On Computing and Updating Triangulations

Much attention has been given to triangulating sets of points and polygons (see [1] for a survey) but the problem of triangulating line segments has not been previously explored. It is well known that a polygon can always be triangulated and a simple proof of this can be found in [2]. Furthermore, efficient algorithms exist for carrying out this task [3,4]. Thus at first glance one may wonder why not just construct a simple polygon through the set of line segments and subsequently apply the algorithms of [3] or [4]. Unfortunately a set of line segments does not necessarily admit a simple circuit [5]. The reader can easily construct such an example with three parallel line segments. In the following section we provide optimal O(nlogn) algorithm for triangulating a set of n line segments. Optimality follows from the fact that Ω(nlogn) time is a lower bound for triangulating a set of points [6, p. 187] which is a set of line segments of zero length. Section 3 is devoted to presenting algorithms for inserting and deleting edges from triangulations.