Quaternions are used to avoid the gimbal lock problem that occurs when a degree of freedom is lost due to a degeneracy when using these quantities.
Wigner's D-matrix formalism is parametrized in terms of these quantities. Crystallographic textures are represented by a so-called orientation distribution function, which depends directly on these quantities, and 9 equations are necessary to convert these quantities into Miller indices. A problem with the use of these quantities occurs because the map from the 3-torus to the real projective space RP3 is not a covering map. These quantities represent a choice of coordinates on the SO(3) (*) configuration space such that a problem with three degrees of freedom can be reduced to a problem of one degree of freedom when two of the degrees are made cyclic. Quaternions are used to avoid the gimbal lock problem that occurs when a degree of freedom is lost due to a degeneracy when using these quantities. These quantities are defined relative to the nodal plane, the vertical, and the horizontal plane. For 10 points, identify these quantities used to represent rotations of a rigid body such that any rotation can be expressed as a rotation by one of these quantities.
Wigner's D-matrix formalism is parametrized in terms of these quantities.
Euler angles [do not accept "Euler axes"]