# Complete geodesics on hyperbolic a pair of pants

I have asked this question on MSE. But I think Mo is a better place to ask my question. Here is the link to my question on MSE. I will rewrite it here:

I am trying to understand the article by Maryam Mirzakhani about Simple geodesics and Weil-Petersson volumes. In the third section of this article, the following proposition is stated. And I want to know why this proposition is correct. I mention the proposition:

Proposition: Let $$P$$ be a hyperbolic pair of pants with (non-empty)geodesic boundary components $$\beta_1 , \beta_2, \beta_3$$ of lenghts $$x_1, x_2, x_3$$ respectively. Then $$P$$ contains $$5$$ complete geodesics disjoint from $$\beta_2 , \beta_3$$ and orthogonal to $$\beta_1$$. More precisley, two of these geodesics meet $$\beta_1$$ respectively at $$y_1,y_2$$ and spiral to $$\beta_3$$, the other two meet $$\beta_1$$ respectively at $$z_1,z_2$$ and spiral to $$\beta_2$$. there is also a unique common geodesic perpendicular from $$\beta_1$$ to itself meeting $$\beta_1$$ perpendicularly at two points $$w_1, w_2$$.

• Very nice explanation, but the picture is a bit sloppy. If someone unfamiliar with hyperbolic geometry sees it, they might get wrong ideas. For example, two geodesics perpendicular to the geodesic $\beta_1$ cannot have a common endpoint at infinity unless they are equal. The shaded area seems to be a fundamental domain for the pair of pants with ends attached to them (like a "pair of flares", to stay in jargon). – Sebastian Goette Sep 16 '20 at 19:49