# 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

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

and also a trigonometric

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

The former yields the rational

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

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

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

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

Title | unit hyperbola |
---|---|

Canonical name | UnitHyperbola |

Date of creation | 2015-02-04 11:10:22 |

Last modified on | 2015-02-04 11:10:22 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 24 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 51N20 |

Related topic | HyperbolicFunctions |

Related topic | AreaFunctions |

Related topic | ConjugateHyperbola |