blender/doc/python_api/examples/mathutils.py

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import mathutils
from math import radians
vec = mathutils.Vector((1.0, 2.0, 3.0))
mat_rot = mathutils.Matrix.Rotation(radians(90.0), 4, 'X')
mat_trans = mathutils.Matrix.Translation(vec)
mat = mat_trans * mat_rot
mat.invert()
mat3 = mat.to_3x3()
quat1 = mat.to_quaternion()
quat2 = mat3.to_quaternion()
angle = quat1.difference(quat2)
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print(angle)