scholarly journals Line-of-Sight Pursuit in Monotone and Scallop Polygons

2019 ◽  
Vol 29 (04) ◽  
pp. 307-351
Author(s):  
Lindsay Berry ◽  
Andrew Beveridge ◽  
Jane Butterfield ◽  
Volkan Isler ◽  
Zachary Keller ◽  
...  
Keyword(s):  
Line Segment ◽  
Winning Strategy ◽  
Line Of Sight ◽  
Unit Distance ◽  
Capture Time ◽  
Sweeping Motion ◽  
Straight Line ◽  
Monotone Polygons ◽  
Simply Connected

We study a turn-based game in a simply connected polygonal environment [Formula: see text] between a pursuer [Formula: see text] and an adversarial evader [Formula: see text]. Both players can move in a straight line to any point within unit distance during their turn. The pursuer [Formula: see text] wins by capturing the evader, meaning that their distance satisfies [Formula: see text], while the evader wins by eluding capture forever. Both players have a map of the environment, but they have different sensing capabilities. The evader [Formula: see text] always knows the location of [Formula: see text]. Meanwhile, [Formula: see text] only has line-of-sight visibility: [Formula: see text] observes the evader’s position only when the line segment connecting them lies entirely within the polygon. Therefore [Formula: see text] must search for [Formula: see text] when the evader is hidden from view. We provide a winning strategy for [Formula: see text] in two families of polygons: monotone polygons and scallop polygons. In both families, a straight line [Formula: see text] can be moved continuously over [Formula: see text] so that (1) [Formula: see text] is a line segment and (2) every point on the boundary [Formula: see text] is swept exactly once. These are both subfamilies of strictly sweepable polygons. The sweeping motion for a monotone polygon is a single translation, and the sweeping motion for a scallop polygon is a single rotation. Our algorithms use rook’s strategy during its pursuit phase, rather than the well-known lion’s strategy. The rook’s strategy is crucial for obtaining a capture time that is linear in the area of [Formula: see text]. For both monotone and scallop polygons, our algorithm has a capture time of [Formula: see text], where [Formula: see text] is the number of polygon vertices.

RADISHCHEV PLOT IN MULTIVARIATE ANALYSIS OF THE PROBLEM OF MULTIPLE TARGETS GROUP PURSUIT

2021 ◽  
pp. 22-31
Author(s):  
A. A. Dubanov
Keyword(s):  
Multivariate Analysis ◽  
Line Segment ◽  
Iterative Process ◽  
Straight Line Segment ◽  
Line Of Sight ◽  
Radius Of Curvature ◽  
Time Step ◽  
Minimum Radius ◽  
Group Pursuit ◽  
Straight Line

This article discusses a kinematic model of the problem of group pursuit of a set of goals. The article discusses a variant of the model when all goals are achieved simultaneously. And also the possibility is considered when the achievement of goals occurs at the appointed time. In this model, the direction of the speeds by the pursuer can be arbitrary, in contrast to the method of parallel approach. In the method of parallel approach, the velocity vectors of the pursuer and the target are directed to a point on the Apollonius circle. The proposed pursuit model is based on the fact that the pursuer tries to follow the predicted trajectory of movement. The predicted trajectory of movement is built at each moment of time. This path is a compound curve that respects curvature constraints. A compound curve consists of a circular arc and a straight line segment. The pursuer's velocity vector applied to the point where the pursuer is located touches the given circle. The straight line segment passes through the target point and touches the specified circle. The radius of the circle in the model is taken equal to the minimum radius of curvature of the trajectory. The resulting compound line serves as an analogue of the line of sight in the parallel approach method. The iterative process of calculating the points of the pursuer’s trajectory is that the next point of position is the point of intersection of the circle centered at the current point of the pursuer’s position, with the line of sight corresponding to the point of the next position of the target. The radius of such a circle is equal to the product of the speed of the pursuer and the time interval corresponding to the time step of the iterative process. The time to reach the goal of each pursuer is a dependence on the speed of movement and the minimum radius of curvature of the trajectory. Multivariate analysis of the moduli of velocities and minimum radii of curvature of the trajectories of each of the pursuers for the simultaneous achievement of their goals i based on the methods of multidimensional descriptive geometry. To do this, the projection planes are entered on the Radishchev diagram: the radius of curvature of the trajectory and speed, the radius of curvature of the trajectory and the time to reach the goal. On the first plane, the projection builds a one-parameter set of level lines corresponding to the range of velocities. In the second graph, corresponding to a given range of speeds, functions of the dependence of the time to reach the target on the radius of curvature. The preset time for reaching the target and the preset value of the speed of the pursuer are the optimizing factors. This method of constructing the trajectories of pursuers to achieve a variety of goals at given time values may be in demand by the developers of autonomous unmanned aerial vehicles.

scholarly journals Constant-Time Complete Visibility for Robots with Lights: The Asynchronous Case

Algorithms ◽  
2021 ◽  
Vol 14 (2) ◽  
pp. 56
Author(s):  
Gokarna Sharma ◽  
Ramachandran Vaidyanathan ◽  
Jerry L. Trahan
Keyword(s):  
Mobile Robots ◽  
Line Segment ◽  
Autonomous Robots ◽  
Straight Line Segment ◽  
Autonomous Mobile Robots ◽  
Asymptotically Optimal ◽  
Time Translation ◽  
Straight Line ◽  
A New Technique ◽  
Colored Lights

We consider the distributed setting of N autonomous mobile robots that operate in Look-Compute-Move (LCM) cycles and use colored lights (the robots with lights model). We assume obstructed visibility where a robot cannot see another robot if a third robot is positioned between them on the straight line segment connecting them. In this paper, we consider the problem of positioning N autonomous robots on a plane so that every robot is visible to all others (this is called the Complete Visibility problem). This problem is fundamental, as it provides a basis to solve many other problems under obstructed visibility. In this paper, we provide the first, asymptotically optimal, O(1) time, O(1) color algorithm for Complete Visibility in the asynchronous setting. This significantly improves on an O(N)-time translation of the existing O(1) time, O(1) color semi-synchronous algorithm to the asynchronous setting. The proposed algorithm is collision-free, i.e., robots do not share positions, and their paths do not cross. We also introduce a new technique for moving robots in an asynchronous setting that may be of independent interest, called Beacon-Directed Curve Positioning.

scholarly journals Using Sequential Photography to Estimate Ice Velocity at the Terminus of Columbia Glacier, Alaska

1986 ◽  
Vol 8 ◽  
pp. 117-123 ◽  
Author(s):  
R.M. Krimmel ◽  
L.A. Rasmussen
Keyword(s):  
Time Resolution ◽  
Principal Direction ◽  
Film Plane ◽  
Time Sequence ◽  
Line Of Sight ◽  
Glacier Surface ◽  
Ice Flow ◽  
Straight Line ◽  
Camera Orientation ◽  
Ice Velocity

The terminus of Columbia Glacier, Alaska, was observed with a single automatic 35 mm camera to determine velocity with a time resolution in the order of a day. The photographic coordinates of the image of a target were then transformed linearly into the direction numbers of the line of sight from the camera to the target. The camera orientation was determined from the film-plane locations of known landmark points by using an adaption of vertical photogrammetry techniques. The line of sight, when intersected with some mathematically-defined glacier surface, defines the true space coordinates of a target, The time sequence of a target’s position was smoothed, first in horizontal x, y space to a straight line, then in y (the principal direction of ice flow) and time with a smoothing cubic spline, and then the x-component was computed from the y-component by considering the inclination of the straight line. This allows daily velocities (about 8 m/day) to be measured at a distance of 5 km, using a 105 mm lens. Errors in daily displacements were estimated to be 1 m. The terminus configuration was also measured using the same photo set.

scholarly journals Pupillary axis of a heterocentric astigmatic eye

2013 ◽  
Vol 72 (1) ◽  
Author(s):  
W. F. Harris
Keyword(s):  
Anterior Chamber ◽  
Contact Lens ◽  
Line Of Sight ◽  
Compound System ◽  
Linear Optics ◽  
Optical Instrument ◽  
Entrance Pupil ◽  
Naked Eye ◽  
Straight Line

The pupillary axis of the eye is a clinically useful concept usually defined as the line through the centre of the entrance pupil that is perpendicular to the cornea. However if the cornea is astigmaticthen, strictly speaking, the entrance pupil is blurred and the pupillary axis is not well defined.  A modified definition is offered in this paper: the pupillary axis is the infinite straight line containing the incident segment of the ray that passes through the centre of the (actual) pupil and is perpendicular to the first surface of the eye.  The definition holds for the naked eye and for an eye with an implant in the anterior chamber.  It also holds for the com-pound system of eye and optical instrument such as a contact lens in front of it if the first surface is interpreted as the first surface of the compound system and the pupil as the limiting aperture of the compound system.  Linear optics is applied to obtain a formula for the position and inclination of the pupillary axis at incidence onto the system; the refracting surfaces may be heterocentric and astigmatic.  The formula allows one to examine the sensitivity of the pupillary axis to displacement of the pupil and any other changes in the anterior eye.  Strictly the pupillary axis depends on the frequency of light but examples show that the dependence is probably negligible.  The vectorized generalization of what is sometimes called angle lambda is easily calculated from the inclination of the pupillary axis and the line of sight. (S Afr Optom 2013 72(1) 3-10)

A Direct Least Squares Method for Camera Pose Estimation Based on Straight Line Segment Correspondences

2015 ◽  
Vol 35 (6) ◽  
pp. 0615003
Author(s):  
李鑫 Li Xin ◽  
张跃强 Zhang Yueqiang ◽  
刘进博 Liu Jinbo ◽  
张小虎 Zhang Xiaohu ◽  
于起峰 Yu Qifeng
Keyword(s):  
Least Squares ◽  
Pose Estimation ◽  
Line Segment ◽  
Least Squares Method ◽  
Straight Line Segment ◽  
Camera Pose Estimation ◽  
Straight Line ◽  
Camera Pose

scholarly journals A Novel Water-Shore-Line Detection Method for USV Autonomous Navigation

Sensors ◽  
2020 ◽  
Vol 20 (6) ◽  
pp. 1682 ◽  
Author(s):  
Xiong Zou ◽  
Changshi Xiao ◽  
Wenqiang Zhan ◽  
Chunhui Zhou ◽  
Supu Xiu ◽  
...  
Keyword(s):  
Line Segment ◽  
Autonomous Navigation ◽  
Field Experiments ◽  
Lower Boundary ◽  
Water Area ◽  
Epipolar Constraint ◽  
Shore Line ◽  
Inland River ◽  
Straight Line ◽  
Selection Of

For the navigation of an unmanned surface vehicle (USV), detection and recognition of the water-shore-line (WSL) is an important part of its intellectualization. Current research on this issue mainly focuses on the straight WSL obtained by straight line fitting. However, the WSL in the image acquired by boat-borne vision is not always in a straight line, especially in an inland river waterway. In this paper, a novel three-step approach for WSL detection is therefore proposed to solve this problem through the information of an image sequence. Firstly, the initial line segment pool is built by the line segment detector (LSD) algorithm. Then, the coarse-to-fine strategy is used to obtain the onshore line segment pool, including the rough selection of water area instability and the fine selection of the epipolar constraint between image frames, both of which are demonstrated in detail in the text. Finally, the complete shore area is generated by an onshore line segment pool of multi-frame images, and the lower boundary of the area is the desired WSL. In order to verify the accuracy and robustness of the proposed method, field experiments were carried out in the inland river scene. Compared with other detection algorithms based on image processing, the results demonstrate that this method is more adaptable, and can detect not only the straight WSL, but also the curved WSL.

Some Hele Shaw flows with time-dependent free boundaries

1981 ◽  
Vol 102 ◽  
pp. 263-278 ◽  
Author(s):  
S. Richardson
Keyword(s):  
Free Boundaries ◽  
Narrow Gap ◽  
Plan View ◽  
Image Domain ◽  
Simply Connected Domain ◽  
Straight Line ◽  
Quarter Plane ◽  
Geometric Character ◽  
Parallel Surfaces ◽  
Simply Connected

We consider a blob of Newtonian fluid sandwiched in the narrow gap between two plane parallel surfaces. At some initial instant, its plan-view occupies a given, simply connected domain, and its growth as further fluid is injected at a number of injection points in its interior is to be determined. It is shown that certain functionals of the domain of a purely geometric character, infinite in number, evolve in a predictable manner, and that these may be exploited in some cases of interest to yield a complete description of the motion.By invoking images, these results may be used to solve certain problems involving the growth of a blob in a gap containing barriers. Injection at a point in a half-plane bounded by a straight line, with an initially empty gap, is shown to lead to a blob whose outline is part of an elliptic lemniscate of Booth for which there is a simple geometrical construction. Injection into a quarter-plane is also considered in some detail when conditions are such that the image domain involved is simply connected.

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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