# Complete geodesics on a hyperbolic pair of pants

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 $$w_1, w_2$$.

• I don't have time right now to draw pictures, but: To see the unique geodesic from $\beta_1$ to itself, think of the pair of pants as a pair of right-angled hexagon, the saddle geodesic will be apparent. To see the spiraling geodesics, develop the pair of pants into $\mathbb{H}^2$, then you will be able to find a geodesic perpendicular to the developing image of $\beta_1$ and asymptotically approaches the developing image of $\beta_3$. This is in 3.9 of Thurston's notes. – Neal Sep 16 at 15:25
• Actually, none of these is a complete geodesic, rather, it is a geodesic ray – Moishe Kohan Sep 19 at 14:02