A New Algorithm for Discrete Area of Convex Polygons with Rational Vertices

Henri-Alex Esbelin (1)

1 LIMOS - Laboratoire d'Informatique, de Modélisation et d'optimisation des Systèmes

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 $\frac{a}{b}$ .

Mots clés: counting; lattice points; rational polygon; Klein polyhedra

Domaines: Mathématique discrète [cs.DM]

Henri-Alex Esbelin. A New Algorithm for Discrete Area of Convex Polygons with Rational Vertices. 2012. ⟨hal-00751492v1⟩