Representing planar domains by polar parameterizations with parabolic parameter lines

Abstract

Polar parameterizations of star-shaped domains are based on the line segments that connect a suitably chosen center point with the points on the domain's boundary. Valid (i.e., regular everywhere except at the center point) polar parameterizations are obtained when choosing a center from the kernel of the domain. Recently, the flexibility of these polar parameterizations has been enhanced by considering so-called arc fibrations (Jüttler et al., 2019), which are polar parameterizations that use circular arcs in order to connect the center with the boundary points. We propose and analyze another generalization of polar parameterizations, which uses parabolic arcs instead of lines or circular arcs. This class of curves is simultaneously simpler (since admitting polynomial parameterizations) and more flexible.