Arrangements of Segments that Share Endpoints: Single Face Results

We provide new combinatorial bounds on the complexity of a face in an arrangement of segments in the plane. In particular, we show that the complexity of a single face in an arrangement of n line segments determined by h endpoints is O(hlogh). While the previous upper bound, O(n (n)), is tight for segments with distinct endpoints, it is far from being optimal when n = (h2 ). Our result shows that, in a sense, the fundamental combinatorial complexity of a face arises not as a result of the number of segments, but rather as a result of the number of endpoints.

• Type: Conference
• Year: 1991
• Where: COMPGEOM
• Authors: Esther M. Arkin, Dan Halperin, Klara Kedem, Joseph S. B. Mitchell, Nir Naor
• Keywords: COMPGEOM 1991, Discrete Geometry, Distinct Endpoints, Line Segments, Single Face