Extended graph rotation systems as a model for cyclic weaving on orientable surfaces

We present an extension of the theory of graph rotation systems, which has been a widely used model for graph imbeddings on topological surfaces. The extended model is quite beyond what is needed to specify graph imbeddings on surfaces, and it can be used to represent and generate link structures immersed on surfaces. Since link structures immersed on surfaces can be viewed as woven images in 3D space, the extended graph rotation systems provide a well-formulated mathematical model for developing a topologically robust graphics system with strong interactive operations for the design of woven images in 3D mesh-modeling and computer-aided sculpting.

[1]  G. C. Shephard,et al.  Tilings and Patterns , 1990 .

[2]  Martti Mäntylä,et al.  Introduction to Solid Modeling , 1988 .

[3]  R. H. Crowell,et al.  Introduction to Knot Theory , 1977 .

[4]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[5]  Ergun Akleman,et al.  Cyclic plain-weaving on polygonal mesh surfaces with graph rotation systems , 2009, ACM Trans. Graph..

[6]  Ergun Akleman,et al.  Guaranteeing the 2-Manifold Property for Meshes with doubly Linked Face List , 1999, Int. J. Shape Model..

[7]  Kunio Murasugi,et al.  Knot theory and its applications , 1996 .

[8]  L. Heffter.  Ueber das Problem der Nachbargebiete , 1891 .

[9]  G. C. Shephard,et al.  Satins and Twills: An Introduction to the Geometry of Fabrics , 1980 .

[10]  Ergun Akleman,et al.  A minimal and complete set of operators for the development of robust manifold mesh modelers , 2003, Graph. Model..

[11]  Bruce G. Baumgart.  Winged edge polyhedron representation. , 1972 .

[12]  J. Gross,et al.  The topological theory of current graphs , 1974 .

[13]  Jianer Chen,et al.  Overlap matrices and total imbedding distributions , 1994, Discret. Math..

[14]  Ergun Akleman,et al.  A New Corner cutting Scheme with tension and Handle-Face Reconstruction , 2001, Int. J. Shape Model..

[15]  Jianer Chen.  The distribution of graph imbeddings on topological surfaces , 1991 .

[16]  Dan Suciu,et al.  Journal of the ACM , 2006 .

[17]  L. Beineke,et al.  Topics in Topological Graph Theory , 2009 .

[18]  Ergun Akleman,et al.  Cyclic twill-woven objects , 2011, Comput. Graph..

[19]  Xiaoya Zha.  Closed 2-cell embeddings of 4 cross-cap embeddable graphs , 1996, Discret. Math..

[20]  Saul Stahl,et al.  Generalized Embedding Schemes , 1978, J. Graph Theory.

[21]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[22]  Ergun Akleman,et al.  A new paradigm for changing topology during subdivision modeling , 2000, Proceedings the Eighth Pacific Conference on Computer Graphics and Applications.

[23]  Ergun Akleman,et al.  Cyclic twill-woven objects , 2010, SIGGRAPH '10.

[24]  Saul Stahl,et al.  Graphs and Surfaces , 2013 .

[25]  Jonathan L. Gross,et al.  Finding a maximum-genus graph imbedding , 1988, JACM.

[26]  Jonathan L. Gross,et al.  Topological Graph Theory , 1987, Handbook of Graph Theory.

[27]  G. Budworth.  The Knot Book , 1983 .