The Complexity of the Outer Face in Arrangements of Random Segments
Outer face segments randomly generated in a square. 
Abstract
We investigate the complexity of the outer face in arrangements of line segments of a fixed length l in the plane, drawn uniformly at random within a square. We derive upper bounds on the expected complexity of the outer face, and establish a certain phase transition phenomenon during which the expected complexity of the outer face drops sharply as a function of the total number of segments. In particular we show that up till the phase transition the complexity of the outer face is almost linear in n, and that after the phase transition, the complexity of the outer face is roughly proportional to sqrt(n). Our study is motivated by the analysis of a practical pointlocation algorithm (socalled walkalong aline pointlocation algorithm) and indeed, it explains experimental observations of the behavior of the algorithm on arrangements of random segments.
Illustrations
The following illustrations demonstrate phase transition in the number of outer face segments randomly generated in a disc/square.
 Almost all of the outer face segments are near the boundary.
 The outer face segments appear both inside and on the boundary.
 Almost all the segments are on the outer face.
Phase transition in the number of outer face segments randomly generated in a disc. 
Phase transition in the number of outer face segments randomly generated in a square. 
Movies
Movies demonstrating the full transition illustrated above are available for download:
Links

Noga Alon, Oren Nechushtan, Dan Halperin, and Micha Sharir.
The Complexity of the Outer Face in Arrangements of Random Segments
In Proceedings of the Twentyfourth Annual Symposium on Computational Geometry (SoCG ), pages 6978, 2008 [link] [bibtex]
Contact
Oren Nechushtan  
Dan Halperin 