Quaternionic arrays

This module subclasses numpy's array type, interpreting the array as an array of quaternions, and accelerating the algebra using numba. This enables natural manipulations, like multiplying quaternions as a*b, while also working with standard numpy functions, as in np.log(q). There is also basic initial support for symbolic manipulation of quaternions by creating quaternionic arrays with sympy symbols as elements, though this is a work in progress.

This package has evolved from the quaternion package, which adds a quaternion dtype directly to numpy. In some ways, that is a better approach because dtypes are built in to numpy, making it more robust than this package. However, that approach has its own limitations, including that it is harder to maintain, and requires much of the code to be written in C, which also makes it harder to distribute. This package is written entirely in python code, but should actually have comparable performance because it is compiled by numba. Moreover, because the core code is written in pure python, it is reusable for purposes other than the core purpose of this package, which is to provide the numeric array type.

Installation

Because this package is pure python code, installation is very simple. In particular, with a reasonably modern installation, you can just run

bash conda install -c conda-forge quaternionic

or

bash python -m pip install quaternionic

These will download and install the package. (Using python -m pip instead of just pip or pip3 helps avoid problems that new python users frequently run into; the reason is explained by a veteran python core contributor here.)

You can also install the package from source if you have pip version 10.0 or greater by running python -m pip install . — or if you have poetry by running poetry install — from the top-level directory.

Note that only python 3.8 or greater is supported. (I have also tried to support PyPy3, although I cannot test this as scipy does not currently install. Pull requests are welcome.) In any case, I strongly recommend installing by way of an environment manager — especially conda, though other managers like virtualenv or pipenv should also work.

For development work, the best current option is poetry. From the top-level directory, you can run poetry run <some command> to run the command in an isolated environment.

Usage

Basic construction

The key function is quaternionic.array, which takes nearly the same arguments as numpy.array, except that whatever array will result must have a final axis of size 4 (and the dtype must be float). As long as these conditions are satisfied, we can create new arrays or just reinterpret existing arrays:

```python import numpy as np import quaternionic

a = np.random.normal(size=(17, 11, 4)) # Just some random numbers; last dimension is 4 q1 = quaternionic.array(a) # Reinterpret an existing array q2 = quaternionic.array([1.2, 2.3, 3.4, 4.5]) # Create a new array ```

In this example, q1 is an array of 187 (17*11) quaternions, just to demonstrate that any number of dimensions may be used, as long as the final dimension has size 4.

Here, the original array a will still exist just as it was, and will behave just as a normal numpy array — including changing its values (which will change the values in q1), slicing, math, etc. However, q1 will be another "view" into the same data. Operations on q1 will be quaternionic. For example, whereas 1/a returns the element-wise inverse of each float in the array, 1/q1 returns the quaternionic inverse of each quaternion. Similarly, if you multiply two quaternionic arrays, their product will be computed with the usual quaternion multiplication, rather than element-wise multiplication of floats as numpy usually performs.

| :warning: WARNING | |:-------------------------------------------------------------------------------------------------| | Because of an unfortunate choice by the numpy developers, the np.copy function will not preserve the quaternionic nature of an array by default; the result will just be a plain array of floats. You could pass the optional argument subok=True, as in q3 = np.copy(q1, subok=True), but it's easier to just use the member function: q3 = q1.copy(). |

Algebra

All the usual quaternion operations are available, including

• Addition q1 + q2
• Subtraction q1 - q2
• Multiplication q1 * q2
• Division q1 / q2
• Scalar multiplication q1 * s == s * q1
• Scalar division q1 / s and s / q1
• Reciprocal np.reciprocal(q1) == 1/q1
• Exponential np.exp(q1)
• Logarithm np.log(q1)
• Square-root np.sqrt(q1)
• Conjugate np.conjugate(q1) == np.conj(q1)

All numpy ufuncs that make sense for quaternions are supported. When the arrays have different shapes, the usual numpy broadcasting rules take effect.

Attributes

In addition to the basic numpy array features, we also have a number of extra properties that are particularly useful for quaternions, including

• Methods to extract and/or set components
  • w, x, y, z
  • i, j, k (equivalent to x, y, z)
  • scalar, vector (equivalent to w, [x, y, z])
  • real, imag (equivalent to scalar, vector)
• Methods related to norms
  • abs (square-root of sum of squares of components)
  • norm (sum of squares of components)
  • modulus, magnitude (equal to abs)
  • absolute_square, abs2, mag2 (equal to norm)
  • normalized
  • inverse
• Methods related to array infrastructure
  • ndarray (the numpy array underlying the quaternionic array)
  • flattened (all dimensions but last are flattened into one)
  • iterator (iterate over all quaternions)

Note that this package makes a distinction between abs and norm — the latter being equal to the square of the former. This version of the norm is also known as the "Cayley" norm, commonly used when emphasizing the properties of an object in an algebra, as opposed to the "Euclidean" norm more common when emphasizing the properties of an object in a vector space — though of course, algebras are vector spaces with additional structure. This choice agrees with the Boost library's implementation of quaternions, as well as this package's forerunner quaternion. This also agrees with the corresponding functions on the C++ standard library's complex numbers. Because this may be confusing, a number of aliases are also provided that may be less confusing. For example, some people find the pair abs and abs2 (meaning the square of abs) to be more sensible.

Rotations

The most common application of quaternions is to representing rotations by means of unit quaternions. Note that this package does not restrict quaternions to have unit norms, since it is usually better for numerical purposes not to do so. For example, whereas rotation of a vector $v$ by a quaternion is usually implemented as $R\, v\, \bar{R}$, it is generally better to drop the assumption that the quaternion has unit magnitude and implement rotation as $R\, v\, R^{-1}$. This is almost always more efficient, and more accurate. That is what this package does by default whenever rotations are involved.

Although this package does not restrict to unit quaternions, there are several converters to and from other representations of rotations. First, we have

• to_vector_part, from_vector_part

These convert between the standard 3-d vector representation and their equivalent quaternions, which allows them to be manipulated as vectors — as in R * from_vector_part(v) * R.conjugate(). However, note that you may not need to convert to/from quaternions. For example, to rotate vectors v by R, you can use python R.rotate(v) It may also be relevant to consider a vector as a "generator" of rotations, in which case the actual rotation is obtained by applying exp to the generator. This does require conversion to a quaternionic array. We also have converters that deal with standard representations of rotations:

• to_rotation_matrix, from_rotation_matrix
• to_transformation_matrix (for non-unit quaternions)
• to_axis_angle, from_axis_angle
• to_euler_angles, from_euler_angles (though using Euler angles is almost always a bad idea)
• to_euler_phases, from_euler_phases (see above)
• to_spherical_coordinates, from_spherical_coordinates
• to_angular_velocity, from_angular_velocity
• to_minimal_rotation

Note that the last two items relate to quaternion-valued functions of time. Converting to an angular velocity requires differentiation, while converting from angular velocity requires integration (as explored in this paper). The "minimal rotation" modifies an input rotation-function-of-time to have the same effect on the z axis, while minimizing the amount of rotation that actually happens.

For these converters, the "to" functions are properties on the individual arrays, whereas the "from" functions are "classmethod"s that take the corresponding objects as inputs. For example, we could write

python q1 = quaternionic.array(np.random.rand(100, 4)).normalized m = q1.to_rotation_matrix

to obtain the matrix m from a quaternionic array q1. (Here, m is actually a series of 100 3x3 matrices corresponding to the 100 quaternions in q1.) On the other hand, to obtain a quaternionic array from some matrix m, we would write

python q2 = quaternionic.array.from_rotation_matrix(m)

Also note that, because the unit quaternions form a "double cover" of the rotation group (meaning that quaternions q and -q represent the same rotation), these functions are not perfect inverses of each other. In this case, for example, q1 and q2 may have opposite signs. We can, however, prove that these quaternions represent the same rotations by measuring the "distance" between the quaternions as rotations:

python np.max(quaternionic.distance.rotation.intrinsic(q1, q2)) # Typically around 1e-15

Also note the classmethod

• random

This constructs a quaternionic array in which each component is randomly selected from a normal (Gaussian) distribution centered at 0 with scale 1, which means that the result is isotropic (spherically symmetric). It is also possible to pass the normalize argument to this function, which results in truly random unit quaternions.

Distance functions

The quaternionic.distance contains four distance functions:

• rotor.intrinsic
• rotor.chordal
• rotation.intrinsic
• rotation.chordal

The "rotor" distances do not account for possible differences in signs, meaning that rotor distances can be large even when they represent identical rotations; the "rotation" functions just return the smaller of the distance between q1 and q2 or the distance between q1 and -q2. So, for example, either "rotation" distance between q and -q is always zero, whereas neither "rotor" distance between q and -q will ever be zero (unless q is zero). The "intrinsic" functions measure the geodesic distance within the manifold of unit quaternions, and is somewhat slower but may be more meaningful; the "chordal" functions measure the Euclidean distance in the (linear) space of all quaternions, and is faster but its precise value is not necessarily as meaningful.

These functions satisfy some important conditions. For each of these functions d, and for any nonzero quaternions q1 and q2, and unit quaternions q3 and q4, we have

• symmetry: d(q1, q2) = d(q2, q1)
• invariance: d(q3*q1, q3*q2) = d(q1, q2) = d(q1*q4, q2*q4)
• identity: d(q1, q1) = 0
• positive-definiteness:
  • For rotor functions d(q1, q2) > 0 whenever q1 ≠ q2
  • For rotation functions d(q1, q2) > 0 whenever q1 ≠ q2 and q1 ≠ -q2

Note that the rotation functions also satisfy both the usual identity property d(q1, q1) = 0 and the opposite-identity property d(q1, -q1) = 0.

See Moakher (2002) for a nice general discussion.

Interpolation

Finally, there are also capabilities related to interpolation, for example as functions of time:

• slerp (spherical linear interpolation)
• squad (spherical quadratic interpolation)

Caching

By default, the compiled code generated by numba is cached so that the compilation only needs to take place on the first import. If you want to disable this caching, for example in a high-performance computing environment where it may be preferable to compile the code than try to load a cache from disk, set the environment variable QUATERNIONIC_DISABLE_CACHE to 1 before importing this package.

Related packages

Other python packages with some quaternion features include

• quaternion (core written in C; very fast; adds quaternion dtype to numpy; named numpy-quaternion on pypi due to name conflict)
• clifford (very powerful; more general geometric algebras)
• rowan (many features; similar approach to this package; no acceleration or overloading)
• pyquaternion (many features; pure python; no acceleration or overloading)
• quaternions (basic pure python package; no acceleration; specialized for rotations only)
• scipy.spatial.transform.Rotation.as_quat (quaternion output for Rotation object)
• mathutils (a Blender package with python bindings)
• Quaternion (extremely limited capabilities; unmaintained)

Also note that there is some capability to do symbolic manipulations of quaternions in these packages:

• galgebra (more general geometric algebras; analogous to clifford, but for symbolic calculations)
• sympy.algebras.quaternion