The Dual Half-Arc Data Structure: Towards the Universal B-rep Data Structure

Abstract

In GIS, the use of efficient spatial data structures is becoming increasingly important, especially when dealing with multidimensional data. The existing solutions are not always efficient when dealing with big datasets, and therefore, research on new data structures is needed. In this chapter, we propose a very general data structure for storing any real or abstract cell complex in a minimal way in the sense of memory space utilization. The originality and quality of this novel data structure is to be the most compact data structure for storing the geometric topology of any geometric object, or more generally, the topology of any topological space. For this purpose, we generalize an existing data structure from 2D to 3D and design a new 3D data structure that realizes the synthesis between an existing 3D data structure (the Dual Half-Edge (See Footonote 1) data structure) and the generalized 3D Quad-Arc data structure, (See Footonote 2) and at the same time, improves the Dual Half-Edge towards a simpler and more effective representation of cell complexes through B-rep structures. We generalize the idea of the Quad-Arc data structure from 2D to 3D, but instead of transforming a simple edge of the Quad-Edge data structure to an arc with multiple points along it, we group together primal edges of the Dual Half-Edge that have the same dual Half-Edge vertex tags (volume tags) into one Dual Half-Arc whose dual is the common Dual Half-Edge and primal faces corresponding to dual. This corresponds to grouping together straight line segment edges into arcs. This allows us to transform the Dual Half-Edge data structure into a 3D data structure for cell complexes with fewer Dual Half-Edges. Since the input/output operations are the most costly on any computer (even with solid state disks), this will result in a much more efficient data structure, where computation of topological relationships is much easier and efficient, like cell complex homologies (See Footonote 3) are easier to compute than their simplicial counterparts. This new data structure, thanks to its efficiency, could have a positive impact on applications that need near real time response, like mapping for natural disasters, emergency planning, evacuation, etc.KeywordsCell ComplexGeometric ObjectBoundary RepresentationDual GraphTopological RelationshipThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.