Annie Lacasse

The discrete Green Theorem and some applications in discrete geometry

The discrete version of Greens Theorem and bivariate difference calculus provide a general and unifying framework for the description and generation of incremental algorithms. It may be used to compute various statistics about regions bounded by a finite and closed polygonal path. More specifically, we illustrate its use for designing algorithms computing many statistics about polyominoes, regions whose boundary is encoded by four letter words: area, coordinates of the center of gravity, moment of inertia, set characteristic function, the intersection with a given set of pixels, hook-lengths, higher order moments and also q-statistics for projections.

A note on a result of daurat and nivat

We consider paths in the square lattice and use a valuation called the winding number in order to exhibit some combinatorial properties on these paths. As a corollary, we obtain a characteristic property of self-avoiding closed paths, generalizing in this way a recent result of Daurat and Nivat (2003) on the boundary properties of polyominoes concerning salient and reentrant points.

PROPERTIES OF THE CONTOUR PATH OF DISCRETE SETS

We consider paths in the square lattice and use a valuation called the winding number in order to exhibit some combinatorial properties on these paths. As a corollary, we obtain a characteristic property of non-crossing closed paths, generalizing in this way a result of Daurat and Nivat (2003) on the boundary properties of polyominoes concerning salient and reentrant points. Moreover we obtain a similar result for hexagonal lattices and show that there is no other regular lattice having that property.

Incremental Algorithms Based on Discrete Green Theorem

By using the discrete version of Green’s theorem and bivariate difference calculus we provide incremental algorithms to compute various statistics about polyominoes given, as input, by 4-letter words describing their contour. These statistics include area, coordinates of the center of gravity, moment of inertia, higher order moments, size of projections, hook lengths, number of pixels in common with a given set of pixels and also q-statistics.

Algorithms for polyominoes based on the discrete Green theorem

The use of Greens theorem and bivariate difference calculus provides a general and unifying framework for the description and generation of incremental algorithms. The method is applied in order to provide algorithms computing various statistics about polyominoes coded by 4-letter words describing their contour. These statistics include area, coordinates of the center of gravity, moment of inertia, size of projections, hook lengths, number of pixels in common with a given set of pixels, in particular the intersection of two polyominoes and also q-statistics for projections.

A study of Jacobi-Perron boundary words for the generation of discrete planes

The construction of a Sturmian word, and thus of a discrete line, from a continued fraction development generalizes to higher dimensions. Given any vector v@?R^3, a list of 6-connected points approximating the line defined by v may be obtained via a generalized continued fraction algorithm. By duality, a discrete plane with normal vector v can also be generated using a related technique. We focus on such discrete planes, more precisely on the finite patterns generated at each step of the process. We show that the choice of the Jacobi-Perron algorithm as a higher dimension generalization of Euclids algorithm together with the specific substitutions deduced from it allows us to guaranty the simple connectedness of those patterns.

On minimal moment of inertia polyominoes

We analyze the moment of inertia I(S), relative to the center of gravity, of finite plane lattice sets S. We classify these sets according to their roundness: a set S is rounder than a set T if I(S) < I(T). We show that roundest sets of a given size are strongly convex in the discrete sense. Moreover, we introduce the notion of quasi-discs and show that roundest sets are quasi-discs. We use weakly unimodal partitions and an inequality for the radius to make a table of roundest discrete sets up to size 40. Surprisingly, it turns out that the radius of the smallest disc containing a roundest discrete set S is not necessarily the radius of S as a quasi-disc.

Discrete sets with minimal moment of inertia

We analyze the moment of inertia I(S), relative to the center of gravity, of finite plane lattice sets S. We classify these sets according to their roundness: a set S is rounder than a set T if I(S)