scholarly journals Digital straight lines in the Khalimsky plane

2005 ◽  
Vol 96 (1) ◽  
pp. 49 ◽  
Author(s):  
Erik Melin
Keyword(s):  
Line Segment ◽  
Straight Line Segment ◽  
Integer Points ◽  
Straight Line ◽  
Digital Plane ◽  
Straight Lines ◽  
Khalimsky Topology

We consider the digital plane of integer points equipped with the Khalimsky topology. We suggest a digitization of straight lines such that the digitized image is homeomorphic to the Khalimsky line and a digitized line segment is a Khalimsky arc. It is demonstrated that a Khalimsky arc is the digitization of a straight line segment if and only if it satisfies a generalized version of the chord property introduced by Rosenfeld.

Three-dimensional extension of Bresenham's algorithm and its application in straight-line interpolation

2002 ◽  
Vol 216 (3) ◽  
pp. 459-463 ◽  
Author(s):  
X-W Liu ◽  
K Cheng
Keyword(s):  
Line Segment ◽  
Three Dimensional ◽  
Straight Line Segment ◽  
Numerical Control ◽  
Interpolation Algorithm ◽  
Straight Line ◽  
Actual Error ◽  
Potential Applications ◽  
Sample Position ◽  
Straight Lines

Conventional straight-line generating algorithms, such as the digital differential analyser (DDA), Bresenham's algorithm and the mid-point algorithm, are suitable only for planer straight lines on the coordinate planes, of which Bresenham's algorithm is the most efficient. In this paper, the authors have extended Bresenham's algorithm to spatial straight lines. Given a spatial straight-line segment with two end-points, the authors have applied Bresenham's algorithm to the projections of the line segment on two of the three coordinate planes, which is determined by the largest of the coordinate lengths of the line segment, thereby obtaining a three-dimensional extension of the algorithm. In a case study, the authors calculated the distance between each sample position and the given line segment. The result reveals that the actual error at each sample position is smaller than the maximum theoretical error, and the performance of the three-dimensional extension of Bresenham's algorithm is as good as that of Bresenham's original planer algorithm. One of its potential applications is the three-dimensional step straight-line interpolation used in computer numerical control (CNC) systems of machine tools and rapid prototyping machines. Application of the algorithm is contrasted with that of the traditional DDA step straight-line interpolation algorithm. The result confirms that the three-dimensional extension of Bresenham's algorithm is much better than the DDA straight-line interpolation algorithm.

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.

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.

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

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.

scholarly journals DESIGN AND IMPLEMENTATION OF POTENTIOMETER-BASED NONLINEAR TRANSDUCER EMULATOR

2011 ◽  
Vol 12 (1) ◽  
pp. 77-87
Author(s):  
Sheroz Khan
Keyword(s):  
Line Segment ◽  
Nonlinear Response ◽  
Response Pattern ◽  
Input Voltage ◽  
Straight Line Segment ◽  
Straight Line ◽  
Three Stages ◽  
Complete Circuit ◽  
Nonlinear Curve ◽  
Range Selection

This work attempts to design and implement in hardware a transducer with a nonlinear response using potentiometer. Potentiometer is regarded as a linear transducer, while a the response of a nonlinear transducer can be treated as a concatenation of linear segments made out of the response curve of an actual nonlinear transducer at the points of inflections being exhibited by the nonlinear curve. Each straight line segment is characterized by its slope and a constant, called the y-intercept, which is ultimately realized by a corresponding electronic circuit. The complete circuit diagram is made of three stages: (i) the input stage for range selection, (ii) a digital logic to make appropriate selection, (iii) a conditioning circuit for realizing a given straight-line segment identified by its relevant slope and reference voltage. The simulation of the circuit is carried using MULTISIM, and the designed circuit is afterward tested to verify that variations of the input voltage give us an output voltage very close to the response pattern envisaged in the analytical stage of the design. The utility of this work lies in its applications in emulating purpose built transducers that could be used to nicely emulate a transducer in a real world system that is to be controlled by a programmable digital system.

Lower bounds for point-to-point wandering exponents in Euclidean first-passage percolation

2000 ◽  
Vol 37 (04) ◽  
pp. 1061-1073
Author(s):  
C. Douglas Howard
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Line Segment ◽  
Passage Time ◽  
Straight Line Segment ◽  
First Passage Percolation ◽  
First Passage ◽  
Straight Line ◽  
Point To Point

In first-passage percolation models, the passage time T(0,L) from the origin to a point L is expected to exhibit deviations of order |L|χ from its mean, while minimizing paths are expected to exhibit fluctuations of order |L|ξ away from the straight line segment . Here, for Euclidean models in dimension d, we establish the lower bounds ξ ≥ 1/(d+1) and χ ≥(1-(d-1)ξ)/2. Combining this latter bound with the known upper bound ξ ≤ 3/4 yields that χ ≥ 1/8 for d=2.

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