ON THE FARTHEST LINE-SEGMENT VORONOI DIAGRAM

2013 ◽  
Vol 23 (06) ◽  
pp. 443-459 ◽  
Author(s):  
EVANTHIA PAPADOPOULOU ◽  
SANDEEP KUMAR DEY
Keyword(s):  
Convex Hull ◽  
Voronoi Diagram ◽  
Line Segment ◽  
Euclidean Plane ◽  
Line Segments ◽  
Polygonal Curve ◽  
Number Of Faces ◽  
Farthest Point ◽  
Construction Algorithms ◽  
Standard Techniques

The farthest line-segment Voronoi diagram illustrates properties surprisingly different from its counterpart for points: Voronoi regions may be disconnected and they are not characterized by convex-hull properties. In this paper we introduce the farthest hull and its Gaussian map as a closed polygonal curve that characterizes the regions of the farthest line-segment Voronoi diagram, and derive tighter bounds on the (linear) size of this diagram. With the purpose of unifying construction algorithms for farthest-point and farthest line-segment Voronoi diagrams, we adapt standard techniques to construct a convex hull and compute the farthest hull in O(n log n) or output sensitive O(n log h) time, where n is the number of line-segments and h is the number of faces in the corresponding farthest Voronoi diagram. As a result, the farthest line-segment Voronoi diagram can be constructed in output sensitive O(n log h) time. Our algorithms are given in the Euclidean plane but they hold also in the general Lp metric, 1 ≤ p ≤ ∞.

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.

Some properties of line segment processes

1976 ◽  
Vol 13 (1) ◽  
pp. 96-107 ◽  
Author(s):  
Philip Parker ◽  
Richard Cowan
Keyword(s):  
Random Process ◽  
Line Segment ◽  
Euclidean Plane ◽  
Line Segments ◽  
Total Length ◽  
Number Of Segments

This paper formulates the random process of line-segments in the Euclidean plane. Under conditions more general than Poisson, expressions are obtained, for Borel A ⊂ R2, for the first moments of M(A), the number of segment mid-points in A; N(A), the number of segments which intersect with convex A; S(A), the total length within A of segments crossing A; and C(A) the number of segment-segment crossings within A. In the case of Poisson mid-points, the distribution of the rth nearest line-segment to a given point is found.

Some properties of line segment processes

1976 ◽  
Vol 13 (01) ◽  
pp. 96-107 ◽  
Author(s):  
Philip Parker ◽  
Richard Cowan
Keyword(s):  
Random Process ◽  
Line Segment ◽  
Euclidean Plane ◽  
Line Segments ◽  
Total Length ◽  
Number Of Segments

This paper formulates the random process of line-segments in the Euclidean plane. Under conditions more general than Poisson, expressions are obtained, for Borel A ⊂ R 2, for the first moments of M(A), the number of segment mid-points in A; N(A), the number of segments which intersect with convex A; S(A), the total length within A of segments crossing A; and C(A) the number of segment-segment crossings within A. In the case of Poisson mid-points, the distribution of the rth nearest line-segment to a given point is found.

MINIMUM POLYGON TRANSVERSALS OF LINE SEGMENTS

1995 ◽  
Vol 05 (03) ◽  
pp. 243-256 ◽  
Author(s):  
DAVID RAPPAPORT
Keyword(s):  
Convex Hull ◽  
General Problem ◽  
Line Segment ◽  
Simple Polygon ◽  
Fixed Number ◽  
Line Segments ◽  
Geometric Objects ◽  
Finite Set ◽  
Minimum Perimeter ◽  
Set Of Points

Let S be used to denote a finite set of planar geometric objects. Define a polygon transversal of S as a closed simple polygon that simultaneously intersects every object in S, and a minimum polygon transversal of S as a polygon transversal of S with minimum perimeter. If S is a set of points then the minimum polygon transversal of S is the convex hull of S. However, when the objects in S have some dimension then the minimum polygon transversal and the convex hull may no longer coincide. We consider the case where S is a set of line segments. If the line segments are constrained to lie in a fixed number of orientations we show that a minimum polygon transversal can be found in O(n log n) time. More explicitely, if m denotes the number of line segment orientations, then the complexity of the algorithm is given by O(3mn+log n). The general problem for line segments is not known to be polynomial nor is it known to be NP-hard.

Problems with handling spatial data — the voronoi approach

CISM journal ◽  
1991 ◽  
Vol 45 (1) ◽  
pp. 65-80 ◽  
Author(s):  
Christopher M. Gold
Keyword(s):  
Error Estimation ◽  
Voronoi Diagram ◽  
Spatial Data ◽  
Euclidean Plane ◽  
Interpolation Error ◽  
Line Segments ◽  
Definition Of ◽  
Incremental Construction

Experience with the handling of spatial data on a computer led to the identification of a variety of “awkward” problems, including interpolation, error estimation and dynamic polygon building and editing. Many of the problems encountered could be classified as “spatial adjacency” issues. The Voronoi diagram of points and line segments in the Euclidean plane is shown to give a functional definition of spatial adjacency. The basic operations for incremental construction and maintenance of this Voronoi diagram and its dual are described, and a variety of applications are outlined.

Less is more: Simple stimuli provoke more variable search strategies.

2020 ◽  
Author(s):  
Anna Nowakowska ◽  
Alasdair D F Clarke ◽  
Jessica Christie ◽  
Josephine Reuther ◽  
Amelia R. Hunt
Keyword(s):  
Line Segment ◽  
Search Strategies ◽  
Complex Objects ◽  
Search Behaviour ◽  
Line Segments ◽  
Surface Level ◽  
Efficient Search ◽  
Less Is More ◽  
Dramatic Shift

We measured the efficiency of 30 participants as they searched through simple line segment stimuli and through a set of complex icons. We observed a dramatic shift from highly variable, and mostly inefficient, strategies with the line segments, to uniformly efficient search behaviour with the icons. These results demonstrate that changing what may initially appear to be irrelevant, surface-level details of the task can lead to large changes in measured behaviour, and that visual primitives are not always representative of more complex objects.

An Efficient Improved Algorithm of Convex Hull

2014 ◽  
Vol 602-605 ◽  
pp. 3104-3106
Author(s):  
Shao Hua Liu ◽  
Jia Hua Zhang
Keyword(s):  
Convex Hull ◽  
Line Segment ◽  
Main Idea ◽  
Discrete Point ◽  
Directed Line ◽  
Generation Algorithm ◽  
Simple Program ◽  
Point Set ◽  
High Computational Efficiency ◽  
Improved Algorithm

This paper introduced points and directed line segment relation judgment method, the characteristics of generation and Graham method using the original convex hull generation algorithm of convex hull discrete points of the convex hull, an improved algorithm for planar discrete point set is proposed. The main idea is to use quadrilateral to divide planar discrete point set into five blocks, and then by judgment in addition to the four district quadrilateral internally within the point is in a convex edge. The result shows that the method is relatively simple program, high computational efficiency.

Explanation for Neon Color Effect in Achromatic Line Segments on Chromatic Inducers Based on the Multiple Interpretation Hypothesis

2005 ◽  
Vol 101 (1) ◽  
pp. 267-282
Author(s):  
Seiyu Sohmiya
Keyword(s):  
Visual System ◽  
Line Segment ◽  
Color Contrast ◽  
Achromatic Color ◽  
Line Segments ◽  
Color Effect ◽  
Complementary Color ◽  
Multiple Interpretation ◽  
Color Induction ◽  
Simultaneous Color Contrast

In van Tuijl's neon configurations, an achromatic line segment on a blue inducer produces yellowish illusory color in the illusory area. This illusion has been explained based on the idea of the complementary color induced by the blue inducer. However, it is proposed here that this illusion can be also explained by introducing the assumption that the visual system unconsciously interprets an achromatic color as information that is constituted by transparent and nontransparent colors. If this explanation is correct, not only this illusion, but also the simultaneous color contrast illusion can be explained without using the idea of the complementary color induction.

ALPHA-REGULAR STICK UNKNOTS

2012 ◽  
Vol 21 (06) ◽  
pp. 1250059 ◽  
Author(s):  
CHRISTOPHER FRAYER ◽  
CHRISTOPHER SCHAFHAUSER
Keyword(s):  
Line Segment ◽  
Line Segments

Suppose Pn is a regular n-gon in ℝ2. An embedding f : Pn ↪ ℝ3 is called an α-regular stick knot provided the image of each side of Pn under f is a line segment of length 1 and any two consecutive line segments meet at an angle of α. The main result of this paper proves the existence of α-regular stick unknots for odd n ≥ 7 with α in the range [Formula: see text]. All knots constructed will have trivial knot type, and we will show that any non-trivial α-regular stick knot must have [Formula: see text].

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