  One of the most animated themes of multidimensional persistence is the
comparison between invariants. The matching distance between persistent Betti
numbers functions (or rank invariants), is among the most studied metrics in
this context, particularly in 2-parameter persistence. The main reason for this
interest is that, in the 2-parameter case, the foliation method allows for an
effective computation of the matching distance, based on filtering the space
along lines of positive slope. Our work provides a qualitative analysis, based
on a construction called extended Pareto grid, of the filtering lines that
actually contribute to the computation of the matching distance. Under certain
genericity assumptions, we show that these lines must either be horizontal,
vertical, of slope 1 or belong to a finite collection of special lines
associated with discontinuity phenomena.
