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tf-ORDER-BASED TOPOLOGICAL RELATIONS OF GEO-ENTITIES IN 3D RASTER 
SPACE 
GUO Jiateng 3 ’*, WU Lixin 3 ’ b , MA Hongbin 3 , CHE Defu 3 
a Institute for Geoinformatics & Digital Mine Research, Northeastern University, Shenyang 110004, China, - 
guojiateng@mail.neu.edu.cn 
b Institute for GIS/RS/GPS & Subsidence Research, China University of Mining & Technology, Beijing 100083, China, 
- awulixin@263.net 
KEY WORDS: GIS, Model, Spatial, Raster, Three-dimensional, Algorithms 
ABSTRACT: 
The theoretical foundations and application availability of topological spatial relationship research for three dimensional Geo-entities 
in raster space are discussed. The digital topology theory is applied to define the interior, boundary and border six neighborhood of a 
3D geo-entity which is represented with regular hexahedron tessellation. A new 9-intersection model, which replaces the original 
interior I, the boundary B and the exterior E of an entity in 9-1 model with I 6 , B 6 and E k 6 respectively, is presented. A computable 
method based on relation database query language SQL (Structured Query Language) is adopted for the model computation, 
topological query and analysis. The experimental analytic results show that the topology analysis model can be properly used for 
topological relations query and analysis of 3D Geo-entities represented in raster space, and it can also be applied for the reasoning of 
metric relation, direction relation and other spatial relationships. 
1. INTRODUCTION 
Topological relations denote the unchanged characteristics 
under the topological transform, such as translation, rotation and 
scale (Egenhofer 1989). Early in 1988, the American National 
Science Foundation (NSF) began to sponsor the American 
National Center for Geographic Information & Analysis 
(NCGIA) on researches on several theoretical issues of spatial 
relationship (NCGIA, 1989). Topological relations are one of 
the most foundational and most important aspects in spatial 
relationships, which play the key roles in many fields such as 
spatial data modeling, spatial query, spatial analysis, spatial 
reasoning and cartographic generalization. In the latter twenty 
years, researches on topological relations have attracted many 
specialists and a large number of papers have been published, 
which mainly concentrate on topological data model (Wu 
Lixin, 2004) and formal representation framework (Egenhofer, 
1991; Li Chenming, Chen Jun, 1997) of topological relations. 
The topological data model usually embodies local topological 
relations between inner parts of an entity, while the formal 
representation framework of topological relations, which is 
named as Entity Topology, emphasizes on global topological 
relations between geological entities. Currently, the 
9-intersection model presented by Egenhofer is widely used in 
two dimensional space. However, there are some imperfections 
in theory and application in this model, such as linear 
dependency between three sets of geometric elements of an 
entity, difficult definition for the exterior of an entity and worse 
computability. Hence, the 9-intersection model can not be 
directly used for Geo-entity topology research unless some 
tessellation algorithm, simple data structure, smaller data 
storage and strict neighborhood definition, especially has good 
performance in indicating the spatial structured variety of 
Geo-entities. Besides, it has already been widely used as a three 
dimensional block model and has been proved with good 
improvements are made (Zhao Renliang, 2002; Chen Jun, Guo 
Wei, 1998). 
With continuous development of human beings cognition and 
behaviors, the human beings has gradually changed its attention 
on geo-objects from two dimensional plane to three dimensional 
space, and from Earth surface to underground space. As a result 
of human beings activities, the entities on Earth surface are 
usually artificial entities with regular structure, accurate position 
and unambiguous boundary, which can be properly represented 
with vector data structure. While the underground space is 
usually invisible and continuous; many kinds of entities that 
embodied in underground space are usually with fuzzy 
boundary and may invade into each other. In this situation, the 
boundaries of underground Geo-entities can not be properly 
represented with geometric elements; especially the trend 
relation between inner parts of an entity and the relation 
between adjacent Geo-entities can not be indicated. However, 
the digital space based on raster tessellation is much more 
suitable for the representation of the continuity and structured 
variety of underground space. 
2. 3D RASTER TOPOLOGY 
2.1 Raster Data Model 
There are several kinds of raster data model for Geo-entity 
tessellation such as tetrahedron, regular hexahedron, three 
Dimensional Voronoi etc. Compared to other data models, 
regular hexahedron has many good characteristics such as easy 
applicability in attribute interpolation and resources estimation. 
A three dimensional spatial interpolation method such as 
inverse distance weighting method or Kriging method can be 
applied to combine the attribute variety of Geo-entities 
intensively with the spatial position of each hexahedron to 
* Corresponding author. Tel.: +8624 83683612. E-mail addresses: guojiateng@mail.neu.edu.cn (GUO Jiateng)