Multi-dimensional Unified GIS Data Model Based On Geometric Algebra

Supporting the intergrated expression, modeling and simulation of complex geometric phonemena is one of the hottest topics of GIS and geographical researches. It is clearly the trend that the GIS research dimensions will be extended from two dimensions to three dimensions and even to higher. Because the geographical phonemena itself are complex, existing data models should have various kinds of coordinations system and heterogenous computational framework, which will occur difficulties when the dimesions are extended. Commonly used data models are still limited in the area of adaptive expression of multidimensional objects, the spatial index of solid objects, the unified multidimensional spatial analysis methods and algorithms, the system implementation and application of multidimensional GIS systems. Make innovations on the theoretical foundations and implementation new technologies, based on new mathematical theories, will provide a potensial way to break through the above limitations of existed GIS spatial data models and construct unified multidimensional expression and computational framework.The Clifford Algebra is introduced to constructed new unified multidimensional spatial data model, referenced on its successfully application at mathematical and physical areas. The adaptive expression and intergrated modeling of multidimensional objects and according data managementare first researched. The storing and indexing mechanism, associated multidimensional unified computational model and protype software system based on the above results are also proposed. The above researches will benefit and innovate in the GIS spatial data model theory and technology, which provided new technogical supporting for the expression, modeling and simulation of geographical phonemena. The research also provdes new potentially ways to extend the unfied multidimensional spatial analysis methods. The main contents and results of this paper are summarized as follow:(1) The Grassmann structure is consisted with the dimensional structure of geometrical expression under conformal geometric algebra (CGA). Based on this, the adaptive geometric expression model by inner and outer product, which intergrates the dimensional construction level and metric relations, is proposed. The dual relations between the inner and outer product expression and the reciprocal transformation of different geometric objects based on their geometric meannings are discussed. Both inner and outer product expression of common geometric objects, the mapping relations between geographical objects of different dimensions and their associated basic elements of geometric algbra (Blades) and the unified expression and storage of multidimensional objects based on multivector data structures are given. Based on these, we constructed the main architecture of our multidimesion spatial data model based on geometric algebra. The storage structure, the editing and updating mechanism of multidimensional data are discussed. The application on several standard geometric objects and two 3D geographical sences of different scales suggest that our data model have simple structure and clear geometric meaning, which reduce the space occupation as well as support for mathematical analysis.(2) The sphere expression under geometric algebra has clear geometric meaning, dynamical updateable parameters and simple geometric relations calculation. Taking advantage of these, the Boundry restrictednon-overlap sphere tree spatial index (BRNO-ST) were proposed. The boundary restruction and the sphere filling algorithm based on non-overlaping spheres, which achieved the blance between granularity of subdivision and expression precision, are proposed. The volume adjusted adaptive batch Neural Gas herichical cluster algorithm (Vol-BGNG-GA), that can quickly, roubustly and relatively uniformly classify the filling sphere, and the level structure of the index trees are then constructed. The query mechanism of the any loction and ranges on and in the solid objects based on the BRNO-ST index are provided. With the help of geometric algebra operators, the hierarcal approximatical nearest linkage distance and dynamical overlapping query under limited time restrictions are also implemented.(3) Existing geometric algebra operators are filtered and extended to construct the operators and operations that suited for GIS spatial analysis and geographical analysis. The computational mechanism of geometric metric, spatial relations and spatial topology are also discussed. Take advantage of the unified expression of multidimensional geometric objects and metric & spatial relations, the batch multidimensional geometric and topological relations algorithms are proposed. The basic GIS geometric and topological analysis methods are also constructed. We also constructed the unified transformation and expression based on the versor equations for the motion expression and analysis. The motion interpolation algorithm are constructed based on the exponentional expression and their geometric meaning of the even grade versor. The unified multidimensional GIS spatial relation computation framework is proposed based on the unified operator and algorithm libraries and their implementation framework. The case studies on the three dimensional housing estate suggest the framework are simple and efficient in geometric and topology relation computations, which has the potential to support large scale GIS analysis.(4) The unified multidimensional GIS spatial analysis propo type system is constructed based on the data model, spatial index and spatial relations computation framework proposed above. The main architecture, layer structures and main function moduals are discussed. The core geometric algebra computation engine and the according data interface with other common GIS spatial data files are implementated to fulfill the characteristics of geographical data and the needs of geometric algebra computation. A pluging system is also constructed for implementing the constructionand intergration the multidimensional spatial analysis framework. Main functional moduals, such as the space transformation from geographicalspace in geometric algebra space, data I/O and data management, dynamical sence simulation etc. are impelemented and demonstrated. Finally, a comprehensive application based on the intergrated sea-land-ice coupled dynamical interaction and evolutionary process are proposed, which suggest that our approach can well intergrated the processes from data organization, storage, search and query, algorithem construction and the geographical analysis.The studies proposed in this paper suggest that geometric algebra can express geometric objects in different dimensions effectively, which also can support simply and effectively computation in multidimesioal space in a coordination free geometric computation. The data model constructed in this paper is simple and of clear geometric meaning, which can also support for unified multidimensional coordinate-free geometric computation. Based on specific problems and requires, we can constructed suited computational space and using basic operators to abstract ad construct the algorithms. Our work can form a basic framework and application platform that will intergrated the underline theories, data models, data analysis and expressions, which will imporove the new GIS analysis technology that are multidimensional unified.