Non-manifold Multiresolution Modeling (some preliminary results) Leila De Floriani DISI University of Genova Genova (Italy)

Non-Manifold Multiresolution Modeling A mathematical framework for describing non-manifold d-dimensional objects as assembly of simpler components (that we call quasi-manifold ). Topological data structures for non-manifold meshes in arbitrary dimensions. Multiresolution models for meshes with a non-manifold and non-regular domain for CAD applications: Data structures Query algorithms (extract topological data structures).

Non-Manifold Multiresolution Modeling A Euclidean d-manifold M [with boundary] is a subspace of the Euclidean space Ed such that the neighborhood of any point of M is homeomorphic to a d-dimensional open ball [intersected with a closed d-dimensional half-space in Ed]. Spatial objects which do not fulfill the above condition are called non-manifolds. Spatial objects composed of parts of different dimensionalities are called non-regular.

Non-Manifold Modeling: Issues Non-manifolds are not well-understood and classified from a mathematical point of view: The class of combinatorial manifolds is not decidable in high dimensions. The homeomorphism problem is completely solved only in two dimensions All 3-manifolds can be triangulated There is no known algorithm to decide if a complex describes 5-manifold (the problem is recognizing the 4-sphere)

Non-Manifold Modeling: Issues Non-manifold cell complexes are difficult to encode and manipulate: Decomposition of a non-manifold object into manifold components done only in two dimensions Topological data structures have been proposed for two- dimensional complexes, but do not scale well with the degree of “non-manifoldness” of the complex. General, but verbose and redundant, representations: incidence graph, and chains of n-Gmaps.

Non-Manifold modeling in arbitrary dimensions A mathematical framework for describing non-manifold simplicial complexes in arbitrary dimensions as assembly of simpler quasi-manifold components (De Floriani et al., DGCI, 2002). An algorithm for decomposing a d-complex into a natural assembly of quasi-manifolds of dimension h<=d. A dimension-independent data structure for representing the resulting decomposition (on-going work).

Abstract simplicial complexes Let V be a finite set of vertices. An abstract simplicial complex on V is a subset  of the set of the non-empty part of V such that {v}  for every v  V, if g  V is an element of , then every subset of g is also an element of . Dimension of a cell g = | g|-1. Order d of a complex is the maximum of the dimensions of its cells: simplicial d-complex A d-complex in which all k-cells, k=0,1,…,d-1, are on the boundary of some d-cell is called a regular complex. An example of a non-regular complex

Pseudo-manifolds A (d-1)-cell g in a regular d-complex is a manifold cell iff there exist at most two d-cells incident at g. A regular complex which has only manifold cells is called a combinatorial pseudo-manifold. A 2-pseudo-manifold A regular 2-complex which is not a pseudo-manifold

Quasi-manifolds A regular simplicial 1-complex is a regularly-adjacent complex. A regular abstract simplicial complex is regularly-adjacent iff the link of each of its vertices is a connected regularly-adjacent (d-1)-complex . A complex is a quasi-manifold iff it is both a pseudo-manifold and a regularly-adjacent complex. A 3-pseudo-manifold which is not regularly-adjacent A 3-quasi-manifold (which is not a triangulation of a 3-manifold)

An example of a decomposition Quasi-manifolds  Triangulations of 2-manifolds in E2

Non-Manifold Multi-Triangulation (NMT) Extension of a Multi-Triangulation to deal with simplicial meshes having a non-manifold, non-regular domain Dimension-independent and application-independent definition of a modification An example of a modification

Non-Manifold Multi-Triangulation: An Example

Non-Manifold Multi-Triangulation (NMT) A compact data structure for a specific instance of the NMT in which each modification is a vertex expansion (vertex expansion = inverse of vertex-pair contraction, i.e., contraction of a pair of vertices to a new vertex). A topological data structure for non-regular, non-manifold 2D simplicial complexes, which scales to the manifold case with a small overhead Algorithms for performing vertex-pair contraction and vertex expansion (basic ingredients for performing selective refinement) on the topological representation of the complex.