Comparison of three bounding regions with cubic convergence to planar freeform curves

Abstract

We compare the relative performance of bounding regions generated by three different curve-bounding methods with cubic convergence to planar freeform curves: spiral fat arcs (SFA) (Barton and Elber in Graph Models 73(2):50---57, 2011), bilens (Kumosenko in Comput Aided Geom Des 30(3):310---330, 2013), and bounding circular arcs (BCA) (Meek and Walton in J Comput Appl Math 59(2):221---231, 1995). For quantitative comparison, we consider three different criteria: geometric complexity (the number of circular arcs and line segments), construction time, and numerical stability. The BCA construction after one-step refinement (producing four circular arcs) is almost comparable to the other two methods in geometric complexity: the SFA with two circular arcs and two line segments, and the bilens with four circular arcs. In other comparison criteria, the BCA approach is more efficient and stable than the other two methods in producing a hierarchy of bounding regions that approximate a family of freeform planar curves within a given error bound.