Simploidal sets: A data structure for handling simploidal Bézier spaces.
Saved in:
| Title: | Simploidal sets: A data structure for handling simploidal Bézier spaces. |
|---|---|
| Authors: | Peltier, Samuel1 samuel.peltier@univ-poitiers.fr, Lienhardt, Pascal1 pascal.lienhardt@univ-poitiers.fr |
| Source: | Computer Aided Geometric Design. May2018, Vol. 62, p44-62. 19p. |
| Subjects: | Algebraic topology, Data structures, Geometric modeling, CAD/CAM systems, Computer graphics, Combinatorics, Computational topology |
| Abstract: | Simplicial sets and cubical sets are combinatorial structures which have been studied for a long time in Algebraic Topology. These structures describe sets of regular cells, respectively simplices and cubes, and any kind of assembly of cells can be represented. They are used for many applications in Computational Topology, Geometric Modeling, CAD/CAM, Computer Graphics, etc. For instance, simplicial and cubical sets are ”naturally” associated with simplicial and cubical Bézier spaces. In this paper, a new combinatorial structure, namely simploidal sets is defined for representing and handling Bézier simploids. Simploidal sets describe sets of simploids, which are also regular cells corresponding to Cartesian product of simplices. Simplices and cubes are then particular simploids. In order to associate shapes with structures, structural relations between simploidal sets and simploidal Bézier spaces are also stated. In fact, simploidal sets generalize and homogenize simplicial and cubical sets. Construction operations are also defined, extending all those of simplicial and cubical sets: cone, Cartesian product, degeneracy, and identification. In their basic version, the first three operations allow to create any simploid, and the last one, to create any assembly of simploids. It is then possible to simultaneously handle through a single formalism any assembly of simplices, cubes, and other simploids, with a very low additional cost, regarding space (data structure), time (construction or computation operations) or software development. [ABSTRACT FROM AUTHOR] |
| Copyright of Computer Aided Geometric Design is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| Abstract: | Simplicial sets and cubical sets are combinatorial structures which have been studied for a long time in Algebraic Topology. These structures describe sets of regular cells, respectively simplices and cubes, and any kind of assembly of cells can be represented. They are used for many applications in Computational Topology, Geometric Modeling, CAD/CAM, Computer Graphics, etc. For instance, simplicial and cubical sets are ”naturally” associated with simplicial and cubical Bézier spaces. In this paper, a new combinatorial structure, namely simploidal sets is defined for representing and handling Bézier simploids. Simploidal sets describe sets of simploids, which are also regular cells corresponding to Cartesian product of simplices. Simplices and cubes are then particular simploids. In order to associate shapes with structures, structural relations between simploidal sets and simploidal Bézier spaces are also stated. In fact, simploidal sets generalize and homogenize simplicial and cubical sets. Construction operations are also defined, extending all those of simplicial and cubical sets: cone, Cartesian product, degeneracy, and identification. In their basic version, the first three operations allow to create any simploid, and the last one, to create any assembly of simploids. It is then possible to simultaneously handle through a single formalism any assembly of simplices, cubes, and other simploids, with a very low additional cost, regarding space (data structure), time (construction or computation operations) or software development. [ABSTRACT FROM AUTHOR] |
|---|---|
| ISSN: | 01678396 |
| DOI: | 10.1016/j.cagd.2018.03.010 |
