Tesseral amalgamators and hierarchical tessellations

摘要

Hierarchical tesselations are used in many fields, for example image processing and geographic information systems (GIS). This paper describes a general method of generating hierarchical tilings1 on a two-dimensional, optionally square lattice using the Tesseral Amalgamator Theory (TAT). Each hierarchy has an hierarchical address which may be used in the same way as that of the quadtree2–4, and an arithmetic for geometric transforms5. Thousands of new hierarchies have been listed. Examples demonstrating how the new hierarchies and TAT can be used to increase the efficiency of hierarchical methods are given, including an account of three recent successful benchmarks using TAT. References are given to the generalization of TAT to three and four dimensions.