Graduate Thesis Or Dissertation
 

Branched Covering Space Construction and Visualization

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https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/0g354j017

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  • Branched covering spaces are a mathematical concept which originates from complex analysis and topology and has found applications in tensor field topology and geometry re-meshing. Given a manifold surface and an N-way rotational symmetry field, a branched covering space is a manifold surface that has an N-to-1 map to the original surface except at the so-called ramification points, which correspond to the singularities in the rotational symmetry field. Understanding the notion and mathematical properties of branched covering spaces is important to researchers in tensor field visualization and geometry processing. The theory of covering space is used in many research areas in computer science such as vector and tensor field topological simplification, parametrization, and re-meshing. In this research, we provide a framework to construct and visualize the branched covering space (BCS) of an input mesh surface and a rotational symmetry field defined on it. In our framework, the user can visualize not only the BCSs but also their construction process. In addition, our system allows the user to design the geometric realization of the BCS using mesh deformation techniques. This enables the user to verify important facts about BCSs such as they are manifold surface around singularities as well as the Riemann-Hurwitz formula which relates the Euler characteristic of the BCS to that of the original mesh. By enabling the design of BCS,our system can be used to generate meshes with a relatively large number of self-intersections.
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