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| Title of object: |
unit hyperbola |
| Canonical Name: |
UnitHyperbola |
| Type: |
Definition |
| Created on: |
2004-07-12 18:43:06 |
| Modified on: |
2004-07-13 04:19:44 |
| Classification: |
msc:51N20 |
Preamble:
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Content:
The {\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 {\em transversal semiaxis} $a$ and the {\em conjugate semiaxis} $b$ have length equal to 1. \,The unit hyperbola is {\em rectangular} (i.e. the asymptotes \,$x^2-y^2 = 0$\, are at right angles to each other).
The unit hyperbola has the well-known parametric representation
$$x = \pm\cosh{t}, \quad y = \sinh{t},$$
also a trigonometric
$$x = \sec{t}, \quad y = \tan{t}$$
and a rational parametric representation
$$x = \frac{t^2-1}{2t}, \quad y = \frac{t^2+1}{2t}.$$ |
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