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Event Type:

Applied Mathematics and Computation Seminar

Date/Time:

Friday, March 3, 2017 - 12:00 to 13:00

Location:

GLK 113

Guest Speaker:

Institution:

CSEE

Abstract:

Branched covering spaces are a mathematical concept which originates from complex analysis and topology and has found applications in tensor field topology and geometry remeshing. 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 {\em 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. In this paper, 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 BCS 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 BCS such that they are manifold surface around singularities as well as the {\em Riemann-Hurwitz formula} which relates the Euler characteristic of the BCS to that of the original mesh.