Constant of a hyperbola

Question

Hyperbolas are a companion to a circle, sharing many properties when it comes to their trig functions and equation. But, if the circle has $\pi$ as a constant relation, does a hyperbola have some constant relation as well?

Answer 1

Also for an hyperbola there is a link with the number $\pi$, but it can be see only using complex numbers.

The coordinates of a point of a circle of radius $1$ satisfy the equation $ x^2+y^2=1$ and from this we define the trigonometric functions such that $\cos^2 \theta +\sin^2 \theta=1$.

From a unit hyperbola of equation $x^2-y^2=1$ we can define ( in a symylar way) the hyperbolic functions such that $\cosh^2 \theta -\sinh^2 \theta=1$.

Now, using complex numbers we can see that such functions have a nice link with the ''magic'' numbers $\pi$, $i$ and $e$, given by;

$$ \cos^2 \theta +\sin^2 \theta=1=e^{2i\pi} $$ $$ \cosh^2 \theta -\sinh^2 \theta=1=e^{2i\pi} $$

Answer 2

Not an answer. This was my thought in college.

A real $ \theta $ makes $ 2 \pi $ in one real rotation by virtue of a real Pythagoras thm

$$ \cos^2 \theta +\sin^2 \theta=1=e^{2i\pi} $$

Since

$ \sin i \theta = $i$ \, \sinh \theta, \cos i \theta = \cosh \theta, $

An imaginary $ i \theta$ should make a full imaginary rotation by an imaginary version of Pythagoras thm:

$$ \cosh^2 \theta -\sinh^2 \theta=1=e^{2i\pi}. $$