Are Your Polyhedra the Same as My Polyhedra?

Grünbaum, Branko

doi:10.1007/978-3-642-55566-4_21

Cited by 25 publications

(26 citation statements)

References 26 publications

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“…There is no single definition for a solid or a polyhedron (notice that these two terms are often used interchangeably in the scientific literature). This is best illustrated by Grünbaum () who states that even in the field of mathematics opinions differ as to what constitutes the term “polyhedron.” Some use it only for a regular polyhedron, or only for a convex one, and some consider nonplanar faces as part of the definition. De Berg et al () characterize the term as “difficult to define,” and give a simple definition that is probably the most common one: a polyhedron is a 3D solid bounded by planar faces.…”

Section: What Is a Solid? And The Implications For Its Validationmentioning

confidence: 99%

On the Validation of Solids Represented with the International Standards for Geographic Information

Ledoux

2013

Computer aided Civil Eng

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The international standards for geographic information provide unambiguous definitions of geometric primitives, with the aim of fostering exchange and interoperability in the geographical information system (GIS) community. In two dimensions, the standards are wellaccepted and there are algorithms (and implementations of these) to validate primitives, i.e. given a polygon, they ensure that it respects the standardised definition (and if it does not a reason is given to the user). However, while there exists an equivalent definition in three dimensions (for solids), it is ignored by most researchers and by software vendors. Several different definitions are indeed used, and none is compliant with the standards: e.g. solids are often defined as 2-manifold objects only, while in fact they can be non-manifold objects. Exchanging and converting datasets from one format/platform to another is thus highly problematic. I present in this paper a methodology to validate solids according to the international standards. It is hierarchical and permits us to validate the primitives of all dimensionalities. To understand and study the topological relationships between the different parts of a solid (the shells) the concept of Nef polyhedron is used. The methodology has been implemented in a prototype, and I report on the main engineering decisions that were made and on its use for the validation of real-world three-dimensional datasets.

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Section: What Is a Solid? And The Implications For Its Validationmentioning

confidence: 99%

On the Validation of Solids Represented with the International Standards for Geographic Information

Ledoux

2013

Computer aided Civil Eng

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“…According to Grünbaum (2003b), there are two distinct reasons for the failure of this idea as applied to the uniform polyhedra, to obtain isohedral ones by polarity with respect to the circumsphere:…”

Section: What Now?mentioning

confidence: 99%

Polygons: Meister was right and Poinsot was wrong but prevailed

Grünbaum

2011

Beitr Algebra Geom

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The definitions of the term "polygon" as given and used by Meister (1724Meister ( -1788 in 1770 and by Poinsot (1777Poinsot ( -1859 in 1810 are discussed. Since it is accepted that mathematicians are free to define concepts whichever way they like, the claim that one of them is right and the other wrong may appear strange. The following pages should justify the assertion of the title by pointing out some of the errors and inconsistencies in Poinsot's work, and-more importantly-show the undesirable and harmful consequences resulting from it.

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“…Конвексни полиедри и полиедарске структуре (Платонова и Архимедова тела, фамилија призми и антипризми, Џонсонова тела и варијације Џонсонових тела) су већ испитани [2], [4], [11], [12], [13], [14], [32], [57], [63], [67], [69].…”

Section: примарни извориunclassified

Constructive-geometric generating of cupolae with concave polyhedral surfaces

Mišić¹

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Are Your Polyhedra the Same as My Polyhedra?

Cited by 25 publications

References 26 publications

On the Validation of Solids Represented with the International Standards for Geographic Information

On the Validation of Solids Represented with the International Standards for Geographic Information

Polygons: Meister was right and Poinsot was wrong but prevailed

Constructive-geometric generating of cupolae with concave polyhedral surfaces

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