on the 2-sphere. A point at rotated by an angle about the -axis is specified simply by . While hyperspherical coordinates are also useful in dealing with 4D rotations, an even more useful coordinate system for 4D is provided by Hopf coordinates , which are a set of three angular coordinates specifying a position on the 3-sphere. For example: Because , the points lie on the 3-sphere. In 4D space, every rotation about the origin has two invariant planes which are completely orthogonal to each other and intersect at the origin, and are rotated by two independent angles and . Without loss of generality, we can choose, respectively, the - and -planes as these invariant planes. A rotation in 4D of a point through angles and is then simply expressed in Hopf coordinates as . Visualization of 4D rotations Every rotation in 3D space has a fixed axis unchanged by rotation. The rotation is completely specified by specifying the axis of rotation and the angle of rotation about that axis. Without loss of generality, this axis may be chosen as the -axis of a Cartesian coordinate system, allowing a simpler visualization of the rotation. In 3D space, the spherical coordinates may be seen as a parametric expression of the 2-sphere. For fixed they describe circles on the 2-sphere which are perpendicular to the -axis and these circles may be viewed as trajectories of a point on the sphere. A point on the sphere, under a rotation about the -axis, will follow a trajectory as the angle varies. The trajectory may be viewed as a rotation parametric in time, where the angle of rotation is linear in time: , with being an "angular velocity". Analogous to the 3D case, every rotation in 4D space has at least two invariant axis-planes which are left invariant by the rotation and are completely orthogonal (i.e. they intersect at a point). The rotation is completely specified by specifying the axis planes and the angles of rotation about them. Without loss of generality, these axis planes may be chosen to be the - and -planes of a Cartesian coordinate system, allowing a simpler visualization of the rotation. In 4D space, the Hopf angles parameterize the 3-sphere. For fixed they describe a torus parameterized by and , with being the special case of the Clifford torus in the - and -planes. These tori are not the usual