**Polar Coordinates:**

$$\mathbf{x}\_0 = \cosh \upsilon,\tag{56a}$$

*x*1= sinh *v* sin *ψ*, (*v* 0, *ψ* ∈ [0, <sup>2</sup>*π*)), (56b)

$$\mathbf{x}\_2 = \sinh \upsilon \cos \psi. \tag{56c}$$

It is easy to verify that *x*20 − *x*21 − *x*22 = cosh<sup>2</sup> *v* − sinh<sup>2</sup> *v* = 1, and *x*0 1.

**Figure 4.** Upper sheet <sup>2</sup>*X*+2of the real two-sheteed hyperboloid. Re *z* is directed along *u*1 and Im *z* along *u*2.
