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.

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.

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 ≤ ∞.

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.

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].

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 Minimum Cell Connection in Line Segment Arrangements

2017 ◽  
Vol 27 (03) ◽  
pp. 159-176
Author(s):  
Helmut Alt ◽  
Sergio Cabello ◽  
Panos Giannopoulos ◽  
Christian Knauer
Keyword(s):  
Linear Time ◽  
Optimal Solution ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Constant Number ◽  
Line Segments ◽  
Straight Line ◽  
Minimum Number ◽  
Number Of Segments ◽  
Connection Problems

We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points [Formula: see text] and [Formula: see text] in different cells of the induced arrangement: [(i)] compute the minimum number of segments one needs to remove so that there is a path connecting [Formula: see text] to [Formula: see text] that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting [Formula: see text] to [Formula: see text] must stay inside a given polygon [Formula: see text] with a constant number of holes, the segments are contained in [Formula: see text], and the endpoints of the segments are on the boundary of [Formula: see text]. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution.

A Duplicate Chinese Document Image Retrieval System Based on Line Segment Feature in Character Image Block

2011 ◽  
pp. 14-23
Author(s):  
Yung-Kuan Chan ◽  
Tung-Shou Chen ◽  
Yu-An Ho
Keyword(s):  
Image Retrieval ◽  
Line Segment ◽  
Document Image ◽  
Experimental Results ◽  
Document Images ◽  
Rapid Progress ◽  
Image Block ◽  
Line Segments ◽  
Image Retrieval System ◽  
On Line

With the rapid progress of digital image technology, the management of duplicate document images is also emphasized widely. As a result, this paper suggests a duplicate Chinese document image retrieval (DCDIR) system, which uses the ratio of the number of black pixels to that of white pixels on the scanned line segments in a character image block as the feature of the character image block. Experimental results indicate that the system can indeed effectively and quickly retrieve the desired duplicate Chinese document image from a database.

Plate and line segment processes

1978 ◽  
Vol 15 (03) ◽  
pp. 494-501 ◽  
Author(s):  
N. A. Fava ◽  
L. A. Santaló
Keyword(s):  
Line Segment ◽  
Integral Geometry ◽  
Random Variables ◽  
Random Processes ◽  
Line Segments ◽  
Expected Values

Random processes of convex plates and line segments imbedded in R 3 are considered in this paper, and the expected values of certain random variables associated with such processes are computed under a mean stationarity assumption, by resorting to some general formulas of integral geometry.

Planar Lattices

1975 ◽  
Vol 27 (3) ◽  
pp. 636-665 ◽  
Author(s):  
David Kelly ◽  
Ivan Rival
Keyword(s):  
Line Segment ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Straight Line Segment ◽  
Line Segments ◽  
Small Circle ◽  
Partially Ordered ◽  
Straight Line ◽  
Finite Partially Ordered Set ◽  
Planar Lattices

A finite partially ordered set (poset) P is customarily represented by drawing a small circle for each point, with a lower than b whenever a < b in P, and drawing a straight line segment from a to b whenever a is covered by b in P (see, for example, G. Birkhoff [2, p. 4]). A poset P is planar if such a diagram can be drawn for P in which none of the straight line segments intersect.

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