多维统一GIS构建与实现的几何代数方法

针对时空GIS发展所面临的时空表达不统一,高维扩展困难等问题,基于以多维统一和坐标无关为特征的几何代数理论,建立多维统一GIS构建的层次体系与方法。利用几何代数对象构造中几何维度与Grassmann维度的一致性,实现整合维度结构与几何关系的多维地理对象自适应表达。利用几何代数算子的特征内蕴与结构明晰特性,构建多维统一计算模型以及可支撑复杂地理计算的多维统一分析框架,进而建立基于几何代数的多维统一GIS系统架构与实现方案。对上述关键技术加以实现并进行了相关应用案例分析。结果显示,基于几何代数构建的多维统一GIS有望突破现有时空GIS的不足,可为以多维统一表达与运算为特征GIS的发展提供新的数学工具与实现途径。

Geometric algebra-based construction and implementation methods for multidimension-unified GIS

High dimensional data analysis ability,computation complexity and extensibility have become the handicap of existing GIS.There still exist problems such as lack of unification in spatio-temporal expression and high dimensional extention in present spatio-temporal GIS.Geometric algebra(GA)theory,featuring multidimension unification and coordinate-free properties,is introduced to construct the multidimension-unified GIS.The GA method can connect different geometric and algebra systems,and provide rigorous and elegant foundation for expression,modeling and analysis in GIS.The layer structure of multidimension-unified GIS based on GA is proposed.With the help of the outer and inner product expressions,and the consistance of the geometric dimensions and the Grassmann dimensions in object expression in GA,adaptive expression methods that integrate the dimensional structure and geometric relations are defined.The GA operators are extended and constructed for multidimensional GIS analysis.And the computational model,which have intrinsic characteristic and clear structure,are constructed with these operators.And then,a multidimension-unified analysis framework that can support complex geo-computation is discussed.The implementation methods of multidimension-unified GIS based on GA, including key technologies such as multidimension-unified GIS data models,spatio-temporal indexes and analysis methods are also studied.Finally,the prototype system is implemented and demonstrated as case studies.This work shows that GA provides a new mathematical tool to the development of GIS characterized as multidimension unified expression and computation.It is equipped with possible development of multidimension-unified GIS based on GA and suggests a possible way to avoid leakage of existing spatio-temporal GIS.