Results 1  10
of
29
Topologically Sweeping Visibility Complexes via Pseudotriangulations
, 1996
"... This paper describes a new algorithm for constructing the set of free bitangents of a collection of n disjoint convex obstacles of constant complexity. The algorithm runs in time O(n log n + k), where k is the output size, and uses O(n) space. While earlier algorithms achieve the same optimal run ..."
Abstract

Cited by 85 (9 self)
 Add to MetaCart
This paper describes a new algorithm for constructing the set of free bitangents of a collection of n disjoint convex obstacles of constant complexity. The algorithm runs in time O(n log n + k), where k is the output size, and uses O(n) space. While earlier algorithms achieve the same optimal running time, this is the first optimal algorithm that uses only linear space. The visibility graph or the visibility complex can be computed in the same time and space. The only complicated data structure used by the algorithm is a splittable queue, which can be implemented easily using redblack trees. The algorithm is conceptually very simple, and should therefore be easy to implement and quite fast in practice. The algorithm relies on greedy pseudotriangulations, which are subgraphs of the visibility graph with many nice combinatorial properties. These properties, and thus the correctness of the algorithm, are partially derived from properties of a certain partial order on the faces of th...
Tight degree bounds for pseudotriangulations of points
, 2003
"... We show that every set of n points in general position has a minimum pseudotriangulation whose maximum vertex degree is five. In addition, we demonstrate that every point set in general position has a minimum pseudotriangulation whose maximum face degree is four (i.e., each interior face of this p ..."
Abstract

Cited by 32 (9 self)
 Add to MetaCart
(Show Context)
We show that every set of n points in general position has a minimum pseudotriangulation whose maximum vertex degree is five. In addition, we demonstrate that every point set in general position has a minimum pseudotriangulation whose maximum face degree is four (i.e., each interior face of this pseudotriangulation has at most four vertices). Both degree bounds are tight. Minimum pseudotriangulations realizing these bounds (individually but not jointly) can be constructed in O(n log n) time.
PseudoTriangulations  a Survey
 CONTEMPORARY MATHEMATICS
"... A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory an ..."
Abstract

Cited by 25 (5 self)
 Add to MetaCart
A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of current literature includes combinatorial properties and counting of special classes, rigidity theoretical results, representations as polytopes, straightline drawings from abstract versions called combinatorial pseudotriangulations, algorithms and applications of pseudotriangulations.
Counting Triangulations and PseudoTriangulations of Wheels
 IN PROC. 13TH CANAD. CONF. COMPUT. GEOM
, 2001
"... Motivated by several open questions on triangulations and pseudotriangulations, we give closed form expressions for the number of triangulations and the number of minimum pseudotriangulations of n points in wheel configurations, that is, with n  1 in convex position. Although the numbers of trian ..."
Abstract

Cited by 23 (5 self)
 Add to MetaCart
Motivated by several open questions on triangulations and pseudotriangulations, we give closed form expressions for the number of triangulations and the number of minimum pseudotriangulations of n points in wheel configurations, that is, with n  1 in convex position. Although the numbers of triangulations and pseudotriangulations vary depending on the placement of the interior point, their difference is always the (n2)nd Catalan number. We also prove an inequality #PT # 3 i #T for the numbers of minimum pseudotriangulations and triangulations of any point configuration with i interior points.
Convexity Minimizes PseudoTriangulations
 COMPUTATIONAL GEOMETRY 28 (2004) 3–10
, 2004
"...
..."
(Show Context)
On the number of pseudotriangulations of certain point sets
 J. Combin. Theory Ser. A
, 2007
"... We pose a monotonicity conjecture on the number of pseudotriangulations of any planar point set, and check it on two prominent families of point sets, namely the socalled double circle and double chain. The latter has asymptotically 12 n n Θ(1) pointed pseudotriangulations, which lies significant ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
(Show Context)
We pose a monotonicity conjecture on the number of pseudotriangulations of any planar point set, and check it on two prominent families of point sets, namely the socalled double circle and double chain. The latter has asymptotically 12 n n Θ(1) pointed pseudotriangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far. ⋆ Parts of this work were done while the authors visited the Departament de
HOMOTOPIC FRÉCHET DISTANCE BETWEEN CURVES OR, WALKING YOUR DOG IN THE WOODS IN POLYNOMIAL TIME
, 2008
"... The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, which requi ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, which requires the leash itself to move continuously over time. For example, for curves in the punctured plane, the leash cannot pass through or jump over the obstacles (“trees”). We describe a polynomialtime algorithm to compute the homotopic Fréchet distance between two given polygonal curves in the plane minus a given set of polygonal obstacles.
Segment Endpoint Visibility Graphs are Hamiltonian
 COMPUT. GEOM
, 2002
"... We show that the segment endpoint visibility graph of any finite set of disjoint line segments in the plane admits a simple Hamiltonian polygon, if not all segments are collinear. This proves a conjecture of Mirzaian. ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
We show that the segment endpoint visibility graph of any finite set of disjoint line segments in the plane admits a simple Hamiltonian polygon, if not all segments are collinear. This proves a conjecture of Mirzaian.
A Sum of Squares Theorem for Visibility Complexes and Applications
, 2001
"... We present a new method to implement in constant amortized time the ip operation of the socalled Greedy Flip Algorithm, an optimal algorithm to compute the visibility graph or the visibility complex of a collection of pairwise disjoint bounded convex sets of constant complexity. The method uses sim ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
We present a new method to implement in constant amortized time the ip operation of the socalled Greedy Flip Algorithm, an optimal algorithm to compute the visibility graph or the visibility complex of a collection of pairwise disjoint bounded convex sets of constant complexity. The method uses simple data structures and only the leftturn or counterclockwise predicate; it relies, among other things, on a sum of squares like theorem for visibility complexes stated and proved in this paper. (The sum of squares theorem for a simple arrangement of lines states that the average value of the square of the number of vertices of a face of the arrangement is a O(1).)