I am currently reading the article Computing geodesics and minimal surfaces via graph cuts.I have difficulties with understanding the Cauchy-Crofton formula posed for 2D Riemannian space with continuously varying metric $D$:
$$\int \frac{\text{det}D}{2 (u^T_L \cdot D \cdot u_L )^{3/2}} n_c d \mathcal{L} = 2 |C|_{R}$$
where $u_L$ is the unit vector in the direction of the line L. $C_R$ is Riemannian length of contour $C$.
In particular I am interested in applying this formula for Poincare disc, with metric: $$ds^2 = l^2_{\text{AdS}} (d\rho ^2 + \text{sinh}^2 \rho d \theta^2) $$ here $l_{\text{AdS}}$ is a const that defines curvature of the space.
In order to check myself I am trying to evaluate length of a circle of radius $p_{*}$ which center coincides with the center of hyperbolic disc. (Following picture also represents my view on position and direction of unit vectors $u_{L}$):
These are my assumptions about this formula:
1)In my case metric D is given by metric of Poincare disc.
2) In phrase "$u_L$ is the unit vector in the direction of the line L" L is geodesic characterized by $(p; \phi); p \in (-\infty; +\infty), \phi \in (0,2\pi)$
3)By def of $u_{L}$ expression $(u^T_L \cdot D \cdot u_L) $ always equals 1. Why do we need to write it explicitly in the formula?
4) I assume that $d \mathcal{L} = dp d\phi$
5) For the circle defined above I'll have $n_c = 2, p \in (-p_{*};p_{*}), $ and $n_c = 0$ for other values of $p$ ($n_c$ is the same for all values of $\phi$)
If I plug all this information into original formula I obtain what seems to be the wrong answer.
I have also obtained length of this circle in two other ways. Both approaches give the same result which is different from aforementioned formula's result:
A) $|\gamma| = \int ds = \int \sqrt{d\rho^2 + \text{sinh}^2 \rho d\theta^2} = 2\pi \cdot \text{sinh}p_* $
B) This approach is taken from the article Integral Geometry and Holography.This is actually also Cauchy-Crofton formula, yet written in a different form: $$|\gamma| = \frac{1}{4} \int_{-p_{*}} ^{p_{*}} \int_0 ^{2\pi} 2 \text{cosh}p dp d\theta = 2\pi \cdot \text{sinh}p_* $$
Could someone give me hints about what is the correct way to understand the original formula? Or may be there are other misunderstandings which I do not realize? I would be glad to hear any feedback about the question.