By Guillaume Damiand, Pascal Lienhardt.

"Although they're much less widely recognized than different types, combinatorial maps are very robust info constructions and will be valuable in lots of functions, together with special effects and snapshot processing. The ebook introduces those info constructions, describes algorithms and information buildings linked to them, makes connections to different universal buildings, and demonstrates how one can use those buildings in geometric modeling Read more...

Introduction Subdivisions of Geometric gadgets specific Representations of Subdivisions a number of constructions mobile buildings initial Notions easy Topological Notions Paper Surfaces type of Paper Surfaces Manifolds, Quasi-Manifolds, Pseudo-Manifolds, Complexes Discrete constructions occurrence Graphs Intuitive Presentation n-maps n-Gmaps n-Gmaps uncomplicated Definitions simple Operations Completeness, Multi-Incidence information buildings, Iterators, and Algorithms enhances n-maps simple Definitions easy Operations Completeness, Multi-Incidence information buildings, Iterators, and Algorithms enhances Operations Closure elimination Contraction Insertion enlargement Chamfering Extrusion Triangulation Embedding for Geometric Modeling and photograph Processing Embedding Geometric Modeling snapshot Processing mobile buildings as dependent Simplicial constructions Simplicial buildings Numbered Simplicial constructions and mobile constructions a few results comparability with different mobile facts buildings heritage of Combinatorial Maps orientated mobile Quasi-Manifolds Orientable and Nonorientable mobile Quasi-Manifolds Concluding feedback Bibliography Index

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Surface boundary A free edge is incident to a single face; it is sewn2 when it is incident to two faces. The boundaries of a subdivided surface are made by the free edges: moreover, they are cycles of free edges. When a surface has no free edge, it is without boundary, for instance the surface of a cube, made by gluing six square 2 The use of the word “sew” comes from the fact that, for constructing paper surfaces, instead of gluing faces by sticking two edges into one edge, it is also possible to sew the faces together around the resulting edge.

Denote subclasses of combinatorial maps. 4 The basics of incidence graphs are more related to our usual intuition about subdivisions, although this intuition can be misleading, for instance when multi-incidence occurs. Introduction 13 • efficient data structures have been conceived, based on combinatorial maps, together with operations for handling these structures, for geometric modeling, computational geometry, discrete geometry, computer graphics, image processing and analysis purposes; based on such structures and operations, kernels of geometric modeling softwares can be conceived for handling subdivisions; • several specialized classes of combinatorial maps can be deduced in order to represent subdivisions satisfying some structural properties, illustrating optimization mechanisms; • the formal links between combinatorial maps and ordered models, combinatorial maps and incidence graphs, combinatorial maps and simplicial structures, are well-known; it is thus possible to conceive conversion operations between all these structures.

Figs. 20(d), (e) and (f). Note that we can define several notions as that of free cells, sewn cells, boundaries of quasimanifolds, as for the 2-dimensional case. This construction of quasi-manifolds can be generalized for taking into account more general cells: cf. chapter 8. e. curves and surfaces: there is no difference between these two notions for dimensions up to 2. Let us now define a n-dimensional manifold without boundary as an ndimensional quasi-manifold without boundary, such that each point has a neighborhood homeomorphic to an n-dimensional open ball (this definition can be extended for manifolds with boundaries).