Branched covering spaces are a mathematical concept which originates from complex analysis and
topology and has 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 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, and their application areas. In
this paper, we provide a framework to interactively design 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 as well as connecting tubes. This enables the user to verify important facts
about BCSs such as that they are manifold surfaces around singularities, as well as the
Riemann-Hurwitz formula which relates the Euler characteristic of the BCS to that of the original
mesh. Our system is evaluated by student researchers in scientific visualization and geometry
processing as well as faculty members in mathematics at our university who teach topology. We
include their evaluations and feedback in the paper.
Eugene Zhang received the PhD degree in computer science from the Georgia Institute of
Technology in 2004. He is currently a Professor with Oregon State University, where he is a member
in the School of Electrical Engineering and Computer Science. During part of 2011 and 2012, he was
a guest professor with the Free University of Berlin and the Max-Planck-Institute in informatics.
His research interests include computer graphics, scientific visualization, geometric modeling, and
computational topology. He received the National Science Foundation CAREER award in 2006. He has
also served as a program co-chair for CAD/CG 2013. He is a senior member of the IEEE and the
University of Stuttgart, Powerwall Room, VISUS, cellar
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