Blender

Rotation formalisms

See Rotation formalisms in three dimensions. Blender’s user interface only allows for rotations to be entered as as Quaternions, Euler rotations, or an Euler axis-angle. If you have a rotation in some other form, how do you enter it in Blender? See Rotation — SciPy Manual:

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

def show_rot(rot: R):
    print('Matrix:')
    display(rot.as_matrix())
    print('\nBlender "Quaternion":')
    display(rot.as_quat(canonical=True, scalar_first=True))
    print('\nBlender "XYZ Euler":')
    display(rot.as_euler('xyz', degrees=True))
    print('\nBlender "Axis Angle":')
    rot_vec = rot.as_rotvec(degrees=True)
    norm = np.linalg.norm(rot_vec)
    display(norm, rot_vec / norm)

show_rot(R.from_matrix([
[ 0.000, -0.643, -0.766],
[ 0.000, -0.766,  0.643],
[-1.000,  0.000,  0.000],]))

Matrix:

array([[-2.77555756e-17, -6.42962682e-01, -7.65897506e-01],
       [ 0.00000000e+00, -7.65897506e-01,  6.42962682e-01],
       [-1.00000000e+00,  0.00000000e+00, -2.77555756e-17]])

Blender "Quaternion":

array([ 0.2419207 , -0.66443538,  0.2419207 ,  0.66443538])

Blender "XYZ Euler":

/opt/conda/lib/python3.12/site-packages/IPython/core/interactiveshell.py:3577: UserWarning: Gimbal lock detected. Setting third angle to zero since it is not possible to uniquely determine all angles.
  exec(code_obj, self.user_global_ns, self.user_ns)

array([-139.98690433,   90.        ,    0.        ])

Blender "Axis Angle":

np.float64(152.00014158445592)

array([-0.68477595,  0.24932669,  0.68477595])

show_rot(R.from_matrix([
[ 0.000,  0.000,  1.000],
[-1.000,  0.000,  0.000],
[ 0.000, -1.000,  0.000],]))

Matrix:

array([[ 0.,  0.,  1.],
       [-1.,  0.,  0.],
       [ 0., -1.,  0.]])

Blender "Quaternion":

array([ 0.5, -0.5,  0.5, -0.5])

Blender "XYZ Euler":

array([-90.,   0., -90.])

Blender "Axis Angle":

np.float64(119.99999999999999)

array([-0.57735027,  0.57735027, -0.57735027])

See also Rotation Modes - Blender 4.5 LTS Manual. Many Blender tutorials try to explain Gimbal lock, but the best references still seem to be Euler (gimbal lock) Explained - YouTube and Gimbal lock - Wikipedia.

In Eigen, quaternion-vector multiplication is just multiplication. In Eigen the coeffs command produces XYZW, but internally the data is XYZW. If you install the Eigen pretty printers, it’s explicit:

$1 = Eigen::Quaternion<double> (data ptr: 0x7fffffffd7d0) = {[x] = 0, [y] = 0.7071067811865841, [z] = 0, [w] = 0.70710678118651094}