I already asked [this question](https://math.stackexchange.com/questions/2668737/what-is-an-half-cusps-in-hyperbolic-geometry) on math.stackexchange, but it was [suggested](https://math.stackexchange.com/questions/2668737/what-is-an-half-cusps-in-hyperbolic-geometry#comment5512579_2668737) that I post it here as well.


The paper [Devadoss, Heath, and Vipismakul - Deformations of bordered Riemann surfaces and associahedral polytopes](https://arxiv.org/abs/1002.1676) defines bordered Riemann surfaces with geodesic boundaries, punctures, and marked points on the boundaries.


Then it is stated that



> 
> Indeed, any stable marked bordered Riemann surface has a unique hyperbolic metric such that it is compatible with the complex structure, where all the boundary circles are geodesics, all punctures are cusps, and all boundary marked points are half cusps.
> 
> 
> 


I understand this statement for the boundaries and the punctures, but not for the marked points on the boundaries.


Can someone explain this to me, maybe with some neat example? Any reference with good examples?



What is a half cusp in hyperbolic geometry?
This means that the boundary is geodesic with cusps in the marked points.


The easiest example is a disk with 3 marked points on its boundary. In this case the hyperbolic metric is given by identification with an ideal triangle in ${\mathbb H}^2$, i.e., a triangle with its three vertices in $\partial\_\infty{\mathbb H}^2$ and the geodesics between these ideal vertices as its boundary.


