In the areas of geometry and biology, there are a number of modelling problems that require the creation and manipulation of discrete surfaces that behave dynamically. For example, in geometric modelling there are surface subdivision algorithms that require the repeated insertion of vertices into a polygon mesh. In biological modelling there is the question of modelling growing surfaces, such as a growing flower or a growing tissue of cells. In these cases, there is the open question of how to model dynamical systems with a dynamical structure of a 2-manifold topology, discrete surfaces that have components that change in character, connectivity and number over time.
However, the selection of available tools for modelling dynamical surfaces is limited. There have been some proposed solutions for limited cases, such as cell systems for modelling cells. But there is still a need for a methodology and tools for dealing with dynamical surfaces in general.
In this dissertation, I present a methodology for modelling dynamical systems with a dynamical structure of a 2-manifold topology. This methodology is comprised of the vertex-vertex data structure and algebra and is implemented in the vertex-vertex software environment. I also demonstrate its application with examples in the domains of geometric and biological modelling.
Colin Smith. On Vertex-Vertex Systems and Their Use in Geometric and Biological Modelling. Ph.D. dissertation, University of Calgary, January 2006.
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