I want to find the relationship between arc length $A$ of a hyperbolic function and it's corresponding horizontal location $a$ relative to the $y$ axis. In this case: the arc length is the input and $a$ is the output.
To find the arc length $A$ of a function $f$ between $x=0$ and $a$, one uses the formula:
$$A(a)=\int_0^a\sqrt{1+(\frac{d}{dx}f(x))^2}dx$$
Though, when calculating it for a hyperbolic function $\sqrt{x^2-1}$, the integral is non-elementary.
$$A(a)=\int_0^a\sqrt{1+\frac{x^2}{x^2-1}}dx$$
Yet I still want to find the inverse relationship between $A$ and $a$ (closed form or not).
i.e:
$$a(A)=?$$
Best Answer
First, you will have to start measuring from somewhere other than $x=0$ because the hyperbola does not reach $x=0$. It only covers $(-\infty,-1]\cup[1,\infty)$. It would be a little cleaner to regard $x$ as a function of $y$. The positive branch goes through $(1,0)$ and has a horizontal instead of vertical slope at the point $(1,0)$. Now the function is $x=\sqrt{1+y^2}$ Your arc length becomes $\int_0^a \sqrt{1+\frac {y^2}{y^2+1}}dy$ which Alpha can do using an elliptic integral of a hyperbolic sine.