Rotation.as_davenport(self, axes, order, degrees=False)#

Represent as Davenport angles.

Any orientation can be expressed as a composition of 3 elementary rotations.

For both Euler angles and Davenport angles, consecutive axes must be are orthogonal (axis2 is orthogonal to both axis1 and axis3). For Euler angles, there is an additional relationship between axis1 or axis3, with two possibilities:

  • axis1 and axis3 are also orthogonal (asymmetric sequence)

  • axis1 == axis3 (symmetric sequence)

For Davenport angles, this last relationship is relaxed [1], and only the consecutive orthogonal axes requirement is maintained.

A slightly modified version of the algorithm from [2] has been used to calculate Davenport angles for the rotation about a given sequence of axes.

Davenport angles, just like Euler angles, suffer from the problem of gimbal lock [3], where the representation loses a degree of freedom and it is not possible to determine the first and third angles uniquely. In this case, a warning is raised, and the third angle is set to zero. Note however that the returned angles still represent the correct rotation.

axesarray_like, shape (3,) or ([1 or 2 or 3], 3)

Axis of rotation, if one dimensional. If two dimensional, describes the sequence of axes for rotations, where each axes[i, :] is the ith axis. If more than one axis is given, then the second axis must be orthogonal to both the first and third axes.


If it belongs to the set {‘e’, ‘extrinsic’}, the sequence will be extrinsic. If if belongs to the set {‘i’, ‘intrinsic’}, sequence will be treated as intrinsic.

degreesboolean, optional

Returned angles are in degrees if this flag is True, else they are in radians. Default is False.

anglesndarray, shape (3,) or (N, 3)

Shape depends on shape of inputs used to initialize object. The returned angles are in the range:

  • First angle belongs to [-180, 180] degrees (both inclusive)

  • Third angle belongs to [-180, 180] degrees (both inclusive)

  • Second angle belongs to a set of size 180 degrees, given by: [-abs(lambda), 180 - abs(lambda)], where lambda is the angle between the first and third axes.



Shuster, Malcolm & Markley, Landis. (2003). Generalization of the Euler Angles. Journal of the Astronautical Sciences. 51. 123-132. 10.1007/BF03546304.


Bernardes E, Viollet S (2022) Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE 17(11): e0276302. 10.1371/journal.pone.0276302


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

Davenport angles are a generalization of Euler angles, when we use the canonical basis axes:

>>> ex = [1, 0, 0]
>>> ey = [0, 1, 0]
>>> ez = [0, 0, 1]

Represent a single rotation:

>>> r = R.from_rotvec([0, 0, np.pi/2])
>>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True)
array([90.,  0.,  0.])
>>> r.as_euler('zxy', degrees=True)
array([90.,  0.,  0.])
>>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape

Represent a stack of single rotation:

>>> r = R.from_rotvec([[0, 0, np.pi/2]])
>>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True)
array([[90.,  0.,  0.]])
>>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape
(1, 3)

Represent multiple rotations in a single object:

>>> r = R.from_rotvec([
... [0, 0, 90],
... [45, 0, 0]], degrees=True)
>>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True)
array([[90.,  0.,  0.],
       [ 0., 45.,  0.]])
>>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape
(2, 3)