Cyclic Plain-Weaving on Polygonal Mesh Surfaces with Extended Graph Rotation Systems

In this paper, we show how to create plain-weaving over an arbitrary surface. To create a plain-weaving on a surface, we need to create cycles that cross other cycles (or themselves) by alternatingly going over and under. We prove that it is possible to create such cycles, starting from any given manifold-mesh surface, by simply twisting every edge of the manifold mesh. Our proof is based on our extended theory of graph rotation systems, which is also first introduced in this paper. Our extended theory relates non-orientable meshes with links. We have developed a new method that converts plain-weaving cycles to 3D thread structures. Using this method, it is possible to cover a surface without large gaps between threads by controlling the sizes of the gaps. We have developed a system that converts any manifold mesh to a plain-woven object, by interactively controlling the shapes of the threads with a set of parameters. We have demonstrated that by using this system, we can create a wide variety of plain-weaving patterns, some of which may not have been seen before. CR Categories: G.2.2 [Graph Theory]: Topological Graph Theory—Graph Rotation Systems; I.3.5 [Computational Geometry and Object Modeling]: Geometric Algorithms—Links, Knots and Weaving.