BeCOTS: Bent Corner-Operated Tran-Similar Maps and Lattices. (December 2020) - Kurzeja, Kelsey; Rossignac, Jarek
Abstract: A field (2-parameter family of similarities) S ( x, y ) is Tran-Similar (TS) if there exist similarities U and V such that S ( x, y ) = U x ⋅ V y and U ⋅ V = V ⋅ U . The recently proposed COTS map, M, is defined in terms of a planar TS field, S, as M ( x, y ) = S ( x, y ) ⋅ A and is Corner-Operated (CO), i.e., controlled by the images A, B, C, and D of the four corners of the unit-square in parameter space. The CO property enables intuitive design of planar warps. The TS property has several benefits: (1) The COTS map distributes distortion evenly; (2) It may be used to create pleasing patterns in which consecutive instances along any given row or column are related by the same similarity; and (3) Point-Membership Classification and Integral-Property Computation may be performed in constant time. However, the COTS map is only defined when the four corners are coplanar. To remove this restriction, we propose the Bent COTS (BeCOTS) generalization of the COTS map to three dimensions. It is CO (operated by its four non-coplanar corners) and TS (inheriting the properties of COTS mentioned above). We discuss potential benefits of BeCOTS for supporting the design and parameterization of highly-complex, bent lattices, for accelerating their processing, and for reducing the cost of manufacturing a class of large architectural structures. Highlights: Non-planar generalization of the 2-directional Corner-Operated Tran-Similar Maps. Non-planar patterns where all neighbor elements are related by the same similarity. Design patterns of solids with constant-time point-membership and integral queries. Architectural lattice structures with possible manufacturing cost reduction. 3-directional lattices with constant-time point-membership queries.