I need to prove that the connected sum of two orientable (top.) manifolds is orientable. To do so, I need to find an atlas for the connected sum, and to find it, I need to provide an explicit orientation preserving homeomorphism between half closed disk and waxing moon-shaped surface (see figure)
Obviously they are homeo, but I cannot find an explicit homeo between the two, I find some problems even in finding a kind of parametrisation of the waxing-moon surface.
I tried with cartesian and polar coordinate, but once I found two separate parametrisations for the arcs, I do not know how to combine them in an unique parametrisation
Can somebody provide any hints?
addendum the upper arc of the waxing moon is a circular arc
