Arc fibrations of planar domains

Abstract

It is well known that star-shaped domains possess particularly simple polar parameterizations, which are formed by the line segments that connect a suitably chosen center with the points on the domain's boundary. The polar parameterization is valid (i.e., regular everywhere except for the center) if the center is located in the kernel of the domain. In the case of a domain with a smooth free-form boundary curve, the kernel is a convex region which is enclosed by the curve and (some of) the boundary's inflection tangents. These parameterizations possess numerous applications, most recently also including domain parameterization in isogeometric analysis.Since the class of star-shaped domains is quite limited, we propose to increase the flexibility of the underlying polar parameterizations by considering circular arcs that connect the center with the points on the domain's boundary. Parameterizations that are regular everywhere except at the center are said to form an arc fibration of a planar domain. We analyze the existence of an arc fibration with a given center and present an algorithm that computes it in the affirmative case. In addition, we explore the arc fibration kernel that contains the suitable center points.