The unit hyperbola (cf. the unit circle) is the special case

of the hyperbola

where both the transverse semiaxis $a$ and the conjugate semiaxis $b$ have length equal to 1.  The unit hyperbola is rectangular, i.e. its asymptotes ($y=\pm x$) are at right angles to each other.

The unit hyperbola has the well-known parametric representation

and also a trigonometric representation

The former yields the rational representation

when one substitutes  $e^{t}=u$, and the latter, via the substitution  $\tan\frac{t}{2}=u$, the rational representation

(which does not give the left apex of the hyperbola).