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Corner-operated Tran-similar (COTS) Maps, Patterns, and Lattices

The planar COTS map proposed here takes the unit square to a region R bounded by four log-spiral edges. It is Corner-operated (controlled by the four corners of R) and Tran-similar (it maps translations to similarities). The tiles of the COTS map of a regular pattern are similar to each other. It may facilitate intuitive design and algorithmic optimization of procedural models of complex, possibly multi-resolution, lattices, because it affords constant-cost algorithms for Point-in-Lattice testing and for Total-Area-Calculations. We provide simple, closed-form expressions for evaluating the COTS map and its inverse from the positions of its corners. We conjecture that the COTS map may be useful in a variety of applications in Engineering, Architecture, and Art, and we provide a few illustrative examples of its possibilities. We compare it to related, previously proposed, planar maps and discuss several variations and extensions.