Fixed axis vs. fixed point The end result of any sequence of rotations of any object in 3D about a fixed point is always equivalent to a rotation about an axis. However, an object may physically rotate in 3D about a fixed point on more than one axis simultaneously, in which case there is no single fixed axis of rotation - just the fixed point. However, these two descriptions can be reconciled - such a physical motion can always be re-described in terms of a single axis of rotation, provided the orientation of that axis relative to the object is allowed to change moment by moment. Axis of 2 dimensional rotations 2 dimensional rotations, unlike the 3 dimensional ones, possess no axis of rotation. This is equivalent, for linear transformations, with saying that there is no direction in the plane which is kept unchanged by a 2 dimensional rotation, except, of course, the identity. The question of the existence of such a direction is the question of existence of an eigenvector for the matrix A representing the rotation. Every 2D rotation around the origin through an angle in counterclockwise direction can be quite simply represented by the following matrix: A standard eigenvalue determination leads to the characteristic equation , which has as its eigenvalues. Therefore, there is no real eigenvalue whenever , meaning that no real vector in the plane is kept unchanged by A. Rotation angle and axis in 3 dimensions Knowing that the trace is an invariant, the rotation angle for a proper orthogonal 3x3 rotation matrix is found by Using the principal arc-cosine, this formula gives a rotation angle satisfying . The corresponding rotation axis must be defined to point in a direction that limits the rotation angle to not exceed 180 degrees. (This can always be done because any rotation of more than 180 degrees about an axis can always be written as a rotation having if the axis is replaced with .) Every proper rotation in 3D space has an axis of rotation, which is defined such that any vector that is aligned with the rotation axis will not be affected by rotation. Accordingly, , and the rotation axis therefore corresponds to an eigenvector of the rotation matrix associated with an eigenvalue of 1. As long as the rotation angle is nonzero (i.e., the rotation is not the identity tensor), there is one and only