Geometric algebra method for multidimensionally-unified GIS computation

Abstract

Seamless multidimensional handling and coordinate-free characteristics of geometric algebra (GA) provide means to construct multidimensionally-unified GIS computation models. Using the multivector representation for basic geometric objects within GA, we are able to construct adaptable unified geometric-topological structural models of a multidimensional geographical scene. Multidimensional operators found within the geometry, topology and GIS analysis are developed with basic GA operators. A unified computational framework is proposed, it unifies expressions and operation structures, as well as supports the analysis of multidimensional complex scenes. Finally, we illustrate modelling a three-dimensional residential district, which shows that GAbased multidimensionally-unified computation models can effectively represent and analyze complex and multidimensional geographical scenes. The development of the proposed GIS multidimensionally-unified representation, analysis, and modeling enhances current GIS algorithms and geographical models. The representation and arrangement of complex geographical objects and scenes are identified as important in the application and usability of geographic information systems (GIS) [1][2][3][4]. Object expression and code construction of operations is an important step in advancing GIS research from static characteristics to a more comprehensive research on temporal and spatially distributed structures, and modeling and forecasting of dynamic processes and mechanisms. It is also an indispensable process in achieving model integration and comprehensive use in GIS and professional areas [5][6][7][8]. There still exist many limitations within traditional GIS, based on Euclidean geometry, in maintaining complex representations of geographical objects, multidimensional spatial relations, and topographical analysis [9]. Constructing models that support multidimensionally-unified geographical object representations and operations based on new mathematical theories is one possible way to reduce the multidimensional complexity of current GIS and to improve architectural efficiencies in its operations and analysis [10,11]. Geometric algebra (GA), known as a unifying descriptive language, can integrate algebra with geometry, mathematics with physics. By defining the spatial operation sets (e.g. geometry, measurement, topology) with GA operators, Euclidean, homogeneous, and conformal spaces can be described and converted among each other, this provide a unification and representation of varied algebraic structures and geometric systems [12,13]. Calculus, geometry, and signal processing (e.g. neural network and wavelet) methods can be reconstructed and expanded under the GA framework [14,15], which then provides a foundation for expansion and reconstruction of topological models. Here, based on GA's multivectors and operators, we construct a geometric framework that enables multidimensionallyunified computation modeling of multiple geometric objects

3D Cadastral Data Model Based on Conformal Geometry Algebra

“…Different dimensional geometry objects, such as the lines and planes shown in Figure 8, can be uniformly expressed with the outer product in CGA. In Euclidean space, representations of geometry objects are primarily based on coordinates, which causes problems such as complicated expression structure, non-uniform expression form in different dimensions, and unclear geometry meaning [47]. Geometric objects are expressed in CGA, and the outer product differs from the traditional methods, which are based on Euclidean geometry.…”

Change detection for 3D vector data: a CGA-based Delaunay–TIN intersection approach

“…DTINs and associated elements (points, segments, and triangles) can be uniformly represented with the multivector structure (Yuan et al 2011). Then the multivector representation can support multidimensionally unified computing with CGA operators (Yuan et al 2012).…”

“…The result of the meet operator on two multivectors is also a multivector, which indicates the geometry of the intersection result. The intersection relations of geometric objects would adaptively change as the geometry types and positions change (Yuan et al 2012). Since the intersection with the meet product is unified and directly computed in a single mathematical framework, it can link the geometric representation and algebraic computation seamlessly.…”

“…The data structure stores the DTIN and the associated elements in a hierarchical structure, with both the dimensional construction structure and the topological relation structure inherited. Therefore, the data structure can be easily processed in the multidimensionally unified computation framework with CGA operators (Yuan et al 2012). The compact representation with meaningful symbolical representations also helps to reduce the space cost during the data organization and computation (Yuan et al 2013)…”

Single boson realizations of the deformed angular momentum algebra of Witten’s types

“…For each kind, the unitary realization, the non-unitary realization, and their connection by the corresponding similarity transformation are respectively discussed. Lie algebras have played an exciting and significant role in the development of various branches of physics and many other areas of science and technology [1][2][3][4][5][6][7][8][9][10][11]. In 1983, Kulish et al [12] showed that the algebra that governs the XXZHeisenberg spin model was a deformation of the Lie algebra SU(2), called nowadays SU q (2).…”