Cyclic plain-weaving on polygonal mesh surfaces with graph rotation systems

Abstract

Figure 1: Examples of woven objects constructed with ribbons: Venus consists of five distinct cycles. The bunny has eight cycles, the rocker arm has only two cycles, and the genus-three object has 16 cycles. The first three models are created by the Quadcover method [Kalberer et al. 2007], courtesy of Wenping Wang and Li Yupei. The genus-three object is created using TopMod3D [Akleman et al. 2008]. AbstractIn 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 use the fact that it is possible to create such cycles, starting from any given manifold-mesh surface by simply twisting every edge of the manifold mesh. 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.

Paper-Strip Sculptures

“…We have revised some existing surgery theorems of topological graph theory based on our observations from paper strips. Our theoretical results in this regard appear in [2] and [3].…”

“…Akleman et al noticed that the boundary walks induced by a graph rotation system define a link in 3D-space, and they used this property to construct plainweaving cycles on arbitrary polygonal mesh surfaces [3], [2]. This insight can be realized visually using paperstrip sculptures, as shown in Figure 4.…”

Turning Hild’s Sculptures into Single-Sided Surfaces

“…His demonstration tool is readily available on the web [35], and by choosing the right smoothing option, one can produce visualization models that can also stand on their own as pleasing sculptural shapes. Other mathematical concepts, where visualization models can readily turn into art, concern non-orientable surfaces-such as Boy's surface (Figure 5a) [36,37] or Klein bottles (Figure 5b) [38,39]-and the depiction of regular maps [40][41][42], regular meshes [43,44], woven surfaces [45,46], knot theory [47][48][49], or high-genus objects (Figure 5d) [50,51]. [36]; (b) a Klein bottle [38]; (c) a regular map [40]; and (d) a high-genus object [50].…”

A Celtic Framework for Knots and Links

“…Cyclic plain-weaving is a more general form of computer-assisted artwork, and, as observed by [ACXG09], the graphics it creates are alternating projections of links onto various surfaces in 3-space. From a topological perspective, Celtic knots and links are a special case of cyclic plain-weaving.…”