Convex-Arc Drawings of Pseudolines

A weak pseudoline arrangement is a topological generalization of a line arrangement, consisting of curves topologically equivalent to lines that cross each other at most once. We consider arrangements that are outerplanar---each crossing is incident to an unbounded face---and simple---each crossing point is the crossing of only two curves. We show that these arrangements can be represented by chords of a circle, by convex polygonal chains with only two bends, or by hyperbolic lines. Simple but non-outerplanar arrangements (non-weak) can be represented by convex polygonal chains or convex smooth curves of linear complexity.Comment: 11 pages, 8 figures. A preliminary announcement of these results was made as a poster at the 21st International Symposium on Graph Drawing, Bordeaux, France, September 2013, and published in Lecture Notes in Computer Science 8242, Springer, 2013, pp. 522--52

Finding Pairwise Intersections Inside a Query Range

We study the following problem: preprocess a set O of objects into a data structure that allows us to efficiently report all pairs of objects from O that intersect inside an axis-aligned query range Q. We present data structures of size $O(n({\rm polylog} n))$ and with query time $O((k+1)({\rm polylog} n))$ time, where k is the number of reported pairs, for two classes of objects in the plane: axis-aligned rectangles and objects with small union complexity. For the 3-dimensional case where the objects and the query range are axis-aligned boxes in R^3, we present a data structures of size $O(n\sqrt{n}({\rm polylog} n))$ and query time $O((\sqrt{n}+k)({\rm polylog} n))$. When the objects and query are fat, we obtain $O((k+1)({\rm polylog} n))$ query time using $O(n({\rm polylog} n))$ storage

On finding widest empty curved corridors

Open archive-ElsevierAn α-siphon of width w is the locus of points in the plane that are at the same distance w from a 1-corner polygonal chain C such that α is the interior angle of C. Given a set P of n points in the plane and a fixed angle α, we want to compute the widest empty α-siphon that splits P into two non-empty sets.We present an efficient O(n log3 n)-time algorithm for computing the widest oriented α-siphon through P such that the orientation of a half-line of C is known.We also propose an O(n3 log2 n)-time algorithm for the widest arbitrarily-oriented version and an (nlog n)-time algorithm for the widest arbitrarily-oriented α-siphon anchored at a given point