Rotation.as_quat(self, canonical=False, *, scalar_first=False)#

Represent as quaternions.

Rotations in 3 dimensions can be represented using unit norm quaternions [1].

The 4 components of a quaternion are divided into a scalar part w and a vector part (x, y, z) and can be expressed from the angle theta and the axis n of a rotation as follows:

w = cos(theta / 2)
x = sin(theta / 2) * n_x
y = sin(theta / 2) * n_y
z = sin(theta / 2) * n_z

There are 2 conventions to order the components in a quaternion:

  • scalar-first order – (w, x, y, z)

  • scalar-last order – (x, y, z, w)

The choice is controlled by scalar_first argument. By default, it is False and the scalar-last order is used.

The mapping from quaternions to rotations is two-to-one, i.e. quaternions q and -q, where -q simply reverses the sign of each component, represent the same spatial rotation.

canonicalbool, default False

Whether to map the redundant double cover of rotation space to a unique “canonical” single cover. If True, then the quaternion is chosen from {q, -q} such that the w term is positive. If the w term is 0, then the quaternion is chosen such that the first nonzero term of the x, y, and z terms is positive.

scalar_firstbool, optional

Whether the scalar component goes first or last. Default is False, i.e. the scalar-last order is used.

quatnumpy.ndarray, shape (4,) or (N, 4)

Shape depends on shape of inputs used for initialization.



>>> from scipy.spatial.transform import Rotation as R
>>> import numpy as np

A rotation can be represented as a quaternion with either scalar-last (default) or scalar-first component order. This is shown for a single rotation:

>>> r = R.from_matrix(np.eye(3))
>>> r.as_quat()
array([0., 0., 0., 1.])
>>> r.as_quat(scalar_first=True)
array([1., 0., 0., 0.])

When multiple rotations are stored in a single Rotation object, the result will be a 2-dimensional array:

>>> r = R.from_rotvec([[np.pi, 0, 0], [0, 0, np.pi/2]])
>>> r.as_quat().shape
(2, 4)

Quaternions can be mapped from a redundant double cover of the rotation space to a canonical representation with a positive w term.

>>> r = R.from_quat([0, 0, 0, -1])
>>> r.as_quat()
array([0. , 0. , 0. , -1.])
>>> r.as_quat(canonical=True)
array([0. , 0. , 0. , 1.])