Hyperbola - Hyperbolic Functions and Equations | Hyperbolic Functions Equations

Hyperbolic Functions and Equations

Just as the sine and cosine functions give a parametric equation for the ellipse, so the hyperbolic sine and hyperbolic cosine give a parametric equation for the hyperbola.

$\cosh^2 \mu - \sinh^2 \mu= 1$

one has for any value of that the point

$x = a\ \cosh\ \mu$

$y = b\ \sinh\ \mu$

satisfies the equation

which is the equation of a hyperbola relative its canonical coordinate system.

When μ varies over the interval one gets with this formula all points on the right branch of the hyperbola.

The left branch for which is in the same way obtained as

$x = -a\ \cosh\ \mu$

$y = b\ \sinh\ \mu$

In the figure the points given by

$x_k = -a\ \cosh \mu _k$

$y_k = b\ \sinh \mu _k$

for

on the left branch of a hyperbola with eccentricity 1.2 are marked as dots.

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