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# A Mathematical Theory and Applications of Non-Manifold Geometric Modeling

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... In this paper we focus on a new invariant property, rather than data structures. Invariant formulae are typically implemented by using a more general formulation that includes non-manifold elements, such as the cusps, disks, zones, regions and walls proposed by Gursoz et al [9] or the shells, complexes, cavities and holes of various types suggested by Masuda et al [10] and others. Application of Euler characteristics and topology in design is also discussed by Lear [11] and Lee et al [4]. ...
... While works on invariants tend to provide increasingly general formulations, none has concentrated on specific modeling tasks, such as modeling of thin-walled parts. These parts are usually nonmanifold and thus comply with general formulae such as those discussed by Gursoz et al [9] and Masuda et al [10]; nonetheless, they are still confined to a relatively narrow topological domain and may therefore use simpler relationships. For example, they cannot directly include detached ('dangling') edges and vertices. ...
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This paper proposes the use of a thin-walled primitive for modeling the geometry of inherently thin objects. The authors suggest a topological invariant supporting both manifold and thin-walled non-manifold objects based on this primitive. The paper establishes a general topological invariant s + e -v -g nm + m -f = 0 regarding the number of components, edges, vertices, holes, volumes, and faces, respectively. Corresponding Euler operators are derived, providing a basis for a modeling system for thin-walled objects. The validity of the proposed invariant constitutes a necessary condition for the validity of a geometrical representation of thin-walled products from a topological point of view. The paper also discusses merging the proposed general formula with standard manifold topology. It specifically proposes the use of non-integer values in the standard Euler-PoincarF formula for representing non-manifold components, thus permitting the use of thin-walled primitives with topological coherence to traditional solid geometry schemes.
... For non-manifold models it is required to be able to construct more complex structures: to create non-manifold cell complexes the standard Euler operators should be extended to manage connections between cells and to include operations like joining two cells by a shared face, edge or vertex. Extended Euler operators for non-manifold modelling including cell complexes were proposed by Masuda [11] and Masuda et al. [17]. ...
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There is an increasing need for building models that permit interior navigation, e.g.,for escape route analysis. This paper presents a non-manifold Computer-Aided Design (CAD)data structure, the dual half-edge based on the Poincaré duality that expresses both the geometricrepresentations of individual rooms and their topological relationships. Volumes and faces areexpressed as vertices and edges respectively in the dual space, permitting a model just based onthe storage of primal and dual vertices and edges. Attributes may be attached to all of theseentities permitting, for example, shortest path queries between specified rooms, or to the exterior.Storage costs are shown to be comparable to other non-manifold models, and construction withlocal Euler-type operators is demonstrated with two large university buildings. This is intendedto enhance current developments in 3D Geographic Information Systems for interior and exteriorcity modelling. The Dual Half-Edge—A Topological Primal/Dual Data Structure and Construction Operators for Modelling and Manipulating Cell Complexes. Available from: http://www.mdpi.com/2220-9964/5/2/19
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Topologic is a software modelling library that supports a comprehensive conceptual framework for the hierarchical spatial representation of buildings based on the data structures and concepts of non-manifold topology (NMT). Topologic supports conceptual design and spatial reasoning through the integration of geometry, topology, and semantics. This enables architects and designers to reflect on their design decisions before the complexities of building information modelling (BIM) set in. We summarize below related work on NMT starting in the late 1980s, describe Topologic’s software architecture, methods, and classes, and discuss how Topologic’s features support conceptual design and spatial reasoning. We also report on a software usability workshop that was conducted to validate a software evaluation methodology and reports on the collected qualitative data. A reflection on Topologic’s features and software architecture illustrates how it enables a fundamental shift from pursuing fidelity of design form to pursuing fidelity of design intent.
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The topic of form features is arousing great interest in the field of computeraided design and manufacture. These entities are used to represent certain shape configurations, usually fairly localised, on engineering parts or subassemblies. The applications envisaged for them are essentially of an engineering nature, but it is quite possible that other uses for them will emerge in the future. Although they are being studied with very practical ends in mind, they raise many interesting issues of a theoretical nature, concerned with representation and communication in geometric modelling.
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This study proposes a non-manifold shell operation which uniformly manipulates solids and surfaces as shells, in order to enable designers to realize free generation of shapes. This report first describe definitions of shell and non-manifold shell operations and presents some examples of them. Next, it proposes direct cancel operations for the shell operations to modify generated shapes. The non-manifold shell operation uniformly processes a regularized set operation, an extraction of solids by automatically detecting closed shells from a complex shell which is combination of solids and/or surfaces. Its direct cancel operation is an operation which recovers a shape, independently from a sequence of operations, to one before the designated operation was executed. This cancel operation is applied to efficient parametric design by recovering the shape after changes of parameters.
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This paper discusses representation, maintenance, and editing of form-features based on non-manifold geometric modeling. Form-features are very important to maintain meanings of partial shapes, but in conventional solid modeling, it is difficult to represent various forms of partial shapes and maintain them throughout design. These problems can be solved by using a non-manifold geometric modeling system. In our method, the system maintains topological structure that can represent any form-feature as a collection of topological entities. This is realized by subdividing a modeling space into sets of topological entities, and maintaining the relationships between embedded form-features and the topological entities. Form-features, resultant shapes and compound form-features can be extracted from the topological structure. In addition, editing operations based on this representation can cancel or modify form-features very quickly.
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This paper describes a theory and application of a manifold-based hierarchical non-manifold model. A local restoration function, which restores only the areas concerning with a specified part in the reverse sequence of the operations, is desirable to cope with design changes. It is realized on a manifold-based hierarchical non-manifold model, which is implemented into a computer system with a 'dead shell method.' A boundary edge created by a set-operation or filleting, and its adjacent face has a relation to an edge of a dead shell, which is a 'lost' or hidden shell created by operations. A 'live shell' and 'dead shells' comprise a hierarchical non-manifold as a whole. This method has been proved to be useful in practical use.
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Geometric solid modeling and form feature-based modeling are the two main approaches to model the shape of polyhedra. This paper proposes a unified shape representation for polyhedra. It is a result of some efforts for the last few years to understand the shape of polyhedra, and how current shape models represent and manipulate it.
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Cell complexes have been used in geometric and solid modeling as a discretization of the boundary of 3D shapes. Also, operators for manipulating 3D shapes have been proposed. Here, we review first the work on data structures for encoding cell complexes in two, three and arbitrary dimensions, and we develop a taxonomy for such data structures. We review and analyze basic modeling operators for manipulating complexes representing both manifold and non-manifold shapes. These operators either preserve the topology of the cell complex, or they modify it in a controlled way. We conclude with a discussion of some open issues and directions for future research.
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