I have some doubts about the application of the following. Let be the regular surface $\mathbb{H} =$ {$(x,y,z)$ $\in \mathbb{R^3} | z=0, y >0$}. For each point $p =(x,y,0) \in \mathbb{H} $, define the quadratic form $I_p^{\mathbb{H}} (v) : T_p (\mathbb{H}) \to \mathbb{R}$ as follows: \begin{equation} I_p^{\mathbb{H}}(v)= v^T \begin{pmatrix} \frac{1}{y^2} & 0 & 0 \\ 0 & \frac{1}{y^2} & 0 \\ 0 & 0 & 1 \end{pmatrix} v \end{equation} And let $\alpha : (a,b) \to \mathbb{H} $ a parametrized curve and define the function "hyperbolic arc lenght" as it follows: \begin{equation} S^{\mathbb{H}} (t) = \int_a^t \sqrt{I_{\alpha(r)}^{\mathbb{H}} (\alpha ' (r))} dr \end{equation}.
My question is: How can we prove, as in the case of the curvature of previous units in differential geometry, that the value is invariant to the parameterization of the curve? Do I use the same procedure as when I got it for normal arc length? Another question is that, how can I get the length of a segment with that array? How do I apply, for example, the points to get the distance? Excuse me for so many doubts, surely it must be trivial but I can't see it and I have to do a lot of problems using that definition