The algebraic specification of spatial data types with applications to constructive volume geometry

Spatial objects are modelled as total functions, mapping a topological space of points to a topological algebra of data attributes. High-level operations on these spatial objects form algebras of spatial objects, which model spatial data types. This thesis presents a comprehensive account of the theory of spatial data types. The motivation behind the general theory is Constructive Volume Geometry (CVG). CVG is an algebraic framework for the specification, representation and manipulation of graphics objects in 3D. By using scalar fields as the basic building blocks, CVG gives an abstract representation of spatial objects, with the goal of unifying the many representations of objects used in 3D computer graphics today. The general theory developed in this thesis unifies discrete and continuous spatial data, and the many examples where such data is used from computer graphics to hardware design. Such a theory is built from the algebraic and topological properties of spatial data types. We examine algebraic laws, approximation methods, and finiteness and computability for general spatial data types. We show how to apply the general theory to modelling (i) hardware and (ii) CVG. We pose the question “Which spatial objects can be represented in the alge­ braic framework developed for spatial data types?”. To answer such a question, we analyse the expressive power of our algebraic framework. Applying our results to the CVG framework yields a new result: We show any CVG spatial object can be approximated by way of CVG terms, to arbitrary accuracy.

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