A new algorithm is presented, which computes the number of lattice points lying inside a convex plane polygon from the sequence of the rational coordinates of its vertices. It reduces the general case in a natural way to a fondamental one, namely a triangle with vertices of coordinates $\{(0;0),(n;0),(n;n\frac{a}{b})\}$, where $n$, $a$ and $b$ are positive natural integers. Then it evaluates the discrete area of such a triangle using the Klein polyhedron of slope $\frac{a}{b}$ and the Ostrowski representation of $n$ with the numeration scale of denominators of the convergents of the continued fraction expansion of $\frac{a}{b}$ .