HCC: Small: Collaborative Research: Graph and Pattern Design on Surfaces

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Abstract ? Zhang/WonkaThe research investigates theory and efficient algorithms for pattern design on surfaces. Patterns on surfaces appear in many natural phenomena such as leaves, animal textures, and terrains as well as man-made objects such as origami, glass ornaments, and facades. Patterns can also be used to describe networks, such as street layouts, power grids, aqueducts, and sensor networks. Pattern design has a wide range of applications in art and entertainment, architecture, engineering, medicine, and city planning. In addition, theory and techniques developed in the research can benefit domains such as computational geometry and vector and tensor field visualization. There are several fundamental challenges when it comes to pattern design on surfaces. First, there is a lack of unified mathematical formulations of patterns in terms of both symmetries and orientations contained in the patterns. Consequently, the aforementioned applications are typically addressed as being unrelated despite the intrinsic links between them. Second, many past approaches to these problems lack hierarchical control. This is required so that the user can design high level information down to occasionally low level specifics and the layout algorithms fill in the rest procedurally. In this research, the investigators explore a unified framework that allows hierarchical design of patterns on surfaces. In this framework, orientation and symmetry information is specified everywhere in the domain through tensor field design. Next, the tensor field which contains desired orientation and symmetry information is used to generate a complex which can be a point set, a graph, a tiling, or any combination of them. Finally, additional details are added onto the complex through texture and geometry synthesis, or sub-patterns are added inside the cells of the complex. Ideas from various mathematical domains such as dynamical systems, tensor calculus, differential geometry, and algebraic topology are borrowed and applied in this research.