the Docs.
Every change in this code is automatically
Travis-CI. This service integrates
beautifully with github, detecting each commit and automatically
re-running the tests. The code is downloaded and installed fresh each
time, and then tested, on each of the five different versions of
python. This ensures that no change I make to the code breaks either
installation or any of the features that I have written tests for.
Travis-CI also automatically builds the
pip versions of
the code hosted on
These are all free services for open-source projects like this one.
The work of creating this code was supported in part by the Sherman Fairchild Foundation and by NSF Grants No. PHY-1306125 and AST-1333129.
1 Euler angles are awful
Euler angles are pretty much the worst things ever and it makes me feel bad even supporting them. Quaternions are faster, more accurate, basically free of singularities, more intuitive, and generally easier to understand. You can work entirely without Euler angles (I certainly do). You absolutely never need them. But if you really can't give them up, they are mildly supported.