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

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.

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.

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.

Voronoi diagram of a circle set from Voronoi diagram of a point set: II. Geometry

2001 ◽  
Vol 18 (6) ◽  
pp. 563-585 ◽  
Author(s):  
Deok-Soo Kim ◽  
Donguk Kim ◽  
Kokichi Sugihara
Keyword(s):  
Voronoi Diagram ◽  
Point Set

Computational Micrograph Registration with Sieve Processes

1996 ◽  
Vol 54 ◽  
pp. 440-441
Author(s):  
P.J. Phillips ◽  
J. Huang ◽  
S. M. Dunn
Keyword(s):  
Planar Graph ◽  
Efficient Algorithm ◽  
Spatial Organization ◽  
Spatial Arrangement ◽  
Stereo Image ◽  
Image Model ◽  
Geometric Invariants ◽  
Point Set

In this paper we present an efficient algorithm for automatically finding the correspondence between pairs of stereo micrographs, the key step in forming a stereo image. The computation burden in this problem is solving for the optimal mapping and transformation between the two micrographs. In this paper, we present a sieve algorithm for efficiently estimating the transformation and correspondence.In a sieve algorithm, a sequence of stages gradually reduce the number of transformations and correspondences that need to be examined, i.e., the analogy of sieving through the set of mappings with gradually finer meshes until the answer is found. The set of sieves is derived from an image model, here a planar graph that encodes the spatial organization of the features. In the sieve algorithm, the graph represents the spatial arrangement of objects in the image. The algorithm for finding the correspondence restricts its attention to the graph, with the correspondence being found by a combination of graph matchings, point set matching and geometric invariants.

Almost empty polygons

2003 ◽  
Vol 40 (3) ◽  
pp. 269-286 ◽  
Author(s):  
H. Nyklová
Keyword(s):  
Exact Results ◽  
Point Sets ◽  
Planar Point ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

Model and algorithm for searching for victims in emergency situations and fires using the Voronoi diagram

2019 ◽  
Vol 86 ◽  
pp. 53-61
Author(s):  
N. G. Topolskiy ◽  
◽  
A. V. Mokshantsev ◽  
To Hoang Thanh ◽  
◽  
...  
Keyword(s):  
Voronoi Diagram ◽  
Emergency Situations
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