Planar and non-planar topologically consistent vector map simplification

This article contains a mathematical analysis of strategies for determining topological consistency of vector map simplifications. Such techniques exploit assumptions that can be made regarding the similarity of corresponding objects in successive simplifications. We propose that all topological relationships may be classified as planar or non-planar. A formal analysis of techniques for determining topological consistency of a simplification in terms of such relationships is presented. For each technique we analyse any corresponding constraints that are imposed. This provides a unified understanding of the benefits and limitations of individual techniques and the relationships that exist between techniques. Subsequently, a new strategy for determining the topological consistency of a simplification is proposed. This technique integrates the benefits all methods studied to provide a solution which is subject to less constraints. The effectiveness of this approach is demonstrated through fusion with an existing simplification technique resulting in simplifications that have equal topology and similar shaped features to the original map.