Abstract. For any finite set A of n points in R 2, we define a (3n − 3)dimensional simple polyhedron whose face poset is isomorphic to the poset of “non-crossing marked graphs ” with vertex set A, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of non-crossing graphs on A appears as the complement of the star of a face in that polyhedron. The polyhedron has a unique maximal bounded face, of dimension 2ni + n − 3 where ni is the number of points of A in the interior of conv(A). The vertices of this polytope are all the pseudo-triangulations of A, and the edges are flips of two types: the traditional diagonal flips (in pseudo-triangulations) and the removal or insertion of a single edge. As a by-product of our construction we prove that all pseudo-triangulations are infinitesimally rigid graphs. 1.

In this paper, we show some properties of a pseudotriangle and present three combinatorial bounds: the ratio of the size of minimum pseudotriangulation of a point set S and the size of minimal pseudotriangulation contained in a triangulation T, the ratio of the size of the best minimal pseudotriangulation and the worst minimal pseudotriangulation both contained in a given triangulation T, and the maximum number of edges in any settings of S and T. We also present a linear-time algorithm for finding a minimal pseudotriangulation contained in a given triangulation. We finally study the minimum pseudotriangulation containing a given set of non-crossing line segments.

One-degree-of-freedom mechanisms induced by minimum pseudo-triangulations with one convex hull edge removed have been recently introduced by the author to solve a family of non-colliding motion planning problems for planar robot arms (open or closed polygonal chains). They induce canonical roadmaps in configuration spaces of simple planar polygons with fixed edge lengths. While the combinatorial part...

We present kinetic data structures for detecting collisions between a set of polygons that are moving continuously. Unlike classical collision detection methods that rely on bounding volume hierarchies, our method is based on deformable tilings of the free space surrounding the polygons. The basic shape of our tiles is that of a pseudo-triangle, a shape sufficiently flexible to allow extensive deformation, yet structured enough to make detection of self-collisions easy. We show different schemes for maintaining pseudotriangulations as a kinetic data structure, and we analyze their performance. Specifically, we first describe an algorithm for maintaining a pseudo-triangulation of a point set, and show that the pseudo-triangulation changes only quadratically many times if points move along algebraic arcs of constant degree. In addition, by refining the pseudotriangulation, we show triangulations of points that only change about O(n 7/3) times for linear motion. We then describe an algorithm for maintaining a pseudo-triangulation of a set of convex polygons. Finally, we extend our algorithm to the general case of maintaining a pseudo-triangulation of a set of moving or deforming simple polygons.

We present a new method to implement in constant amortized time the ip operation of the so-called 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 left-turn 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).)

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.

We pose a monotonicity conjecture on the number of pseudo-triangulations of any planar point set, and check it on two prominent families of point sets, namely the so-called double circle and double chain. The latter has asymptotically 12 n n Θ(1) pointed pseudo-triangulations, 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

Abstract. We consider the problem of computing a minimum weight pseudo-triangulation of a set S of n points in the plane. We first present an O(n log n)-time algorithm that produces a pseudo-triangulation of weight O(log n·wt(M(S))) which is shown to be asymptotically worst-case optimal, i.e., there exists a point set S for which every pseudo-triangulation has weight Ω(log n · wt(M(S))), where wt(M(S)) is the weight of a minimum spanning tree of S. We also present a constant factor approximation algorithm running in cubic time. In the process we give an algorithm that produces a minimum weight pseudo-triangulation of a simple polygon. 1

A pseudo-triangle is a simple polygon with exactly three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as planar bar-and-joint 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, straight-line drawings from abstract versions called combinatorial pseudo-triangulations, algorithms and applications of pseudo-triangulations.

Abstract. Topological sweep can contribute to efficient implementations of various algorithms for data analysis.Real data, however, has degeneracies.The modification of the topological sweep algorithm presented here handles degenerate cases such as parallel or multiply concurrent lines without requiring numerical perturbations to achieve general position.Our method maintains the O(n 2) and O(n) time and space complexities of the original algorithm, and is robust and easy to implement. We present experimental results. 1