= Hyperbolic parametrization of a hyperbola
{title2=$(x,y)=(a\cosh t,b\sinh t),\quad x^2/a^2-y^2/b^2=1,\quad x/a\geq1$}
For nonzero real $a,b$, the identity $\cosh^2t-\sinh^2t=1$ eliminates the parameter. The additional condition $x/a=\cosh t\geq1$ is essential: the parametrization traces only the branch with the sign of $a$. Every point of that branch is obtained by $t=\operatorname{arsinh}(y/b)$. Its vertex is $(a,0)$ and its asymptotes are $y=\pm(b/a)x$. An algebraic equation alone would also include the opposite branch, which this real parametrization never reaches.
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