Quaternions

https://en.wikipedia.org/wiki/Quaternion

https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

Stanisław Dawidowicz, Symulacje Monte Carlo ciekłych kryształów nematycznych dwuosiowych, Uniwersytet Jagielloński, Kraków 2018. [Monte Carlo simulations of biaxial nematic liquid crystals]

INTRODUCTION

Quaternions have practical uses in calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis.

Quaternions are present in several Python libraries, but their interface may be different.

https://pypi.org/project/pyrr/

pip install pyrr

https://pypi.org/project/pyquats/

pip install pyquats

PYQUATS

import math
from pyquats.quats import Quat   # quaternions based on Python lists
#from pyquats.numpyquats import Quat   # quaternions based on numpy arrays
from pyquats.qtools import *

zero = Quat()
one = Quat(1)
ii = Quat(0, 1, 0, 0)
jj = Quat(0, 0, 1, 0)
kk = Quat(0, 0, 0, 1)
assert ii * jj == kk
assert jj * ii == -kk
assert ii.is_unit()
assert abs(ii) == 1
assert Quat(0.5, 0.5, 0.5, 0.5).is_unit()
assert ii.conjugate() == -ii
assert ii ** 2 == -one

# Quaternions and spatial_rotations.

rotZ45 = Quat.from_z_rotation(math.pi/4)
# Quat(0.923879532511, 0.0, 0.0, 0.382683432365)

vec1 = [2, 0, 0]
rotate1(vec1, rotZ45)   # [1.414213562373095, 1.4142135623730951, 0.0]

rotY90 = Quat.from_y_rotation(math.pi/2)
# Quat(0.707106781187, 0.0, 0.707106781187, 0.0)

rotate1(vec1, rotY90)   # [4.440892098500626e-16, 0.0, -2.0]