Numpy

The most important part of this package's API is actually numpy's API, because the quaternion type is added as a possible dtype for numpy arrays. This means that everything you can do with numpy arrays will also apply to quaternion arrays.

It can be helpful to think of quaternions as generalized complex numbers; all the operations that you can do with complex numbers are also available for quaternions. This includes all the usual arithmetic operations (addition, subtraction, multiplication, division, and exponentiation), as well as exponential and logarithmic functions.

Here, we will single out a few of the very most important numpy functions that are particularly useful for quaternions.

numpy.exp(q)

This returns the exponential function applied to each quaternion in the array. This is the quaternion analog of the exponential function for complex numbers. The interpretation is that if q is the axis-angle representation of a rotation, then numpy.exp(q) is the quaternion representation of the same rotation.

import numpy as np
import quaternion
q = quaternion.from_float_array([[0, np.pi/4, 0, 0], [0, 0, np.pi/2, 0]])
R = np.exp(q)
print(R)

Here, q contains two quaternions, which are the axis-angle representations of a $\pi/4$ (45°) rotation about the x-axis and a $\pi/2$ (90°) rotation about the y-axis. The output is

[quaternion(0.707107, 0.707107, 0, 0)
 quaternion(0, 0, 1, 0)]

which represent the same things as unit quaternions.

numpy.log(R)

The log function is (almost) the inverse of the exp function. In particular, we can go backwards from the R defined above to get back to q. If we evaluate

print(np.log(R))

we see that the result is precisely the same as q.

ndarray.conjugate()

Just as with complex numbers and the complex conjugate, the quaternion conjugate is extremely important.

R.conjugate()
R.conj()