unit hyperbola


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

x2-y2=1

of the hyperbolaPlanetmathPlanetmath

x2a2-y2b2=1

where both the a and the b have equal to 1.  The unit hyperbola is rectangular, i.e. its asymptotes (y=±x) are at right anglesMathworldPlanetmathPlanetmath to each other.

The unit hyperbola has the well-known parametric

x=±cosht,y=sinht,

and also a trigonometric

x=sect,y=tant.

The former yields the rational

x=u2+12u,y=u2-12u

when one substitutes  et=u, and the latter, via the substitutiontant2=u, the rational

x=1+u21-u2,y=2u1-u2

(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