The four Euclidean coordinates for are redundant since they are subject to the condition that . As a 3-dimensional manifold one should be able to parameterize by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as
latitude and
longitude). Due to the nontrivial topology of it is impossible to find a single set of coordinates that cover the entire space. Just as on the 2-sphere, one must use
at least two
coordinate charts. Some different choices of coordinates are given below.
Hyperspherical coordinates It is convenient to have some sort of
hyperspherical coordinates on in analogy to the usual
spherical coordinates on . One such choice — by no means unique — is to use , where :\begin{align} x_0 &= r\cos\psi \\ x_1 &= r\sin\psi \cos\theta \\ x_2 &= r\sin\psi \sin\theta \cos \varphi \\ x_3 &= r\sin\psi \sin\theta \sin\varphi \end{align} where and run over the range 0 to , and runs over 0 to 2. Note that, for any fixed value of , and parameterize a 2-sphere of radius r\sin\psi, except for the degenerate cases, when equals 0 or , in which case they describe a point. The
round metric on the 3-sphere in these coordinates is given by :ds^2 = r^2 \left[ d\psi^2 + \sin^2\psi\left(d\theta^2 + \sin^2\theta\, d\varphi^2\right) \right] and the
volume form by :dV =r^3 \left(\sin^2\psi\,\sin\theta\right)\,d\psi\wedge d\theta\wedge d\varphi. These coordinates have an elegant description in terms of
quaternions. Any unit quaternion can be written as a
versor: :q = e^{\tau\psi} = \cos\psi + \tau\sin\psi where is a
unit imaginary quaternion; that is, a quaternion that satisfies . This is the quaternionic analogue of
Euler's formula. Now the unit imaginary quaternions all lie on the unit 2-sphere in so any such can be written: :\tau = (\cos\theta) i + (\sin\theta\cos\varphi) j + (\sin\theta\sin\varphi) k With in this form, the unit quaternion is given by :q = e^{\tau\psi} = x_0 + x_1 i + x_2 j + x_3 k 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 of to and then compressing to a ball. This image shows points on and their corresponding fibers with the same color. For unit radius another choice of hyperspherical coordinates, , makes use of the embedding of in . In complex coordinates we write :\begin{align} z_1 &= e^{i\,\xi_1}\sin\eta \\ z_2 &= e^{i\,\xi_2}\cos\eta. \end{align} This could also be expressed in as :\begin{align} x_0 &= \cos\xi_1\sin\eta \\ x_1 &= \sin\xi_1\sin\eta \\ x_2 &= \cos\xi_2\cos\eta \\ x_3 &= \sin\xi_2\cos\eta. \end{align} 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 :S^1 \to S^3 \to S^2.\, '' case. 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 :ds^2 = d\eta^2 + \sin^2\eta\,d\xi_1^2 + \cos^2\eta\,d\xi_2^2 and the volume form by :dV = \sin\eta\cos\eta\,d\eta\wedge d\xi_1\wedge d\xi_2. To get the interlocking circles of the
Hopf fibration, make a simple substitution in the equations above :\begin{align} z_1 &= e^{i\,(\xi_1+\xi_2)}\sin\eta \\ z_2 &= e^{i\,(\xi_2-\xi_1)}\cos\eta. \end{align} 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 :p = \left(\frac{1-\|u\|^2}{1+\|u\|^2}, \frac{2\mathbf{u}}{1+\|u\|^2}\right) = \frac{1+\mathbf{u}}{1-\mathbf{u}} 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 :\mathbf{u} = \frac{1}{1+x_0}\left(x_1, x_2, x_3\right). We could just as well have projected from the point , in which case the point is given by :p = \left(\frac{-1+\|v\|^2}{1+\|v\|^2}, \frac{2\mathbf{v}}{1+\|v\|^2}\right) = \frac{-1+\mathbf{v}}{1+\mathbf{v}} where is another vector in . The inverse of this map takes to :\mathbf{v} = \frac{1}{1-x_0}\left(x_1,x_2,x_3\right). 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 :\mathbf{v} = \frac{1}{\|u\|^2}\mathbf{u} and vice versa. ==Group structure==