Construction and visualization of branched covering spaces

Abstract

Branched covering spaces are a mathematical concept which originates from complex analysis and topology and has found applications in 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 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 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 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 that they are manifold surfaces around singularities and the Riemann-Hurwitz formula which relates the Euler characteristic of the BCS to that of the original mesh.