LABELING A RECTILINEAR MAP WITH SLIDING LABELS

2001 ◽  
Vol 11 (02) ◽  
pp. 167-179 ◽  
Author(s):  
SUNG KWON KIM ◽  
CHAN-SU SHIN ◽  
TAE-CHEON YANG
Keyword(s):  
Approximation Algorithms ◽  
Finite Number ◽  
Vertical Line ◽  
Line Segments ◽  
Sliding Labels

A rectilinear map consists of a set of mutually non-intersecting rectilinear (i.e., horizontal or vertical) line segments, and each segment is allowed to use a rectangular label of height B and length the same as the segment. Sliding labels are not restricted to any finite number of predefined positions but can slide and be placed at any position as long as it intersects the segment. This paper considers three versions of the problem of labeling a rectilinear map with sliding labels and presents efficient exact and approximation algorithms for them.

A NEW METHOD OF PATTERN SHOOTING

Geophysics ◽  
1955 ◽  
Vol 20 (3) ◽  
pp. 539-564 ◽  
Author(s):  
J. O. Parr ◽  
W. H. Mayne
Keyword(s):  
Finite Number ◽  
Finite Length ◽  
Vertical Line ◽  
New Method ◽  
Horizontal Line ◽  
Wavelength Band ◽  
Continuous Band ◽  
Reversed Polarity ◽  
Multiple Detectors ◽  
Horizontal Area

In areas where reflection shooting is difficult, it is often necessary to attenuate the energy in a broad continuous band of disturbing wavelengths to less than a few hundredths of what would be recorded if all units were bunched together. The wavelength band of the attenuated energy should be adjacent to the band of reflection wavelengths received. Attenuation of the undesired energy is best accomplished with multiple detectors or charges. In many areas the pattern should attenuate energy horizontally propagated in all directions, not just in the direction of the detector line. Neither a finite number of uniformly effective, uniformly spaced units in line nor a uniformly effective sheet of finite length will accomplish this result. A system for gradation of the effectiveness of units described in this paper does produce this result (not only for in‐line disturbances but also for disturbances coming in from the side of the line). The attenuation band can be made broad with good attenuation or narrower with still better attenuation, as desired. The variation of effectiveness can be applied to detectors or charges arranged in a horizontal line, over a horizontal area, in a vertical line, or over a vertical area. The principle of varying effectiveness can also be applied to reversed‐polarity detectors in order to accentuate certain apparent wavelengths.

The Effect of Environmental Pitch on Perceived Optic Slant and Eye Level: Lines vs Dots

Perception ◽  
1996 ◽  
Vol 25 (1_suppl) ◽  
pp. 151-151
Author(s):  
A E Stoper ◽  
J Randle ◽  
M M Cohen
Keyword(s):  
Direct Effect ◽  
Experimental Psychology ◽  
Human Perception ◽  
Vertical Line ◽  
Line Segments ◽  
Level Lines ◽  
Pitch Dimension ◽  
Zero Effect

Visually perceived eye level (VPEL) has been shown to be strongly affected by the pitch of the visible environment (Stoper and Cohen, 1989 Perception & Psychophysics46 469 – 475), even if this environment consists of only two luminous lines pitched from the vertical (Matin and Li, 1992 Journal of Experimental Psychology: Human Perception and Performance18 257 – 289). Here, two luminous vertical lines or 32 randomly distributed luminous dots were mounted on a plane that was viewed monocularly and was pitched (slanted in the pitch dimension) 30° forward or backward from the vertical. In addition to measuring the VPEL, we measured the perceived optic slant (rather than the perceived geographic slant) of this plane by requiring each of our ten subjects to set a target to the visually perceived near point (VPNP) of the plane. We found that, for the lines, VPNP shifted 50% and VPEL shifted 26% of the physical pitch of the plane. For the dots, VPNP shifted 28% but VPEL shifted only 8%. The effect of the dots on VPEL was weaker than might have been predicted by their effect on VPNP, which was used to indicate perceived optic slant. The weakness of the effect of the dots on VPEL implies that changes in VPEL result from a direct effect of the stimuli on VPEL, rather than one mediated by the perceived optic slant of the plane. The non-zero effect of the dots shows that pitched from vertical line segments are not necessary to shift VPEL.

Terminator Position Affects the Perceived Speed of Line Segments Tilted with Respect to Direction of Motion

Perception ◽  
1996 ◽  
Vol 25 (1_suppl) ◽  
pp. 141-141 ◽  
Author(s):  
K C Scott-Brown ◽  
D W Heeley
Keyword(s):  
Random Perturbation ◽  
Critical Factor ◽  
Vertical Line ◽  
Vertical Position ◽  
Stimulus Uncertainty ◽  
Line Segments ◽  
Tilted Line ◽  
Motion Measurements ◽  
Speed Discrimination ◽  
Perceived Speed

We investigated factors producing bias in the perceived speed of tilted lines in horizontal translation. The effects of grouping, collinearity, eccentricity, terminator proximity, and stimulus uncertainty on perceived speed were studied. The matched speed of vertical line compared to an inclined line was estimated with the use of a double random interleaved staircase for speed discrimination with a two-alternative forced choice. The speed of the tilted stimulus was held constant while the speed of the vertical stimulus was modified by the subject's response. Stimuli were single lines and groups of lines. The groups of short lines were arranged either in a collinear or in a randomly scattered fashion. The length of the line stimuli ranged from 0.33 deg to 7.0 deg of visual angle. Speed estimates were obtained for angles of tilt ranging from 0° to 90°. For line segments, collinearity was found to be the critical factor in determining perceived speed. Collinear segments showed a similar bias in perceived speed to single lines of the same overall length. However, randomly scattered segments were not subject to a bias in perceived speed. Random perturbation of the length or vertical position of a single line abolished the bias in perceived speed of a tilted line compared to a vertical line. Current models of the integration of motion measurements should be changed to account for the effects of topological arrangement and terminator position on the perceived speed of inclined lines.

THE MINIMUM GUARDING TREE PROBLEM

2014 ◽  
Vol 06 (01) ◽  
pp. 1450011 ◽  
Author(s):  
ADRIAN DUMITRESCU ◽  
JOSEPH S. B. MITCHELL ◽  
PAWEL ŻYLIŃSKI
Keyword(s):  
Approximation Algorithms ◽  
Nonempty Subset ◽  
Parallel Line ◽  
Minimum Length ◽  
Np Hard ◽  
Orthogonal Axis ◽  
Line Segments ◽  
Parallel Lines ◽  
Simple Alternative ◽  
Axis Parallel

Given a set ℒ of non-parallel lines in the plane and a nonempty subset ℒ′ ⊆ ℒ, a guarding tree for ℒ′ is a tree contained in the union of the lines in ℒ such that if a mobile guard (agent) runs on the edges of the tree, all lines in ℒ′ are visited by the guard. Similarly, given a connected arrangement 𝒮 of line segments in the plane and a nonempty subset 𝒮′ ⊆ 𝒮, we define a guarding tree for 𝒮′. The minimum guarding tree problem for a given set of lines or line segments is to find a minimum-length guarding tree for the input set. We provide a simple alternative (to [N. Xu, Complexity of minimum corridor guarding problems, Inf. Process. Lett.112(17–18) (2012) 691–696.]) proof of the problem of finding a guarding tree of minimum length for a set of orthogonal (axis-parallel) line segments in the plane. Then, we present two approximation algorithms with factors 2 and 3.98, respectively, for computing a minimum guarding tree for a subset of a set of n arbitrary non-parallel lines in the plane; their running times are O(n8) and O(n6 log n), respectively. Finally, we show that this problem is NP-hard for lines in 3-space.

How Can Biomechanical Multibody Models of Scoliosis Be Accurate in Simulating Spine Movement Behavior While Neglecting the Changes of Spinal Length?

2021 ◽  
Author(s):  
Athena Jalalian ◽  
Soheil Arastehfar ◽  
Ian Gibson ◽  
Francis E. H. Tay ◽  
Gabriel Liu
Keyword(s):  
Finite Number ◽  
Spinal Column ◽  
Local Optimization ◽  
Movement Behavior ◽  
Accuracy Improvement ◽  
Line Segments ◽  
Fixed Length ◽  
Curve Approximation ◽  
Polygonal Chains ◽  
Spine Movement

Abstract This paper studies how biomechanical multibody models of scoliosis can neglect the changes of spinal length and yet be accurate in reconstructing spinal columns. As these models with fixed length comprise rigid links interconnected by rotary joints, they resemble polygonal chains that approximate spine curves with a finite number of line segments. In mathematics, using more segments with shorter length can result in more accurate curve approximations. This raises the question of whether more accurate spine curve approximations by increasing the number of links/joints can yield more accurate spinal column reconstructions. For this, the accuracy of spine curve approximation was improved consistently by increasing the number of links/joints, and its effects on the accuracy of spinal column reconstruction were assessed. Positive correlation was found between the accuracy of spine reconstruction and curve approximation. It was shown that while increasing the accuracy of curve approximations, the representation of scoliosis concavity and its side-to-side deviations were improved. Moreover, reconstruction errors of the spine regions separated by the inflection vertebrae had minimal impacts on each other. Overall, multibody scoliosis models with fixed spinal length can benefit from the extra rotational joints that contribute towards the accuracy of spine curve approximation. The outcome of this study leads to concurrent accuracy improvement and simplification of multibody models; joint-link configurations can be independently defined for the regions separated by the inflection vertebrae, enabling local optimization of the models for higher accuracy without unnecessary added complexity to the whole model.

scholarly journals Rectangular Seifert circles and arcs system

2014 ◽  
Vol 23 (08) ◽  
pp. 1450041
Author(s):  
Tatsuo Ando ◽  
Chuichiro Hayashi ◽  
Miwa Hayashi
Keyword(s):  
Negative Integer ◽  
Vertical Line ◽  
Horizontal Line ◽  
Link Diagram ◽  
Line Segments ◽  
Oriented Link ◽  
Link Diagrams ◽  
Ambient Isotopy

Rectangular diagrams of links are link diagrams in the plane ℝ2 such that they are composed of vertical line segments and horizontal line segments and vertical segments go over horizontal segments at all crossings. Cromwell and Dynnikov showed that rectangular diagrams of links are useful for deciding whether a given link is split or not, and whether a given knot is trivial or not. We show in this paper that an oriented link diagram D with c(D) crossings and s(D) Seifert circles can be deformed by an ambient isotopy of ℝ2 into a rectangular diagram with at most c(D) + 2s(D) vertical segments, and that, if D is connected, at most 2c(D) + 2 - w(D) vertical segments, where w(D) is a certain non-negative integer. In order to obtain these results, we show that the system of Seifert circles and arcs substituting for crossings can be deformed by an ambient isotopy of ℝ2 so that Seifert circles are rectangles composed of two vertical line segments and two horizontal line segments and arcs are vertical line segments, and that we can obtain a single circle from a connected link diagram by smoothing operations at the crossings regardless of orientation.

Minimal grid diagrams of 11 crossing prime alternating knots

2020 ◽  
Vol 29 (11) ◽  
pp. 2050076
Author(s):  
Gyo Taek Jin ◽  
Hwa Jeong Lee
Keyword(s):  
Plane Curve ◽  
Crossing Number ◽  
Vertical Line ◽  
Horizontal Line ◽  
Line Segments ◽  
Grid Diagrams ◽  
Arc Index ◽  
Closed Plane

The arc index of a knot is the minimal number of arcs in all arc presentations of the knot. An arc presentation of a knot can be shown in the form of a grid diagram which is a closed plane curve consisting of finitely many horizontal line segments and the same number of vertical line segments. The arc index of an alternating knot is its minimal crossing number plus two. In this paper, we give a list of minimal grid diagrams of the 11 crossing prime alternating knots obtained from arc presentations with 13 arcs.

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