How to Parameterize Hyperbola

Question

I am trying to parametrize general hyperbola (shown below) using $x=t$, $y=1/t$. I tried to factor it, but I didn't get to the correct answer.

The hyperbola:

The correct answer:

thanks a lot.

Answer 1

Your equation can be factored as $$ \left({x\over a}+{y\over b}\right)\left({x\over a}-{y\over b}\right)=1. $$ Set then: $$ {x\over a}+{y\over b}=t \quad\text{and}\quad {x\over a}-{y\over b}={1\over t}. $$

Answer 2

I believe a valid parametrisation would be:

$$ x = a * cosh(t)$$ $$ y = b * sinh(t)$$ $$ t \in \mathbb{R} $$

Answer 3

EDIT1: At first consider a hyperbola rotated by $45^{0}$.

Difference of squares being unity is the common criterion. Some are simpler than the others. One possibility is

$$ x/a=\frac12 (t+1/t), y/b =\frac12(t-1/t)$$

which is constructed on Pythagorean triplets. I.e.,take

$$ (t^2+u^2, t^2-u^2, 2tu ) $$

and then divide all by $2tu$ and set $ u=1.$

The following are from circular and hyperbolic trig.

$$ x/a=\sec(t), y/b =\tan(t))$$

$$ x/a=\cosh(t), y/b =\sinh(t))$$

Then we rotate it back by $45^0$ rotation matrix.