PlanetMath: unit hyperbola

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unit hyperbola

(Definition)

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

$\displaystyle x^2-y^2 = 1$

of the hyperbola

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

where both the transverse semiaxis

and the conjugate semiaxis

have length equal to 1. The unit hyperbola is rectangular, i.e. its asymptotes ( $y = \pm x$ ) are at right angles to each other.

$\includegraphics{unithyperola}$

The unit hyperbola has the well-known parametric representation

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

and also a trigonometric representation

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

The former yields the rational representation

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

when one substitutes

, and the latter, via the substitution $\tan\frac{t}{2} = u$ , the rational representation

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

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

"unit hyperbola" is owned by pahio. [ full author list (2) ]

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Cross-references: apex, rational, right angles, asymptotes, hyperbola, unit circle
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This is version 20 of unit hyperbola, born on 2004-07-12, modified 2007-08-25.
Object id is 5996, canonical name is UnitHyperbola.
Accessed 8827 times total.

Classification:

51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)

Pending Errata and Addenda

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