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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 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^{}

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

and also a trigonometric representation

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

The former yields the rational representation

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

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

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

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

Related:

HyperbolicFunctions, AreaFunctions, ConjugateHyperbola

Major Section:

Reference

Type of Math Object:

Definition

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## Mathematics Subject Classification

51N20*no label found*

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