ON UNFOLDING LATTICE POLYGONS/TREES AND DIAMETER-4 TREES

2009 ◽  
Vol 19 (03) ◽  
pp. 289-321
Author(s):  
SHEUNG-HUNG POON
Keyword(s):  
Convex Polygon ◽  
Two Dimensions ◽  
Leaf Node ◽  
Cubic Lattice ◽  
Lattice Polygon ◽  
Straight Line ◽  
Edge Points ◽  
Lattice Tree ◽  
Edge Crossings ◽  
Lattice Polygons

We consider the problems of straightening polygonal trees and convexifying polygons by continuous motions such that rigid edges can rotate around vertex joints and no edge crossings are allowed. A tree can be straightened if all its edges can be aligned along a common straight line such that each edge points "away" from a designated leaf node. A polygon can be convexified if it can be reconfigured to a convex polygon. A lattice tree (resp. polygon) is a tree (resp. polygon) containing only edges from a square or cubic lattice. We first show that a 2D lattice chain or a 3D lattice tree can be straightened efficiently in O(n) moves and time, where n is the number of tree edges. We then show that a 2D lattice tree can be straightened efficiently in O(n2) moves and time. Furthermore, we prove that a 2D lattice polygon or a 3D lattice polygon with simple shadow can be convexified efficiently in O(n) moves and in O(n log n) time. Finally, we show that two special classes of diameter-4 trees in two dimensions can always be straightened.

Numerical Simulation of Hopping Conductivity in Granular Metals

1990 ◽  
Vol 195 ◽  
Author(s):  
L. F. Chen ◽  
Ping Sheng ◽  
B. Abeles ◽  
M. Y. Zhou
Keyword(s):  
Cubic Lattice ◽  
Grain Sizes ◽  
Simple Cubic ◽  
Simple Cubic Lattice ◽  
Straight Line ◽  
Disorder Potential ◽  
Effect Of Disorder ◽  
Log Normal ◽  
Hopping Conductance ◽  
Log Normal Distribution

ABSTRACTElectrical conduction in granular metals is simulated by mapping the hopping conductance between pairs of metal grains onto a simple cubic lattice with bonds between neighbors. By considering a log-normal distribution of grain sizes and the effect of disorder potential, the numerically calculated network conductance exhibit clear deviation from simple activation. Plotting -log a vs. T-½, where σ denotes conductivity and T the temperature, gives good straight line behavior with slopes comparable to those measured experimentally. Our results are noted to differ from those of Adkins et al.

Mapping with Monocular Vision in Two Dimensions

2010 ◽  
Vol 1 (4) ◽  
pp. 56-65 ◽  
Author(s):  
Nicolau Leal Werneck ◽  
Anna Helena Reali Costa
Keyword(s):  
Optical Axis ◽  
Two Dimensions ◽  
Obstacle Detection ◽  
Uniform Motion ◽  
Mapping Algorithms ◽  
Multiple Images ◽  
Line Segments ◽  
Straight Line ◽  
Camera Intrinsic Parameters ◽  
Over Time

This article presents the problem of building bi-dimensional maps of environments when the sensor available is a camera used to detect edges crossing a single line of pixels and motion is restricted to a straight line along the optical axis. The position over time must be provided or assumed. Mapping algorithms for these conditions can be built with the landmark parameters estimated from sets of matched detection from multiple images. This article shows how maps that are correctly up to scale can be built without knowledge of the camera intrinsic parameters or speed during uniform motion, and how performing an inverse parameterization of the image coordinates turns the mapping problem into the fitting of line segments to a group of points. The resulting technique is a simplified form of visual SLAM that can be better suited for applications such as obstacle detection in mobile robots.

Circuit presentation and lattice stick number with exactly four z-sticks

2018 ◽  
Vol 27 (08) ◽  
pp. 1850046
Author(s):  
Hyoungjun Kim ◽  
Sungjong No
Keyword(s):  
Upper Bound ◽  
Line Segment ◽  
Disjoint Union ◽  
Lattice Plane ◽  
Cambridge Philos ◽  
Line Segments ◽  
Cubic Lattice ◽  
Straight Line ◽  
Two Component ◽  
Number Formula

The lattice stick number [Formula: see text] of a link [Formula: see text] is defined to be the minimal number of straight line segments required to construct a stick presentation of [Formula: see text] in the cubic lattice. Hong, No and Oh [Upper bound on lattice stick number of knots, Math. Proc. Cambridge Philos. Soc. 155 (2013) 173–179] found a general upper bound [Formula: see text]. A rational link can be represented by a lattice presentation with exactly 4 [Formula: see text]-sticks. An [Formula: see text]-circuit is the disjoint union of [Formula: see text] arcs in the lattice plane [Formula: see text]. An [Formula: see text]-circuit presentation is an embedding obtained from the [Formula: see text]-circuit by connecting each [Formula: see text] pair of vertices with one line segment above the circuit. By using a two-circuit presentation, we can easily find the lattice presentation with exactly four [Formula: see text]-sticks. In this paper, we show that an upper bound for the lattice stick number of rational [Formula: see text]-links realized with exactly four [Formula: see text]-sticks is [Formula: see text]. Furthermore, it is [Formula: see text] if [Formula: see text] is a two-component link.

CONVEX GRID DRAWINGS OF FOUR-CONNECTED PLANE GRAPHS

2006 ◽  
Vol 17 (05) ◽  
pp. 1031-1060 ◽  
Author(s):  
KAZUYUKI MIURA ◽  
SHIN-ICHI NAKANO ◽  
TAKAO NISHIZEKI
Keyword(s):  
Convex Polygon ◽  
Linear Time ◽  
Time Algorithm ◽  
Plane Graph ◽  
Outer Face ◽  
Plane Graphs ◽  
Line Segments ◽  
Grid Drawing ◽  
Straight Line ◽  
Grid Points

A convex grid drawing of a plane graph G is a drawing of G on the plane such that all vertices of G are put on grid points, all edges are drawn as straight-line segments without any edge-intersection, and every face boundary is a convex polygon. In this paper we give a linear-time algorithm for finding a convex grid drawing of every 4-connected plane graph G with four or more vertices on the outer face. The size of the drawing satisfies W + H ≤ n - 1, where n is the number of vertices of G, W is the width and H is the height of the grid drawing. Thus the area W · H is at most ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋. Our bounds on the sizes are optimal in a sense that there exist an infinite number of 4-connected plane graphs whose convex drawings need grids such that W + H = n - 1 and W · H = ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋.

Manual Tracking in Two Dimensions

2000 ◽  
Vol 83 (6) ◽  
pp. 3483-3496 ◽  
Author(s):  
Kevin C. Engel ◽  
John F. Soechting
Keyword(s):  
Simple Model ◽  
Two Dimensions ◽  
Target Velocity ◽  
Manual Tracking ◽  
Video Monitor ◽  
Hand Motion ◽  
Target Acceleration ◽  
Straight Line ◽  
Perpendicular Component ◽  
Peak Speed

Manual tracking was studied by asking subjects to follow, with their finger, a target moving on a touch-sensitive video monitor. The target initially moved in a straight line at a constant speed and then, at a random point in time, made one abrupt change in direction. The results were approximated with a simple model according to which, after a reaction time, the hand moved in a straight line to intercept the target. Both the direction of hand motion and its peak speed could be predicted by assuming a constant time to intercept. This simple model was able to account for results obtained over a broad range of target speeds as well as the results of experiments in which both the speed and the direction of the target changed simultaneously. The results of an experiment in which the target acceleration was nonzero suggested that the error signals used during tracking are related to both speed and direction but poorly (if at all) to target acceleration. Finally, in an experiment in which target velocity remained constant along one axis but the perpendicular component underwent a step change, tracking along both axes was perturbed. This last finding demonstrates that tracking in two dimensions cannot be decomposed into its Cartesian components. However, an analytical model in a hand-centered frame of reference in which speed and direction are the controlled variables could account for much of the data.

scholarly journals Lane Detection Algorithm Based on Road Structure and Extended Kalman Filter

2020 ◽  
Vol 12 (2) ◽  
pp. 1-20
Author(s):  
Jinsheng Xiao ◽  
Wenxin Xiong ◽  
Yuan Yao ◽  
Liang Li ◽  
Reinhard Klette
Keyword(s):  
Kalman Filter ◽  
Extended Kalman Filter ◽  
Region Of Interest ◽  
Detection Algorithm ◽  
Lane Detection ◽  
Strong Light ◽  
Edge Strength ◽  
Straight Line ◽  
Edge Points ◽  
The Stability

Lane detection still demonstrates low accuracy and missing robustness when recorded markings are interrupted by strong light or shadows or missing marking. This article proposes a new algorithm using a model of road structure and an extended Kalman filter. The region of interest is set according to the vanishing point. First, an edge-detection operator is used to scan horizontal pixels and calculate edge-strength values. The corresponding straight line is detected by line parameters voted by edge points. From the edge points and lane mark candidates extracted above, and other constraints, these points are treated as the potential lane boundary. Finally, the lane parameters are estimated using the coordinates of the lane boundary points. They are updated by an extended Kalman filter to ensure the stability and robustness. Results indicate that the proposed algorithm is robust for challenging road scenes with low computational complexity.

Modelling of circular wave guides

1992 ◽  
Vol 70 (10-11) ◽  
pp. 1092-1098 ◽  
Author(s):  
A. Delage ◽  
K. A. McGreer ◽  
E. Rainville
Keyword(s):  
Integrated Circuit ◽  
Beam Propagation Method ◽  
Two Dimensions ◽  
Effective Index ◽  
Radius Of Curvature ◽  
Index Method ◽  
Photonic Integrated Circuit ◽  
Circular Arcs ◽  
Straight Line ◽  
Wave Guides

In many circumstances the design of interconnects in a photonic integrated circuit can be simplified by using low loss curved wave guides in the shapes of circular arcs. Radiative losses associated with the curvature have been computed as a function of the radius of curvature. The technique takes advantage of the effective index method to reduce the problem from two dimensions to one dimension (1D) and uses a change of coordinate that transforms an arc of circle into a straight line. This transformation results in a monotonous increase of the refractive index as function of r (the distance from the centre of the circle) for original constant index regions. The new system is solved by discretizing this varying effective index onto many small layers of constant index over a window large enough to contain the region where the field is not negligible. A multilayer algorithm in 1D is then used to find complex propagation constants in which the imaginary part is related to the fundamental energy loss owing to the curvature. The solution also gives the shape of the field necessary to match the mode profiles at the junction between the straight and curved part of the wave guide. The basic change of variable has been extended to the finite difference solution of the scalar wave equation and to the beam propagation method.

GENERALIZED ATMOSPHERIC SAMPLING OF KNOTTED POLYGONS

2011 ◽  
Vol 20 (08) ◽  
pp. 1145-1171 ◽  
Author(s):  
E. J. JANSE VAN RENSBURG ◽  
A. RECHNITZER
Keyword(s):  
Chemical Properties ◽  
Dominant Factor ◽  
Conformational Entropy ◽  
Cubic Lattice ◽  
Type K ◽  
Trefoil Knot ◽  
Ring Polymers ◽  
Atmospheric Sampling ◽  
Physical And Chemical ◽  
Lattice Polygons

Self-avoiding polygons in the cubic lattice are models of ring polymers in dilute solution. The conformational entropy of a ring polymer is a dominant factor in its physical and chemical properties, and this is modeled by the large number of conformations of lattice polygons. Cubic lattice polygons are embeddings of the circle in three space and may be used as a model of knotting in ring polymers. In this paper we study the effects of knotting on the conformational entropy of lattice polygons and so determine the relative fraction of polygons of different knot types at large lengths. More precisely, we consider the number of cubic lattice polygons of n edges with knot type K, pn(K). Numerical evidence strongly suggests that [Formula: see text] as n → ∞, where μ0 is the growth constant of unknotted lattice polygons, α is the entropic exponent of lattice polygons, and NK is the number of prime knot components in the knot type K (see the paper [Asymptotics of knotted lattice polygons, J. Phys. A: Math. Gen.31 (1998) 5953–5967]). Determining the exact value of pn(K) is far beyond current techniques for all but very small values of n. Instead we use the GAS algorithm (see the paper [Generalised atmospheric sampling of self-avoiding walks, J. Phys. A: Math. Theor.42 (2009) 335001–335030]) to enumerate pn (K) approximately. We then extrapolate ratios [pn(K)/pn(L)] to larger values of n for a number of given knot types. We give evidence that for the unknot 01 and the trefoil knot 31, there exists a number M01, 31 ≈170000 such that pn (01) > pn (31) if n < M01, 31 and pn (01) ≤pn (31) if n ≥M01, 31. In addition, the asymptotic relative frequencies for a variety of knot types are determined. For example, we find that [pn(31)/pn(41)] → 27.0 ± 2.2, implying that there are approximately 27 polygons of the trefoil knot type for every polygon of knot of type 41 (the figure eight knot), in the asymptotic limit. Finally, we examine the dominant knot types at moderate values of n and conjecture that the most frequent knot types in polygons of any given length n are of the form [Formula: see text] (or its chiral partner), where [Formula: see text] are right- and left-handed trefoils, and N increases with n.

scholarly journals On convex lattice polygons

1976 ◽  
Vol 15 (3) ◽  
pp. 395-399 ◽  
Author(s):  
P.R. Scott
Keyword(s):  
Boundary Points ◽  
Lattice Polygon ◽  
Convex Lattice ◽  
Convex Lattice Polygons ◽  
Interior Points ◽  
Lattice Polygons

Let π be a convex lattice polygon with b boundary points and c (≥ 1) interior points. We show that for any given c, the number b satisfies b ≤ 2c + 7, and identify the polygons for which equality holds.

scholarly journals Convex lattice polygons of minimum area

1990 ◽  
Vol 42 (3) ◽  
pp. 353-367 ◽  
Author(s):  
R.J. Simpson
Keyword(s):  
Integer Lattice ◽  
Minimum Area ◽  
Lattice Polygon ◽  
Convex Lattice ◽  
Convex Lattice Polygons ◽  
Lattice Polygons

A convex lattice polygon is a polygon whose vertices are points on the integer lattice and whose interior angles are strictly less than π radians. We define a(2n) to be the least possible area of a convex lattice polygon with 2n vertices. A method for constructing convex lattice polygons with area a(2n) is described, and values of a(2n) for low n are obtained.

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