Explicit cylindrical maps for general tubular shapes

A solid cylindrical parameterization is a volumetric map between a tubular shape and a right cylinder embedded in the polar coordinate reference system. This paper introduces a novel approach to derive smooth (i.e., harmonic) cylindrical parameterizations for solids with arbitrary topology. Differently from previous approaches our mappings are both fully explicit and bi-directional, meaning that the three polar coordinates are encoded for both internal and boundary points, and that for any point within the solid we can efficiently move from the object space to the parameter space and vice-versa. To represent the discrete mapping, we calculate a tetrahedral mesh that conforms with the solid's boundary and accounts for the periodic and singular structure of the parametric domain. To deal with arbitrary topologies, we introduce a novel approach to exhaustively partition the solid into a set of tubular parts based on a curve-skeleton. Such a skeleton can be either computed by an algorithm or provided by the user. Being fully explicit, our mappings can be readily exploited by off-the-shelf algorithms (e.g., for iso-contouring). Furthermore, when the input shape is made of tubular parts, our method produces low-distortion parameterizations whose iso-surfaces fairly follow the geometry in a natural way. We show how to exploit this characteristic to produce high-quality hexahedral and shell meshes.