Higher Dimensions
It is possible to define parameters analogous to the Euler angles in dimensions higher than three.
The number of degrees of freedom of a rotation matrix is always less than the dimension of the matrix squared. That is, the elements of a rotation matrix are not all completely independent. For example, the rotation matrix in dimension 2 has only one degree of freedom, since all four of its elements depend on a single angle of rotation. A rotation matrix in dimension 3 (which has nine elements) has three degrees of freedom, corresponding to each independent rotation, for example by its three Euler angles or a magnitude one (unit) quaternion.
In SO(4) the rotation matrix is defined by two quaternions, and is therefore 6-parametric (three degrees of freedom for every quaternion). The 4x4 rotation matrices have therefore 6 out of 16 independent components.
Any set of 6 parameters that define the rotation matrix could be considered an extension of Euler angles to dimension 4.
In general, the number of euler angles in dimension D is quadratic in D; since any one rotation consists of choosing two dimensions to rotate between, the total number of rotations available in dimension D is, which for D=2,3,4 yields .
Read more about this topic: Euler Angles
Famous quotes containing the words higher and/or dimensions:
“Dreams have a poetic integrity and truth. This limbo and dust- hole of thought is presided over by a certain reason, too. Their extravagance from nature is yet within a higher nature.”
—Ralph Waldo Emerson (1803–1882)
“Is it true or false that Belfast is north of London? That the galaxy is the shape of a fried egg? That Beethoven was a drunkard? That Wellington won the battle of Waterloo? There are various degrees and dimensions of success in making statements: the statements fit the facts always more or less loosely, in different ways on different occasions for different intents and purposes.”
—J.L. (John Langshaw)