An algorithm for finding homogeneous pairs

AbstractA homogeneous pair in a graph G = (V, E) is a pair Q1, Q2 of disjoint sets of vertices in this graph such that every vertex of V (Q1 ∪ Q2) is adjacent either to all vertices of Q1 or to none of the vertices of Q1 and is adjacent either to all vertices of Q2 or to none of the vertices of Q2. Also ¦Q1¦ ⩾ 2 or ¦Q2¦⩾ 2 and ¦V (Q1 ∪ Q2)¦ ⩾ 2. In this paper we present an O(mn3)-time algorithm which determines whether a graph contains a homogeneous pair, and if possible finds one

On the expected size of the 2d visibility complex

We study the expected size of the 2D visibility complex of randomly distributed objects in the plane. We prove that the asymptotic expected number of free bitangents (which correspond to 0-faces of the visibility complex) among unit discs (or polygons of bounded aspect ratio and similar size) is linear and exhibit bounds in terms of the density of the objects. We also make an experimental assessment of the size of the visibility complex for disjoint random unit discs. We provide experimental estimates of the onset of the linear behavior and of the asymptotic slope and y-intercept of the number of free bitangents in terms of the density of discs. Finally, we analyze the quality of our estimates in terms of the density of discs.

Farthest-Polygon Voronoi Diagrams

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log^3 n) time algorithm to compute it. We also prove a number of structural properties of this diagram. In particular, a Voronoi region may consist of k-1 connected components, but if one component is bounded, then it is equal to the entire region

Interpolation problem with curvature constraints

This book contain a selection of papers presented at the 4th International Conference on Curves & Surfaces, Saint-Malo, 1999. Contribution à un ouvrage.We address the problem of controlling the curvature of a Bézier curve interpolating a given set of data. More precisely, given two points $M$ and $N$, two directions $\vec{u}$ and $\vec{v}$, and a constant $k$, we would like to find two quadratic Bézier curves $\Gamma_1$ and $\Gamma_2$ joined with continuity $G^1$, interpolating the two points $M$ and $N$, such that the tangent vectors at $M$ and $N$ have directions $\vec{u}$ and $\vec{v}$ respectively, the curvature is everywhere upper bounded by $k$, and some evaluating function, the length of the resulting curve for example, is minimized. In order to solve this problem we need first to determine the maximum curvature of quadratic Bézier curves. This problem was solved by Sapidis and Frey in 1992. Here we present a simpler formula that has an elegant geometric interpretation in terms of distances and areas determined by the control points. We then use this formula to solve several problems. In particular, we solve the variant of the curvature control problem in which $\Gamma_1$ and $\Gamma_2$ are joined with continuity $C^1$, where the length $\alpha$ between the first two control points of $\Gamma_1$ is equal to the length between the last two control points of $\Gamma_2$, and where $\alpha$ is the evaluating function to be minimized. We also study the variant where we require a continuity $G^2$, instead of $C^1$, at the junction point. Finally, given two endpoints of a quadratic Bézier curve $\Gamma$, we characterize the locus of control points such that the maximum curvature of $\Gamma$ is prescribed

Interpolation with Curvature Constraints

We address the problem of controlling the curvature of a B{ézier curve interpolating a given set of data. More precisely, given two points $M$ and $N$, two directions \vecu and \vecv, and a constant $k$, we would like to find two quadratic Bézier curves $\Gamma_1$ and $\Gamma_2$ joined with continuity $G^1$ and interpolating the two points $M$ and $N$, such that the tangent vectors at $M$ and $N$ have directions \vecu and $\vec{v}$ respectively, the curvature is everywhere upper bounded by $k$, and some evaluating function, the length of the resulting curve for example, is minimized. In order to solve this problem, we first need to determine the maximum curvature of quadratic Bézier curves. This problem was solved by Sapidis and Frey in 1992. Here we present a simpler formula that has an elegant geometric interpretation in terms of distances and areas determined by the control points. We then use this formula to solve several problems. In particular, we solve the variant of the curvature control problem in which $\Gamma_1$ and $\Gamma_2$ are joined with continuity $C^1$, where the length $\alpha$ between the first two control points of $\Gamma_1$ is equal to the length between the last two control points of $\Gamma_2$, and where $\alpha$ is the evaluating function to be minimized. We also study the variant where we require a continuity $G^2$, instead of $C^1$, at the junction point. Finally, given two endpoints of a quadratic Bézier curve $\Gamma$, we characterize the locus of control points such that the maximum curvature of $\Gamma$ is prescribed

On the Degree of Standard Geometric Predicates for Line Transversals in 3D

International audienceIn this paper we study various geometric predicates for determining the existence of and categorizing the configurations of lines in 3D that are transversal to lines or segments. We compute the degrees of standard procedures of evaluating these predicates. The degrees of some of these procedures are surprisingly high (up to 168), which may explain why computing line transversals with finite-precision floating-point arithmetic is prone to error. Our results suggest the need to explore alternatives to the standard methods of computing these quantities

Transversals to line segments in three-dimensional space

http://www.springerlink.com/We completely describe the structure of the connected components of transversals to a collection of $n$ line segments in $\mathbb{R}^3$. Generically, the set of transversal to four segments consist of zero or two lines. We catalog the non-generic cases and show that $n\geq 3$ arbitrary line segments in $\mathbb{R}^3$ admit at most $n$ connected components of line transversals, and that this bound can be achieved in certain configurations when the segments are coplanar, or they all lie on a hyperboloid of one sheet. This implies a tight upper bound of $n$ on the number of geometric permutations of line segments in $\mathbb{R}^3$

Transversals to Line Segments in R3

Colloque avec actes et comité de lecture. internationale.International audienceWe completely describe the structure of the connected components of transversals to a collection of $n$ line segments in $\mathbb{R}^3$. We show that $n\geq 3$ arbitrary line segments in $\mathbb{R}^3$ admit $0, 1, \ldots, n$ or infinitely many line transversals. In the latter case, the transversals form up to $n$ connected components