Reporting Red--Blue Intersections between Two Sets of Connected Line Segments

During an earthquake, the better performance of segmental tunnel lining, compared to the continuous in-cast concrete lining, is generally related to the joints between segments. In order to better understand the influence of the segment joints, their effect on the internal forces induced in tunnel lining simultaneously with the effects of the other influential parameters should be considered. In this work, the segmental joints were simulated by the representative stiffnesses and effects of these characteristics in relation to the other parameters such as the soil-liner interface behavior, number of segments in each ring and thickness of segments on the internal forces induced in structure were investigated. For this purpose, 2D numerical analyses were performed and the results obtained were discussed. Results showed that under the seismic condition, the components that had the most significant role on the internal axial forces induced in the segmental lining were rotational stiffness and axial stiffness of joints. Also the bending moments were more affected by the rotational stiffness. Generally, the radial joint stiffness had a less effect on the induced internal forces. With increase in the number of segments and their thickness, the effect of joint stiffness on the internal forces increases and the design of joints should be given more attention; however, the effects of joint stiffness and frictional behavior at the soil-liner interface on the maximum induced forces are almost independent from each other. Also in a specified joint behavior, by variation in each one of the other parameters including the soil-liner interface condition, number of segments and their thickness, the absolute magnitude of the maximum induced internal forces sometimes change significantly.

The convex hull of a planer set of points can be defined as the set of vertices of the smallest convex polygon containing all the points. If S is a planer set of points then convex layers of S can be derived by iteratively computing the convex hull of S and then removing it from S. Computation of the convex layers has widely been studied in the static environment where the set of points is fixed. The theoretical lower bound of computing the convex layers of a fixed set of points is O (n log n). The static convex layers algorithms with the optimal time complexity have already been proposed in the literature. A set of points where the points can be inserted or deleted is called a dynamic set of points. The set of convex layers should be reconstructed from the scratch at each insertion or deletion if a static convex layers algorithm was used to handle a dynamic set of points. Therefore, it takes O (n log n) time to handle an insertion or a deletion even for an optimal static convex layers algorithm. A dynamic convex layers algorithm has been proposed recently that can perform an insertion or a deletion by doing a slight modification to the existing set of convex layers. It takes O (n) time for an insertion or a deletion. It assumes that the set of points does not contain collinear points and that assumption is not valid in practical applications. Furthermore, the notion of tangent used in that approach restricts the extension of the algorithm into higher dimensions. This paper proposes a novel dynamic convex layers algorithm to eliminate the drawbacks in the existing algorithm. Salient feature of this algorithm is that it represents each layer as a set of line segments. The layers are modified upon an insertion or a deletion of a point by adding some new line segments and deleting some existing line segments. The proposed algorithm takes O (n 3 /k 2 ) time for an insertion or a deletion where k is the number of convex layers. A computer implementation is also presented. The proposed algorithm can work with set of points with collinear points and coincident points. The notion used in the algorithm can easily be extended to higher dimensions. Suppose the set of convex layers have already been found for a given set of points. Further, suppose that the layers are close to each other and new points are expected to fall within the region bounded by the outermost layer. This kind of situations widely occurs in practice and then the proposed algorithm takes O (n) time for an inclusion or a deletion.

We describe a powerful framework for designing efficient batch algorithms for certain large-scale dynamic problems that must be solved using external memory. The class of problems we consider, which we call colorable eztemaldecomposable problems, include rectangle intersection, orthogonal line segment intersection, range searching, and point location. We are particularly interested in these problems in two and higher dimensions. They have numerous applications in geographic information systems (GIS), spatial databases, and VLSI and CAD design. We present simplied algorithms for problems previously solved by more complicated approaches (such as rectangle intersection), and we present efficient algorithms for problems not previously solved in an efficient way (such as point location and higherdimensional versions of range searching and rectangle intersection). communication is the bottleneck in many large-scale applications such as those arising in VLSI and CAD design, spatial databases, and geographic information systems (GIS). In this paper we consider I/O-efficient algorithms for batched searching problems. We consider both batched static and batched dynamic problems, and use the correspondence which often exists between a ddimensional static problem and a (dl)-dimensional dynamic problem to obtain a number of new d-dimensional algorithms. We give experimental results concerning the running time for our approach applied to the red-blue rectangle intersection problem, which is a key component of the extremely important database operation spatial join. Our algorithm scales well with the problem size, and for large problems sizes it greatly outperforms the well-known sweepline approach. One prominent example of the problems we consider is the rectangle intersection problem, which is a key component in VLSI design rule checking [31] and in the extremely important database operation spatial join [34]. We illustrate the practical significance of our algorithms by comparing the empirical performance of our algorithm for this problem with the well-known sweepline algorithm developed for internal memory. 1.1 Problem definition and memory model

A simple polygon is said to be weakly internally visible from a line segment lying inside it (chord, if the endpoints of the segment lie on the boundary of the polygon), if every point on the boundary of the polygon is visible from some point of this line segment. In this paper we describe a simple linear time algorithm for computing such a chord, when it exists. We also computed all non-redundant components of P which must be intersected by any chord of P from which P is wiv. The most interesting aspects of this algorithm are that we have been able to dispense with the complicated triangulation algorithm of Chazelle [2] as well as the algorithm in [7] for computing all shortest paths from a given vertex in a triangulated polygon. As a consequence of this, we can now use this algorithm to derive an optimal algorithm for triangulating a weakly internally visible polygon. Also these results can now be used in [3] to find a shortest line segment from which a polygon is weakly internally visible in optimal linear time.

Let P be a simple polygon with m edges, which is the disjoint union of k simple polygons, all monotone in a common direction u, and let Q be another simple polygon with n edges, which is the disjoint union of l simple polygons, all monotone in a common direction v. We show that the combinatorial complexity of the Minkowski sum P ź Q is O(klmnź(min{m,n})), where ź(·) is the inverse Ackermann function. Some structural properties of the case k = l = 1 have been (implicitly) studied in [17]. We rederive these properties using a different proof, apply them to obtain the above complexity bound for k = l = 1, obtain several additional properties of the sum for this special case, and then use them to derive the general bound.

But he opened out the hinges, Pushed and pulled the joints and hinges, Till it looked all squares and oblongs Like a complicated figure In the Second Book of Euclid. C. L. Dodgson After discussing some further properties of triangles and quadrangles (or quadrilaterals), we shall approach the domain of projective geometry (and even trespass a bit). A systematic development of that fascinating subject must be left for another book, but four of its most basic theorems are justifiably mentioned here because they can be proved by the methods of Euclid; in fact, three of the four are so old that no other methods were available at the time of their discovery. All these theorems deal either with collinearity (certain sets of points lying on a line) or concurrence (certain sets of lines passing through a point). The spirit of projective geometry begins to emerge as soon as we notice that, for many purposes, parallel lines behave like concurrent lines. Quadrangles; Varignon's theorem A polygon may be defined as consisting of a number of points (called vertices ) and an equal number of line segments (called sides ), namely a cyclically ordered set of points in a plane, with no three successive points collinear, together with the line segments joining consecutive pairs of the points. In other words, a polygon is a closed broken Sine lying in a plane.

The most generally used method for estimating the basin-wide sequence and scaling of first and last occurrences, based on their occurrence in local sections, is Shaw’s graphic correlation method. The key step in this method is the determination of the line of correlation (LOC), which represents the best estimate of the correlation between two local sections, or between a local section and a composite standard. In general, available techniques for fitting the LOC for multiple sections are tedious, subjective, or computationally expensive. A new method employing genetic algorithms can dramatically reduce the effort involved in determining the LOC and produces stable biostratigraphic correlations and composite range charts objectively and efficiently. Genetic algorithms are an artificial intelligence technique that excels in locating the optimum solution from a large number of alternative choices. In the case of the LOC, the alternative choices are the number of line segments comprising the complete line and the positions of each segment’s beginning and end points. For a given number of segments, a wide range of alternative LOCs can be rapidly evaluated and a potential optimum fit determined. It is also possible to estimate the point when no further refinement of the fit by adding line segments is necessary. Genetic algorithms can also be applied to other methods for quantitative biostratigraphy.

A riveted graph G(V,E) is a graph whose vertices are fixed on a plane. Its rectilinear drawing D(G) is a configuration of G on the plane in such a way that every edge is composed of horizontal and vertical line segments. The authors consider the problem of reducing the number of line segments and present an algorithm to obtain such a rectilinear drawing that the total number of line segments is no more than 3 mod E mod . It is noted that a drawing so obtained contains no edge composed of five line segments. The algorithm runs in O( mod E mod ) time and space. >

We consider some problems on red and blue points in the plane lattice. An L-line segment in the plane lattice consists of a vertical line segment and a horizontal line segment having a common endpoint. There are some results on geometric graphs on a set of red and blue points in the plane. We show that some similar results also hold for a set of red and blue points in the plane lattice using L-line segments instead of line segments. For example, we show that if n red points and n blue points are given in the plane lattice in general position, then there exists a noncrossing geometric perfect matching covering them, each of whose edges is an L-line segment and connects a red point and a blue point.

We consider the problem of finding the staircase kernel in orthogonal polygons, with or without holes, in the plane. Orthogonal polygon is a simple polygon in the plane whose sides are either horizontal or vertical. We generalize the notion of visibility in the following way: We say that two points a and b in an orthogonal polygon P are visible to each other via staircase paths if and only if there exist an orthogonal chain connecting a and b and lying entirely in the interior of P. Furthermore, the orthogonal chain should have the property that the angles between the consecutive segments in the chain are either +90 ∘ or −90 ∘ , and these should alternate along the chain. There are two principal types of staircases, NW-SE and NE-SW. The notion of staircase visibility has been studied in the literature for the last three decades. Based on this notion we can generalize the notion of star-shapedness. A polygon P is called star-shaped under staircase visibility, or simply s-star if and only if there is nonempty set of points S in the interior of P, such that any point of S sees any point of P via staircase path. The largest such set of points is called the staircase kernel of P and denoted ker P. Our work is motivated by the work of Breen (1). She proves that the staircase kernel of an orthogonal polygon without holes is the intersection of all maximal orthogonally convex polygons contained in it. We extend Breen's results for the case when the orthogonal polygon has holes. We prove the necessary geometric properties, and use them to derive a quadratic time, O(���� 2 ) algorithm for computing the staircase kernel of an orthogonal polygon with holes, having n vertices in total, including the holes' vertices. The algorithm is based on the plane sweep technique, widely used in Computational Geometry(4). Our result is optimal in the case of orthogonal polygon with holes, since the kernel (as proven) can consist of quadratic number of disjoint regions. In the case of polygon without holes, there is a linear time algorithm by Gewali(3), that is specific to the case of a polygon without holes. We present ex- amples of our algorithm's results.

In this paper are considered planar point sets generated by linear conditions, which are realized in rectangular or Manhattan metric. Linear conditions are those expressed by the finite sum of the products of distances by numerical coefficients. Finite sets of points and lines are considered as figures defining linear conditions. It has been shown that linear conditions can be defined relative to other planar figures: lines, polygons, etc. The design solutions of the following general geometric problem are considered: for a finite set of figures (points, line segments, polygons...) specified on a plane with a rectangular metric, which are in a common position, it is necessary to construct sets that satisfy any linear condition. The problems in which the given sets are point and segment ones have been considered in detail, and linear conditions are represented as a sum or as relations of distances. It is proved that solution result can be isolated points, broken lines, and areas on the plane. Sets of broken lines satisfying the given conditions form families of isolines for the given condition. An algorithm for building isoline families is presented. The algorithm is based on the Hanan lattice construction and the isolines behavior in each node and each sub-region of the lattice. For isoline families defined by conditions for relation of distances, some of their properties allowing accelerate their construction process are proved. As an example for application of the described theory, the problem of plane partition into regions corresponding to a given set of points, lines and other figures is considered. The problem is generalized problem of Voronoi diagram construction, and considered in general formulation. It means the next: 1) the problem is considered in rectangular metric; 2) all given points may be integrated in various figures – separate points, line segments, triangles, quadrangles etc.; 3) the Voronoi diagram’s property of proximity is changed for property of proportionality. Have been represented examples for plane partition into regions, determined by two-point sets.

The circle formation is a good starting point for many symmetric formations like an approximation of a circle, a simple polygon, or a line segment etc in the plane. A distributed manner is presented to fulfill the coordinated task of N mobile agents swarm (which could be swarm robotics, unmanned air/ocean vehicles namely UAV, satellites, mobile sensors, aircraft and spacecraft etc.) being forced to achieve a circle formation while avoiding collision between themselves. The control development method is based on artificial potential function namely APF and Newton-Raphson iteration. Finally, the performance of the method is examined by numerical simulations, the results further indicated that the individual members living in swarm during the course of coordinative motion can realize the stabilization of particular predefined a desired circle or a convex polygon formation.

Given a simple polygon P on n vertices, two points x, y in P are said to be visible to each other if the line segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum-size set S such that every point in P is visible from a point in S. The set S is referred to as guards. Assuming integer coordinates and a specific general position on the vertices of P, we present the first O(log OPT)-approximation algorithm for the point guard problem. This algorithm combines ideas in papers of Efrat and Har-Peled and Deshpande et al. We also point out a mistake in the latter.

Path planning is one of the most important tasks for autonomous mobile robots. The purpose of path planning is to find a collision-free path from a starting position to a target position due to specific criteria such as distance. This paper proposes a modified path improvement algorithm based on the former and later points. In the initial path, some points are randomly placed on the line segments that have the necessary condition. New positions of points in the path are calculated by their adjacent points. This approach explores the most optimal path between different values of angle for the different number of segments. A method is proposed in which the initial path in the line segments that contain the turns adheres to the obstacles in the way. This operation makes a shorter path. Experiments are conducted on six maps and the results indicate that the proposed method achieves a considerable optimization in the final path.

Random polygon Generation from a given set of points is very important for different applications. A heuristic termed as “GRP_AS” has been proposed here to generate a random simple polygon from a given set of `n' points in 2-Dimensional plane. The “2-Opt Move” heuristic with time complexity O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> ) is the best known among the existing heuristics to generate a simple polygon. We recently designed another heuristic “GRP_CH” for the same problem with time complexity O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) which was shown to be better than that of “2-Opt Move” heuristic in terms of random behaviour and time complexity. In this work, our main objective is to design an algorithm with reduced time complexity. The proposed heuristic, “GRP_AS” is based on the principle of random selection of a line segment and angular sorting of the point set. This “GRP_AS” heuristic takes O(nlogn) time which is less than that of “2-Opt Move” heuristic as well as “GRP_CH” heuristic. The lower bound and upper bound on the number of polygons generated in “GRP_AS” heuristic has been computed.