Besides topological relationships and approximate relationships, cardinal directions like north and southwest have turned out to be an important class of qualitative spatial relationships. They are of interdisciplinary interest in fields like cognitive science, robotics, artificial intelligence, and qualitative spatial reasoning. In spatial databases and Geographic Information Systems (GIS) they are frequently used as join and selection criteria in spatial queries. However, the available computational models of cardinal directions suffer a number of problems like the use of too coarse approximations of the two spatial operand objects in terms of single representative points or minimum bounding rectangles, the lacking property of converseness of the cardinal directions computed, and the limited applicability to simple instead of complex regions only. This article proposes and formally defines a novel two-phase model, called the Objects Interaction Matrix (OIM) model, that solves these problems, and determines cardinal directions for even complex regions. The model consists of a tiling phase and an interpretation phase. In the tiling phase , a tiling strategy first determines the zones belonging to the nine cardinal directions of each individual region object and then intersects them. The result leads to a bounded grid called objects interaction grid . For each grid cell the information about the region objects that intersect it is stored in an objects interaction matrix . In the subsequent interpretation phase , a well-defined interpretation method is applied to such a matrix and determines the cardinal direction. Spatial example queries illustrate our new cardinal direction concept that is embedded in a spatial extension of SQL and provides user-defined cardinal direction predicates.

Recently, a wide range of applications like hurricane research, fire management, navigation systems, and transportation has shown increasing interest in managing and analyzing space and time-referenced objects, so-called moving objects, that continuously change their positions over time. In the same way as moving objects can change their location over time, the spatial relationships between them can change over time. An important class of spatial relationships are cardinal directions like north and southeast. In spatial databases and GIS, they characterize the relative directional position between static objects in space and are frequently used as selection and join criteria in spatial queries. Transferred to a spatiotemporal context, the simultaneous location change of different moving objects can imply a temporal evolution of their directional relationships, called development. The goal of this paper is to illustrate, explain, and formally define cardinal direction developments between two moving points. Categories and Subject Descriptors:

Apart from visualization tasks, three-dimensional (3D) data management features are not or only hardly available in current spatial database systems and Geographic Information Systems (GIS). But the increasing demands from application domains like urban planning, geoscience, and soil engineering call for systems that are capable of storing, retrieving, querying, and manipulating the underlying 3D spatial data. Current 3D data models are tailored to specific applications and simple 3D spatial objects only, and available 3D data structures are restricted to main memory representations; thus they lack the ability of handling general and complex 3D spatial objects in a database context. Available algorithms, especially intersection algorithms for 3D objects, usually require special properties like convexity or monotonicity. Therefore, universal intersection algorithms that are capable of handling general 3D spatial objects are currently unknown. This paper proposes a paradigm called slice representation as a general data representation method for complex 3D spatial data types. In particular, data structures are developed applying the paradigm to point3D, line3D, surface, and volume data types. Two intersection algorithms that involve one argument of type point3D are introduced and show the benefit of the slice representation.