Another Proof of Pick's Area Theorem

Blatter, Christian

doi:10.2307/2691260

Cited by 13 publications

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“…One naturally wonders whether a radically different proof of Pick's Theorem might avoid all the difficulties associated with triangulation of polygons, perhaps using results such as the Gauss-Green theorem relating line and plane integrals. Some intriguingly different proofs are given in Kurogi and Yasukura (2005), Diaz and Robins (1995) and Blatter (1997), all of which may generate interesting formalisation challenges. It would also be natural to consider formalising extensions and generalisations of Pick's Theorem, and other related results.…”

Section: Conclusion and Related Workmentioning

confidence: 99%

A formal proof of Pick's Theorem

Harrison

2011

Math. Struct. Comp. Sci.

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Pick's Theorem relates the area of a simple polygon with vertices at integer lattice points to the number of lattice points in its inside and boundary. We describe a formal proof of this theorem using the HOL Light theorem prover. As sometimes happens for highly geometrical proofs, the formalisation turned out to be more work than initially expected. The difficulties arose mostly from formalising the triangulation process for an arbitrary polygon.

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Section: Conclusion and Related Workmentioning

confidence: 99%

A formal proof of Pick's Theorem

Harrison

2011

Math. Struct. Comp. Sci.

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“…El resultado que se alcanza es una fórmula para calcular elárea de los polígonos simples a partir de la relación de sus vértices en orden correlativo. En realidad no es un resultado original, ya que la fórmula se conoce desde hace tiempo, y habitualmente aparece con el nombre de fórmula del agrimensor [1][2][3][4][5], para la que existe una demostración muy elegante, empleando el Teorema de Green. Lo que se pretende en este trabajo no es tanto presentar el resultado matemático, sino el proceso seguido para alcanzarlo.…”

Section: Introductionunclassified

Cálculo del área de un polígono simple

Arteaga

2012

Model. sci. educ. learn.

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En el presente trabajo nos planteamos determinar la posición relativa entre un punto y un polígono simple en un plano, para lo cual nos creamos un modelo del polígono, cuya manipulación nos lleva, de una manera muy natural, a la solución de dicho problema y a plantearnos un problema más ambicioso, basado en el mismo modelo: la determinación delárea del polígono. Este ejemplo geométrico esútil para ilustrar el proceso de creación en matemáticas. Se considera de especial interés la manera en la que la formalización de un problema sencillo, con una notación afortunada, y con grandes dosis de exploración algebraica, llevan a redescubrir un interesante resultado geométrico, de mucho mayor alcance que el objetivo inicial.In the present paper we consider to determine the relative position between a point and a simple polygon in a plane. For this purpose we build a model of the polygon, which manipulation carries us, in a very natural way, to the solution of this problem and it allows us to arise a more ambitious problem based on the same model: the determination of the polygon area. This geometric example is useful to illustrate the creation process in Mathematics. It is especially interesting the way in which the formalization of a simple problem with a successful notation and a great dose of algebraic exploration results in a rediscovery of an interesting geometric result which is much more important than the original objective.

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“…Since 1960, many papers have been published concerning Pick's formula. They contain several proofs of the formula [1,3,4,11,12,16,20,27,28,29] or proof of the equivalence of this result with other ones [5,7,8,15] or generalizations to more general polygons [2,6,9,10,24,25,26,30] to more general lattices [19,23] and also to higher-dimensional polyhedra [13,21,22,25].…”

Section: Introductionmentioning

confidence: 99%

Euler's characteristics and Pick's theorem

Dubeau¹,

Labbé²

2007

ijcms

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Pick's Theorem is used to compute the area of lattice polygons. In this paper we present a very general definition of a polygon to obtain the definition of faces and holes of a polygon. The decomposition of a polygon into triangles is briefly discussed. Then lattices are introduced with their corresponding elementary triangles and triangulations. We recall and prove Pick's Theorem for simple polygons. Then a generalization of Pick's Theorem is proved for very general lattice polygons, generalized lattice polygons and union of generalized lattice polygons using original and modified Euler's characteristics.

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Another Proof of Pick's Area Theorem

Cited by 13 publications

References 0 publications

A formal proof of Pick's Theorem

A formal proof of Pick's Theorem

Cálculo del área de un polígono simple

Euler's characteristics and Pick's theorem

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