Formal study of plane Delaunay triangulation

This article presents the formal proof of correctness for a plane Delaunay triangulation algorithm. It consists in repeating a sequence of edge flippings from an initial triangulation until the Delaunay property is achieved. To describe triangulations, we rely on a combinatorial hypermap specification framework we have been developing for years. We embed hypermaps in the plane by attaching coordinates to elements in a consistent way. We then describe what are legal and illegal Delaunay edges and a flipping operation which we show preserves hypermap, triangulation, and embedding invariants. To prove the termination of the algorithm, we use a generic approach expressing that any non-cyclic relation is well-founded when working on a finite set

Discrete Jordan Curve Theorem: A proof formalized in Coq with hypermaps

This paper presents a formalized proof of a discrete form of the Jordan Curve Theorem. It is based on a hypermap model of planar subdivisions, formal specifications and proofs assisted by the Coq system. Fundamental properties are proven by structural or noetherian induction: Genus Theorem, Euler's Formula, constructive planarity criteria. A notion of ring of faces is inductively defined and a Jordan Curve Theorem is stated and proven for any planar hypermap

Transportation linear referencing toolboxes : a 'reflective practitioner's' design approach

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Urban Studies and Planning, 2000."September, 2000."Includes bibliographical references (p. 395-407).Seventy percent of the data of a typical transportation agency (e.g., bridges, accidents, etc.) has location as a primary reference. A Linear Referencing System (LRS) is the main way of identifying the location of this data and providing a storage key for it in a database. LRS is based on a one-dimensional offset on a predefined network. In theory, it is one of the simplest spatial cases. In reality, it can be spatially and analytically quite complex. LRS to quite recent date has been little formally researched. That research which has occurred has been the construction of large and comprehensive conceptual data models. This thesis is not primarily aimed at new "tool building research". The existing models have been based to only a limited extent on a fuller analysis of the nature of transportation and spatial data; they have not considered relevant field and wider methodological concerns (i.e., they followed a "model-driven" approach). The goal here is to create a more appropriate foundation and base from which LRS tools may be most appropriately built (i.e., a 'field-driven" approach). A "practitioners perspective" view of LRS was sought. Such a more holistic understanding was sought through the adoption of a "layered methodology" of research that involved gaining the perspectives of a variety of disciplinary viewpoints. This research framework was developed especially for this thesis based on the ideas and work of Schon and Reich. The approach involved in short a desk exercise in fundamental consideration of the nature of LRS, a deeper, cross-field synthesis and literature research, four in-depth state DOT LRS case studies, a panel of transportation field experts, a panel of national data model experts, and a limited object-orientated modeling exercise. The conclusion reached is that while LRS in the simple case can be modeled in general forms, it is also an "exception-driven" field. Thus, a "toolkit approach" may be more appropriate for LRS. It is inferred that this may hold for other similar application areas in transportation and planning. Further research would further develop the holistic layered methodology adopted here and further define the proposed LRS transportation application toolboxes.by Simon Lewis.Ph.D