# Pseudo-line arrangements: Duality, algorithms, and applications

Published

Journal Article

A finite collection of x-monotone unbounded Jordan curves in the plane is called a family of pseudo-lines if every pair of curves intersect in at most one point, and the two curves cross each other there. Let L be such a collection of n pseudo-lines, and let P be a set of m points in R 2. Extending a result of Goodman [Discrete Math., 32 (1980), pp. 27-35], we define a duality transform that maps L to a set L* of points in R 2 and P to a set P* of (x-monotone) pseudo-lines in R 2, so that the incidence and the "above-below" relations between the points and the pseudo-lines are preserved. We present an efficient algorithm for computing the dual arrangement A(P*) under an appropriate model of computation. We also present a dynamic data structure for reporting, in O(m ε+k) time, all k points of P that lie below a query arc, which is either a circular arc or a portion of the graph of a polynomial of fixed degree. This result is needed for computing the dual arrangement for certain classes of pseudo-lines arising in several applications, but is also interesting in its own right. We present a few applications of our dual arrangement algorithm, such as computing incidences between points and pseudo-lines and computing a subset of faces in a pseudo-line arrangement. Next, we present an efficient algorithm for cutting a set of circles into arcs so that every pair of arcs intersect in at most one point, i.e., the resulting arcs constitute a collection of pseudo-segments. By combining this algorithm with our algorithm for computing the dual arrangement of pseudo-lines, we obtain efficient algorithms for several problems involving arrangements of circles or circular arcs, such as reporting or counting incidences between points and circles and computing a set of marked faces in arrangements of circles. © 2005 Society for Industrial and Applied Mathematics.

### Full Text

### Duke Authors

### Cited Authors

- Agarwal, PK; Sharir, M

### Published Date

- June 30, 2005

### Published In

### Volume / Issue

- 34 / 3

### Start / End Page

- 526 - 552

### International Standard Serial Number (ISSN)

- 0097-5397

### Digital Object Identifier (DOI)

- 10.1137/S0097539703433900

### Citation Source

- Scopus