Good global behavior of offsets to plane algebraic curves

In [Alcazar, J.G., Sendra, J.R. 2006. Local shape of offsets to rational algebraic curves. Tech. Report SFB 2006-22 (RICAM, Austria); Alcazar, J.G., Sendra, J.R. 2007. Local shape of offsets to algebraic curves. Journal of Symbolic Computation 42, 338–351], the notion of good local behavior of an offset to an algebraic curve was introduced to mean that the topological behavior of the offset curve was locally good, i.e. that the shape of the starting curve and of its offset were locally the same. Here, we introduce the notion of good global behavior to describe that the offset behaves globally well, from a topological point of view, so that it can be decomposed as the union of two curves (maybe not algebraic) each one with the topology of the starting curve. We relate this notion with that of good local behavior, and we give sufficient conditions for the existence of an interval of distances (0, γ ) such that for all d ∈ (0, γ ) the topological behavior of the offset Od (C) is both locally and globally nice. A similar analysis for the trimmed offset is also done. c © 2008 Elsevier Ltd. All rights reserved.