Bobbin lace is a fibre art form in which intricate and delicate patterns are created by braiding together many threads. We present an overview of the making of bobbin lace, illustrated with a simple, traditional bookmark design. We briefly summarize research on the topology of textiles and braid theory, which form a base for the current work. We then define a new mathematical model that supports the enumeration and generation of bobbin lace patterns using an intelligent combinatorial search, and which is capable of producing patterns that have never been seen before. Finally, we apply our new patterns to an original bookmark design and propose future areas for exploration.
Grishanov, Meshkov and Omelchenko have examined the structure of machine-made textiles and classified these textiles using the ambient isotopy invariant of knots @cite_37 . Textiles are typically made by repeating an arrangement of fibers in a periodic manner to cover an indefinitely large area. The arrangement of fibers can be represented as a period parallelogram which is translated in two non-parallel directions to create an edge-to-edge tiling of the plane @cite_31 . Periodic repetition in textiles is a stronger property than just simple translation of a wallpaper decoration: fibers that terminate at the edges of the parallelogram must connect with fibers of adjacent copies. This property, which Grishanov et al call doubly periodic, can be visualized by joining opposite edges of the period parallelogram to form a torus (see Figure ). When wrapped around a torus, the fibers connect, forming a knot or a link as shown in Figure . The toroidal representation also reduces a pattern description from infinite to finite size without loss of information, a key idea which we will revisit when describing our own model in Section .
@cite_37 @cite_31
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This paper proposes a new systematic approach for the description and classification of textile structures based on topological principles. It is shown that textile structures can be considered as a specific case of knots or links and can be represented by diagrams on a torus. This enables modern methods of knot theory to be applied to the study of the topology of textiles. The basics of knot theory are briefly introduced. Some specific matters relating to the application of these methods to textiles are discussed, including enumeration of textile structures and topological invariants of doubly-periodic structures. This paper describes an automatic and efficient approach to construct unstructured tetrahedral and hexahedral meshes for a composite domain made up of heterogeneous materials. The boundaries of these material regions form non-manifold surfaces. In earlier papers, we developed an octree-based isocontouring method to construct unstructured 3D meshes for a single material (homogeneous) domain with manifold boundary. In this paper, we introduce the notion of a material change edge and use it to identify the interface between two or several different materials. A novel method to calculate the minimizer point for a cell shared by more than two materials is provided, which forms a non-manifold node on the boundary. We then mesh all the material regions simultaneously and automatically while conforming to their boundaries directly from volumetric data. Both material change edges and interior edges are analyzed to construct tetrahedral meshes, and interior grid points are analyzed for proper hexahedral mesh construction. Finally, edge-contraction and smoothing methods are used to improve the quality of tetrahedral meshes, and a combination of pillowing, geometric flow and optimization techniques is used for hexahedral mesh quality improvement. The shrink set of pillowing schemes is defined automatically as the boundary of each material region. Several application results of our multi-material mesh generation method are also provided. Decorative reliefs are widely used for e.g. packaging and porcelain design. In periodic reliefs, the relief repeats a pattern, for example all the way around an underlying surface of revolution. Reverseengineering of existing reliefs allows them to be re-applied to different base surfaces; we show here how to segment a single repeat unit of a periodic relief starting from a scanned triangle mesh. We first briefly review how we segment the relief from the background surface using our previous work. The rest of the paper then concentrates on how we extract a single repeat unit from the relief. To do so, the user provides two points on one relief boundary which are in approximate correspondence on consecutive repeats of the relief. We first refine the relative locations of these points, and then determine a third corresponding point using relief boundary information. These are used to determine three initial cutting planes across the relief. Then surface registration strategies are utilised to refine the correspondence between adjacent repeat units. Finally, we refine the exact locations of the cutting planes by considering only surface information close to the cutting planes. This allows a repeat unit of the periodic relief to be extracted. We demonstrate that our algorithm is successful and practical, using various real scanned models: user input can be quite imprecise, and we can cope with hand-made reliefs in which the pattern units are only approximately copies of each other. The interest shown recently in the algorithmic treatment of symmetries, also known as Computational Symmetry, covers several application areas among which textile and tiles design are some of the most significant. Designers make new creations based on the symmetric repetition of motifs that they find in databases of wallpaper images. The existing methods for dealing with these images have several drawbacks because the use of heuristics achieves low recovery rates when images exhibit imperfections due to the fabrication or the handmade process.