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

Abstract : 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 $\frac{a}{b}$ .
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Contributor : Henri-Alex Esbelin <>
Submitted on : Thursday, February 7, 2013 - 3:28:33 PM
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Henri-Alex Esbelin. A New Algorithm for Discrete Area of Convex Polygons with Rational Vertices. 2012. ⟨hal-00751492v2⟩

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