scholarly journals ON THE WIDTH AND ROUNDNESS OF A SET OF POINTS IN THE PLANE

1999 ◽  
Vol 09 (01) ◽  
pp. 97-108 ◽  
Author(s):  
MICHIEL SMID ◽  
RAVI JANARDHAN
Keyword(s):  
Voronoi Diagram ◽  
Intersection Point ◽  
Rigorous Proof ◽  
Minimum Width ◽  
Point Set ◽  
Set Of Points

Let S be a set of points in the plane. The width (resp. roundness) of S is defined as the minimum width of any slab (resp. annulus) that contains all points of S. We give a new characterization of the width of a point set. Also, we give a rigorous proof of the fact that either the roundness of S is equal to the width of S, or the center of the minimum-width annulus is a vertex of the closest-point Voronoi diagram of S, the furthest-point Voronoi diagram of S, or an intersection point of these two diagrams. This proof corrects the characterization of roundness used extensively in the literature.

BASELINE BOUNDED HALF-PLANE VORONOI DIAGRAM

2013 ◽  
Vol 05 (03) ◽  
pp. 1350021 ◽  
Author(s):  
BING SU ◽  
YINFENG XU ◽  
BINHAI ZHU
Keyword(s):  
Voronoi Diagram ◽  
Half Plane ◽  
Line Segment ◽  
Euclidean Plane ◽  
Voronoi Region ◽  
Point Set ◽  
Set Of Points

Given a set of points P = {p1, p2, …, pn} in the Euclidean plane, with each point piassociated with a given direction vi∈ V. P(pi, vi) defines a half-plane and L(pi, vi) denotes the baseline that is perpendicular to viand passing through pi. Define a region dominated by piand vias a Baseline Bounded Half-Plane Voronoi Region, denoted as V or(pi, vi), if a point x ∈ V or(pi, vi), then (1) x ∈ P(pi, vi); (2) the line segment l(x, pi) does not cross any baseline; (3) if there is a point pj, such that x ∈ P(pj, vj), and the line segment l(x, pj) does not cross any baseline then d(x, pi) ≤ d(x, pj), j ≠ i. The Baseline Bounded Half-Plane Voronoi Diagram, denoted as V or(P, V), is the union of all V or(pi, vi). We show that V or(pi, vi) and V or(P, V) can be computed in O(n log n) and O(n2log n) time, respectively. For the heterogeneous point set, the same problem is also considered.

scholarly journals Topological Approach to Mathematical Programs with Switching Constraints

2021 ◽  
Author(s):  
Vladimir Shikhman
Keyword(s):  
Stationary Point ◽  
Mountain Pass Theorem ◽  
Strong Stability ◽  
Complete Characterization ◽  
Stationary Points ◽  
Mathematical Programs ◽  
Linear Independence Constraint Qualification ◽  
Point Set ◽  
Stationary Level

AbstractWe study mathematical programs with switching constraints (for short, MPSC) from the topological perspective. Two basic theorems from Morse theory are proved. Outside the W-stationary point set, continuous deformation of lower level sets can be performed. However, when passing a W-stationary level, the topology of the lower level set changes via the attachment of a w-dimensional cell. The dimension w equals the W-index of the nondegenerate W-stationary point. The W-index depends on both the number of negative eigenvalues of the restricted Lagrangian’s Hessian and the number of bi-active switching constraints. As a consequence, we show the mountain pass theorem for MPSC. Additionally, we address the question if the assumption on the nondegeneracy of W-stationary points is too restrictive in the context of MPSC. It turns out that all W-stationary points are generically nondegenerate. Besides, we examine the gap between nondegeneracy and strong stability of W-stationary points. A complete characterization of strong stability for W-stationary points by means of first and second order information of the MPSC defining functions under linear independence constraint qualification is provided. In particular, no bi-active Lagrange multipliers of a strongly stable W-stationary point can vanish.

Inflectional Convex Space Curves

1984 ◽  
Vol 36 (3) ◽  
pp. 537-549 ◽  
Author(s):  
Tibor Bisztriczky
Keyword(s):  
Convex Hull ◽  
Convex Space ◽  
Intersection Point ◽  
List Type ◽  
Multiple Points ◽  
Space Curves ◽  
Euclidean Three Space ◽  
Set Of Points ◽  
Three Space ◽  
Regular Closed

Let Φ be a regular closed C2 curve on a sphere S in Euclidean three-space. Let H(S)[H(Φ) ] denote the convex hull of S[Φ]. For any point p ∈ H(S), let O(p) be the set of points of Φ whose osculating plane at each of these points passes through p.1. THEOREM ([8]). If Φ has no multiple points and p ∈ H(Φ), then |0(p) | ≧ 3[4] when p is [is not] a vertex of Φ.2. THEOREM ( [9]). a) If the only self intersection point of Φ is a doublepoint and p ∈ H(Φ) is not a vertex of Φ, then |O(p)| ≧ 2.b) Let Φ possess exactly n vertices. Then(1)|O(p)| ≦ nforp ∈ H(S) and(2)if the osculating plane at each vertex of Φ meets Φ at exactly one point, |O(p)| = n if and only if p ∈ H(Φ) is not vertex.

TWO-DIMENSIONAL RANGE SEARCH BASED ON THE VORONOI DIAGRAM

2005 ◽  
Vol 15 (02) ◽  
pp. 151-166
Author(s):  
TAKESHI KANDA ◽  
KOKICHI SUGIHARA
Keyword(s):  
Voronoi Diagram ◽  
Dimensional Space ◽  
Search Problem ◽  
Two Dimensional ◽  
Worst Case ◽  
Average Case ◽  
Range Search ◽  
Almost All ◽  
Set Of Points ◽  
Dimensional Range

This paper studies the two-dimensional range search problem, and constructs a simple and efficient algorithm based on the Voronoi diagram. In this problem, a set of points and a query range are given, and we want to enumerate all the points which are inside the query range as quickly as possible. In most of the previous researches on this problem, the shape of the query range is restricted to particular ones such as circles, rectangles and triangles, and the improvement on the worst-case performance has been pursued. On the other hand, the algorithm proposed in this paper is designed for a general shape of the query range in the two-dimensional space, and is intended to accomplish a good average-case performance. This performance is actually observed by numerical experiments. In these experiments, we compare the execution time of the proposed algorithm with those of other representative algorithms such as those based on the bucketing technique and the k-d tree. We can observe that our algorithm shows the better performance in almost all the cases.

Voronoi diagram of a circle set from Voronoi diagram of a point set: I. Topology

2001 ◽  
Vol 18 (6) ◽  
pp. 541-562 ◽  
Author(s):  
Deok-Soo Kim ◽  
Donguk Kim ◽  
Kokichi Sugihara
Keyword(s):  
Voronoi Diagram ◽  
Point Set

On the Characterization of the Set of Points of λ-Continuity

1923 ◽  
Vol 25 (2) ◽  
pp. 118
Author(s):  
Henry Blumberg
Keyword(s):  
Set Of Points

scholarly journals THE POINT-SET EMBEDDABILITY PROBLEM FOR PLANE GRAPHS

2013 ◽  
Vol 23 (04n05) ◽  
pp. 357-395 ◽  
Author(s):  
THERESE BIEDL ◽  
MARTIN VATSHELLE
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Line Drawing ◽  
Np Hard ◽  
Bounded Treewidth ◽  
Straight Line ◽  
Triangulated Graphs ◽  
Point Set ◽  
Fixed Embedding ◽  
Set Of Points

In this paper, we study the point-set embeddability problem, i.e., given a planar graph and a set of points, is there a mapping of the vertices to the points such that the resulting straight-line drawing is planar? It was known that this problem is NP-hard if the embedding can be chosen, but becomes polynomial for triangulated graphs of treewidth 3. We show here that in fact it can be answered for all planar graphs with a fixed combinatorial embedding that have constant treewidth and constant face-degree. We prove that as soon as one of the conditions is dropped (i.e., either the treewidth is unbounded or some faces have large degrees), point-set embeddability with a fixed embedding becomes NP-hard. The NP-hardness holds even for a 3-connected planar graph with constant treewidth, triangulated planar graphs, or 2-connected outer-planar graphs. These results also show that the convex point-set embeddability problem (where faces must be convex) is NP-hard, but we prove that it becomes polynomial if the graph has bounded treewidth and bounded maximum degree.

CENTROID TRIANGULATIONS FROM k-SETS

2011 ◽  
Vol 21 (06) ◽  
pp. 635-659 ◽  
Author(s):  
WAEL EL ORAIBY ◽  
DOMINIQUE SCHMITT ◽  
JEAN-CLAUDE SPEHNER
Keyword(s):  
Convex Hull ◽  
Voronoi Diagram ◽  
Efficient Algorithm ◽  
Worst Case ◽  
Point Set

Given a set V of n points in the plane, no three of them being collinear, a convex inclusion chain of V is an ordering of the points of V such that no point belongs to the convex hull of the points preceding it in the ordering. We call k-set of the convex inclusion chain, every k-set of an initial subsequence of at least k points of the ordering. We show that the number of such k-sets (without multiplicity) is an invariant of V, that is, it does not depend on the choice of the convex inclusion chain. Moreover, this number is equal to the number of regions of the order-k Voronoi diagram of V (when no four points are cocircular). The dual of the order-k Voronoi diagram belongs to the set of so-called centroid triangulations that have been originally introduced to generate bivariate simplex spline spaces. We show that the centroids of the k-sets of a convex inclusion chain are the vertices of such a centroid triangulation. This leads to the currently most efficient algorithm to construct particular centroid triangulations of any given point set; it runs in O(n log n + k(n - k) log k) worst case time.

Sign in / Sign up

Export Citation Format

Share Document