The Dual Half-Arc Data Structure: Towards the Universal B-rep Data Structure
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 Complex Geometric Object Boundary Representation Dual Graph Topological Relationship
This research is supported by the Ministry of Higher Education in Malaysia (vote no. 4L047, Universiti Teknologi Malaysia). This research was done while the first author was a Visiting Full Professor at the 3D GIS Research Group, Faculty of Geoinformation and Real Estate at Universiti Teknologi Malaysia, for which the first author is very thankful.
- Baumgart BG (1975) A polyhedron representation for computer vision. In: Proceedings of the 19–22 May, 1975, national computer conference and exposition, pp 589–596Google Scholar
- Boguslawski P (2011) Modelling and analysing 3d building interiors with the dual half-edge data structure. PhD thesis, University of GlamorganGoogle Scholar
- Braid IC, Hillyard RC, Stroud IA (1980) Stepwise construction of polyhedra in geometric modelling. In: Brodlie KW (ed) Mathematical methods in computer graphics and design, Academic Press, pp 123–141Google Scholar
- Ellul C, Haklay M (2009) Using a b-rep structure to query 9-intersection topological relationships in 3d gis—reviewing the approach and improving performance. pp 127–151. http://www.scopus.com/inward/record.url?eid=2-s2.0-84879660700partnerID=40md5=890e5e0c556daba8b3e4ccda230106ce cited By (since 1996)2
- Fomenko A (1994) Visual geometry and topology. Springer, New York. http://books.google.dk/books?id=47TvAAAAMAAJ
- Gold C (1998) The quad-arc data structure. In: Poiker T, Chrisman N (eds) Proceedings of 8th international symposium on spatial data handling, pp 713–724Google Scholar
- Gold CM (1991) Problems with handling spatial data—the Voronoi approach. CISM J 45(1):65–80Google Scholar
- Lee J, Zlatanova S (2008) A 3D data model and topological analyses for emergency response in urban areas. In: Zlatanova S, Li J (eds) Geospatial information technology for emergency response. Taylor & Francis, New York, pp 143–168Google Scholar
- Mäntylä M (1988) An introduction to solid modeling. Computer Science Press, New YorkGoogle Scholar
- Stroud I (2006) Boundary representation modelling techniques. Springer-Verlag New York Inc, SecaucusGoogle Scholar
- Webster J (2003) Cell complexes, oriented matroids and digital geometry. Theoret Comput Sci 305(1–3):491–502, doi: 10.1016/S0304-3975(02)00712-0, URL http://dx.doi.org.globalproxy.cvt.dk/10.1016/S0304-3975(02)00712-0, Topology in computer science (Schloß Dagstuhl, 2000)