| Version 3 |
Version 2 |
| The {\em unit hyperbola} (cf. the unit circle) is the special case |
The {\em unit hyperbola} (cf. the unit circle) is the special case |
| $$x^2-y^2 = 1$$ |
$$x^2-y^2 = 1$$ |
| of the hyperbola |
of the hyperbola |
| $$\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$$ |
$$\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$$ |
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where both the transversal semiaxis $a$ and the conjugate semiaxis $b$ have the length equal to 1. \,The unit hyperbola is rectangular (i.e. the asymptotes \,$x^2-y^2 = 0$\, are at right angles to each other).
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where both the transversal semiaxis $a$ and the conjugate semiaxis $b$ have the length equal to 1. \,The unit hyperbola is rectangular (i.e. the asymptotes are at right angles to each other).
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| The unit hyperbola has the well-known parametric representation |
The unit hyperbola has the well-known parametric representation |
| $$x = \pm\cosh{t}, \quad y = \sinh{t},$$ |
$$x = \pm\cosh{t}, \quad y = \sinh{t},$$ |
| also a trigonometric |
also a trigonometric |
| $$x = \sec{t}, \quad y = \tan{t}$$ |
$$x = \sec{t}, \quad y = \tan{t}$$ |
| and a rational parametric representation |
and a rational parametric representation |
| $$x = \frac{t^2-1}{2t}, \quad y = \frac{t^2+1}{2t}.$$ |
$$x = \frac{t^2-1}{2t}, \quad y = \frac{t^2+1}{2t}.$$ |
