Single-cycle plain-woven objects

It has recently been shown that if we twist an arbitrary subset of edges of a mesh on an orientable surface, the resulting extended graph rotation system (EGRS) can be used to induce a cyclic weaving on the surface. In extended graph rotation systems, an edge is viewed as a paper strip that can be twisted. The sides of the paper strips provide ``two strands'' to construct weaving structures. Either these strands are ``parallel'' to the mesh edge for an ``untwisted edge'', or they both cross over the edge and over each other for a ``twisted edge''. If an arbitrary subset of edges of a mesh on an orientable surface is twisted in the same helical sense, then the EGRS induces a cyclic plain-weaving on the surface, which consists of cycles that cross other cycles (or themselves) by alternatingly going over and under. In this paper, we show that it is always possible to create a single-cycle plain-weaving starting from a mesh on an arbitrary surface, by selecting an appropriate subset of edges to be twisted. We also demonstrate how, starting from a mesh, to construct a large number of single-cycle plain-woven objects. Interestingly, the single-cycle solutions with a minimal number of edge twists correspond to plain-woven objects that are visually similar to Celtic knots. For converting plain-weaving cycles to 3D thread structures, we extend the original projection method, which previously worked only when all mesh edges are twisted. With the extension described here, our projection method can also be used to handle untwisted edges. We have developed a system that converts any manifold mesh into single-cycle plain-woven objects, by interactively controlling the proportion of edges that are twisted. The system also allows us to change the shapes of the threads with a set of parameters, interactively in real-time. We demonstrate here that by using this system, we can create a wide variety of single-cycle plain-woven objects.