Quaternion to Euler Angle Conversion for Arbitrary Rotation Sequence Using Geometric Methods

Nomenclature = normalized Euler rotation axis i<1-3> = indices of first, second and third Euler rotation, e.g. if rotation sequence is 3-2-1, i1 is 3, i2 is 2, and i3 is 1 i<1-3>n = next circular index following i<1-3>, e.g. if i2 = 3, i2n = 1 i<1-3>nn = next circular index following i<1-3>n, e.g. if i2n = 3, i2nn = 2 = Quaternion = = operator: rotation of vector by a quaternion vi = unit vector along the i Euler rotation axis vin = unit vector along the in Euler rotation axis = a general 3 component vector; if used in quaternion multiplication, augmented with a fourth element equal to zero = vector cross product operator <1-3> = Euler angles = Euler rotation angle = quaternion multiplication * = superscript operator: conjugate = vector dot product operator