  The problem of finding the number of lattice points in a triangle has a
classical solution if the lattice is $\mathbf{Z}^2$ and the vertices of the
triangle have integer valued coordinates. We consider what happens when we
replace the lattice by $(2 \mathbf{Z})^2$ instead and give an explicit formula
for the number of lattice points inside a triangle in terms of Hardy sums.
Moreover, we give a second geometric interpretation of the Hardy sums as signed
intersection numbers with a certain oriented net of geodesics. Using this
geometric realization, we prove a generalized reciprocity law for Hardy sums by
an elementary argument.
