Euler's formula. Now the unit imaginary quaternions all lie on the unit 2-sphere in so any such can be written: With in this form, the unit quaternion is given by where are as above. When is used to describe spatial rotations (cf. quaternions and spatial rotations), it describes a rotation about through an angle of . Hopf coordinates For unit radius another choice of hyperspherical coordinates, , makes use of the embedding of in . In complex coordinates we write This could also be expressed in as Here runs over the range 0 to , and and can take any values between 0 and 2. These coordinates are useful in the description of the 3-sphere as the Hopf bundle For any fixed value of between 0 and , the coordinates parameterize a 2-dimensional torus. Rings of constant and above form simple orthogonal grids on the tori. See image to right. In the degenerate cases, when equals 0 or , these coordinates describe a circle. The round metric on the 3-sphere in these coordinates is given by and the volume form by To get the interlocking circles of the Hopf fibration, make a simple substitution in the equations above In this case , and specify which circle, and specifies the position along each circle. One round trip (0 to 2) of or equates to a round trip of the torus in the 2 respective directions. Stereographic coordinates Another convenient set of coordinates can be obtained via stereographic projection of from a pole onto the corresponding equatorial hyperplane. For example, if we project from the point we can write a point in as where is a vector in and . In the second equality above, we have identified with a unit quaternion and with a pure quaternion. (Note that the numerator and denominator commute here even though quaternionic multiplication is generally noncommutative). The inverse of this map takes in to We could just as well have projected from the point , in which case the point is given by where is another vector in . The inverse of this map takes to Note that the coordinates are defined everywhere but and the coordinates everywhere but . This defines an atlas on consisting of two coordinate charts or "patches", which together cover all of . Note that the transition function between these two charts on their overlap is given by and vice versa.