unit hyperbolaUnitHyperbola2004-07-12 18:43:062007-08-25 09:14:50Definitionhyperbola% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe {\em 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 \PMlinkescapetext{{\em transverse semiaxis}} $a$ and the \PMlinkescapetext{{\em conjugate semiaxis}} $b$ have \PMlinkescapetext{length} equal to 1.\, The unit hyperbola is {\em rectangular}, i.e. its asymptotes ($y = \pm x$) are at right angles to each other.
\begin{center}
\includegraphics{unithyperola}
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The unit hyperbola has the well-known parametric \PMlinkescapetext{representation}
$$x = \pm\cosh{t}, \quad y = \sinh{t},$$
and also a trigonometric \PMlinkescapetext{representation}
$$x = \sec{t}, \quad y = \tan{t}.$$
The former yields the rational \PMlinkescapetext{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 \PMlinkescapetext{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).