Topological Mesh Modeling
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Introduction: Topological Mesh Modeling

Topological Mesh Modeling is an umbrella term that covers all our work based on extensions the theory of graph rotation systems. It includes (1) Orientable 2-manifol mesh modeling using graph rotation systems and its computer graphics applications, (2) Knot modeling with immersions of non-orientable manifold meshes and (3) Topological constructions that is based on geometric and physical constraints with graph rotation systems. We recently started to work on immersions of 3-manifolds as a representation to develop shape modeling systems. Click links below to go to related papers and manuscripts.
  • Orientable Mesh Modeling: We have provided a solid foundation for orientable 2-manifold mesh modeling using graph rotation systems. Based on this theory, we have developed TopMod , which is is an orientable 2-manifold mesh modeling system. TopMod provides a wide vriety of High Genus Modeling tools, Remeshings & Subdivisions, and Extrusions & Replacements. Using TopMod, one can find a wide variety of ways to create high genus shapes; almost all subdivision algorithms, wide variety of ways to remeshing shapes and new extrusions. These tools are also useful for Architectural applications, Design and Sculpting and Sketch Based Modeling.
  • Knots Modeling: We have developed provided a solid foundation for knot, link and cyclic woven object modeling using extended graph rotation systems. If we twist an arbitrary subset of edges of a mesh on an orientable surface, we can obtain non-orientable surfaces. The resulting extended graph rotation system can be used to induce a cyclic weaving on the original surface, that corresponds a 3-space immedding of a non-orientable surface.
  • Topological Constructions: Discrete Gaussian-Bonnet theorem and Gaussian curvatures related mesh topologic concepts to geometry. Using this relationship, we have developed methods to phsyically construct shapes.
  • Immersions of 3-Manifolds: Using an extension of graph rotation systems it is possible to represent 3-space immersions of 3-manifolds by employing a topological graph theory concept called 3D thickening.