# unit hyperbola

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

 $x^{2}-y^{2}=1$

of the hyperbola

 $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$

where both the $a$ and the $b$ have 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

 $x=\pm\cosh{t},\quad y=\sinh{t},$

and also a trigonometric

 $x=\sec{t},\quad y=\tan{t}.$

The former yields the rational

 $x=\frac{u^{2}+1}{2u},\quad y=\frac{u^{2}-1}{2u}$

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

 $x=\frac{1+u^{2}}{1-u^{2}},\quad y=\frac{2u}{1-u^{2}}$

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

Title unit hyperbola UnitHyperbola 2015-02-04 11:10:22 2015-02-04 11:10:22 pahio (2872) pahio (2872) 24 pahio (2872) Definition msc 51N20 HyperbolicFunctions AreaFunctions ConjugateHyperbola