Verifiable implementation of geometric algorithms using finite precision arithmetic
Artificial Intelligence - Special issue on geometric reasoning
Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
LEDA: a platform for combinatorial and geometric computing
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COSIT '99 Proceedings of the International Conference on Spatial Information Theory: Cognitive and Computational Foundations of Geographic Information Science
A Qualitative Account of Discrete Space
GIScience '02 Proceedings of the Second International Conference on Geographic Information Science
A Small Set of Formal Topological Relationships Suitable for End-User Interaction
SSD '93 Proceedings of the Third International Symposium on Advances in Spatial Databases
Interactive Computer Graphics: A Top-Down Approach using OpenGL (4th Edition)
Interactive Computer Graphics: A Top-Down Approach using OpenGL (4th Edition)
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In order to be able to draw inferences about real world phenomena from a representation expressed in a digital computer, it is essential that the representation should have a rigorously correct algebraic structure. It is also desirable that the underlying algebra be familiar, and provide a close modelling of those phenomena. The fundamental problem addressed in this paper is that, since computers do not support real-number arithmetic, the algebraic behaviour of the representation may not be correct, and cannot directly model a mathematical abstraction of space based on real numbers. This paper describes a basis for the robust geometrical construction of spatial objects in computer applications using a complex called the "Regular Polytope". In contrast to most other spatial data types, this definition supports a rigorous logic within a finite digital arithmetic. The definition of connectivity proves to be non-trivial, and alternatives are investigated. It is shown that these alternatives satisfy the relations of a region connection calculus (RCC) as used for qualitative spatial reasoning, and thus introduce the rigor of that reasoning to geographical information systems. They also form what can reasonably be termed a "Finite Boolean Connection Algebra". The rigorous and closed nature of the algebra ensures that these primitive functions and predicates can be combined to any desired level of complexity, and thus provide a useful toolkit for data retrieval and analysis. The paper argues for a model with two and three-dimensional objects that have been coded in Java and which implement a full set of topological and connectivity functions which is shown to be complete and rigorous.