I googled it and wikipidead it too : but apparently there is no definition of an ideal triangle on a punctured torus ( i.e a compact [ hyperbolic ] surface with one genus and one boundary component, or torus minus an open disk ). How do you define it ? As I know in the case on just one point removed off the torus, we take the standard tringulation in the rectangle forming the fundamental domain of torus, remove the four vertices off the rectangle and consider the quotient by a $ Z+Z $ action.

So, in this case, should we remove a quartar of a disk from each corner of the rectangle ( fundamental domain or universal cover ) and consider its quotient by a $ Z + Z $ action ?