Collinearity and Concurrence

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.

Point-to-line mappings as Hough transforms

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.

Abstract. We present a new line sweep algorithm, HEAPSWEEP, for reporting bichromatic (``purple'') intersections between a red and a blue family of line segments. If the union of the segments in each family is connected as a point set, HEAPSWEEP reports all k purple intersections in time O((n+k) α(n) log 3 n) , where n is the total number of input segments and α(n) is the nearly constant inverse Ackermann function. To achieve these bounds, the algorithm maintains only partial information about the vertical ordering between curves of the same color, using a new data structure called a kinetic queue . In order to analyze the running time of HEAPSWEEP, we also show that a simple polygon containing a set of n line segments can be partitioned into monotone regions by a set of vertical threads cutting these segments O(n log n) times.

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 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.

In this paper, we propose a method of polypointed mappings of line segments in geometric modeling of complex objects, which differs from ordinary polypointed mappings in that you can set straight lines as an inverse image, not in an implicit form, but as a set of points and obtain a transformed object as a set of points. In addition, it is proposed to use weights for each base point, which allows you to control the shape of the transformed curve in such a way that the curve is somehow attracted to a point with a higher weight compared to other base points. Polypoint transformations are used in the design of complex geometric shapes in mechanical engineering. Enhancing the capabilities of polypoint transformations is an important issue at the present time. The aim of the article is the use of line segments in the method of polypoint mappings for geometric modeling of complex objects. The proposed process of weighted political dot mapping can be applied in the implementation of geometric modeling of complex shape objects with the ability to control the shape of a simulated object by changing the weights of the points. As shown in test examples, the shape of the transformed figure changes in such a way that it seems to be attracted to points with increased weight and, conversely, is repelled from points with reduced weight. As the disadvantages of the proposed method, one can indicate an insufficient level of predictability of the result. Further research is proposed in identifying the properties of using weighted polypoint mappings in the direction of increasing the level of predictability of the results for effective use in geometric modeling of complex objects. Key words: discrete-given curve, line segment, polypoint mapping of lines, geometric modeling.

We prove that the integral points are potentially Zariski dense in the complement of a reduced effective singular anticanonical divisor in a smooth del Pezzo surface, with the exception of ${{\mathbb {P}}}^{2}$ minus three concurrent lines (for which potential density does not hold). This answers positively a question raised by Hassett and Tschinkel and, combined with previous results, completes the proof of the potential density of integral points for complements of anticanonical divisors in smooth del Pezzo surfaces. We then classify the complements that are simply connected and for these we prove that the set of integral points is potentially not thin, as predicted by a conjecture of Corvaja and Zannier.

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.

A straight-line drawing of a plane graph G is a planar drawing of G, where each vertex is drawn as a point and each edge is drawn as a straight-line segment. Given a set S of n points on the Euclidean plane, a point-set embedding of a plane graph G with n vertices on S is a straight-line drawing of G, where each vertex of G is mapped to a distinct point of S. The problem of deciding if G admits a point-set embedding on S is NP-complete in general and even when G is 2-connected and 2-outerplanar. In this paper we give an O(n 2logn) time algorithm to decide whether a plane 3-tree admits a point-set embedding on a given set of points or not, and find an embedding if it exists. We prove an Ω(n logn) lower bound on the time complexity for finding a point-set embedding of a plane 3-tree. Moreover, we consider a variant of the problem where we are given a plane 3-tree G with n vertices and a set S of k > n points, and give a polynomial time algorithm to find a point-set embedding of G on S if it exists.

An obstacle representation of a graph G is a set of points in the plane representing the vertices of G, together with a set of polygonal obstacles such that two vertices of G are connected by an edge in G if and only if the line segment between the corresponding points avoids all the obstacles. The obstacle number $${{\mathrm{obs}}}(G)$$ of G is the minimum number of obstacles in an obstacle representation of G. We provide the first non-trivial general upper bound on the obstacle number of graphs by showing that every n-vertex graph G satisfies $${{\mathrm{obs}}}(G) \le n\lceil \log {n}\rceil -n+1$$ . This refutes a conjecture of Mukkamala, Pach, and Pálvölgyi. For n-vertex graphs with bounded chromatic number, we improve this bound to O(n). Both bounds apply even when the obstacles are required to be convex. We also prove a lower bound $$2^{\Omega (hn)}$$ on the number of n-vertex graphs with obstacle number at most h for $$h<n$$ and a lower bound $$\Omega (n^{4/3}M^{2/3})$$ for the complexity of a collection of $$M \ge \Omega (n\log ^{3/2}{n})$$ faces in an arrangement of line segments with n endpoints. The latter bound is tight up to a multiplicative constant.

A polyhedral object in 3-dimensional space is often well represented by a set of points and line segments that act asits features. By a nice viewpoint of an object we mean a projective view in which all (or most) of the features of theobject, relevant for some task, are clearly visible. Such a view is often called a non-degenerate view or projection. Inthis paper we are concerned with computing non-degenerate orthogonal and perspective projections of sets of pointsand line segments (objects) in 3-dimensional space. We outline the areas in which such problems arize, discuss recentresearch on the computational complexity of these problems, illustrate the fundamental ideas used in the design ofalgorithms for computing non-degenerate projections, and provide pointers to the literature where the results can befound.Keywords: Data visualization, nice viewpoints, knot theory, computer vision, graph drawing, computer graphics,degeneracies, general position, orthogonal projections, perspective projections, robustness of algorithms, computa-tional geometry.

Introduction Convex sets play an important role in integral geometry. For this reason we will review here their principal properties, especially those which will be needed in the following sections. In this chapter we consider convex sets in the plane. For convex sets in n -dimensional euclidean space, see Chapter 13. For a more complete treatment, refer to the classical books of Blaschke [50] and Bonnesen and Fenchel [63], or to the more modern texts of Benson [27], Eggleston [162], Grunbaum [247], Jaglom and Boltjanski [320], Hadwiger [270], Hadwiger and coauthors [282], and Valentine [683]. A set of points K in the plane is called convex if for each pair of points A ∈ K , B ∈ K it is true that AB ⊂ K , where AB is the line segment joining A and B . For convenience we shall assume throughout that the convex sets are bounded and closed. A curve with end points P, Q , is called convex if its point set, together with the segment PQ , bounds a convex set. If the convex set K is bounded and has interior points, then the boundary of K is called a closed convex curve . Throughout, we will denote by ∂ K the boundary of the set K . If all the points of K belong to ∂ K , then K is a line segment. We can prove that (a) All convex curves are piecewise differentiable (i.e., they are the union of a countable set of arcs with continuously turning tangent); in other words, convex curves have at most a countable set of corners; (b) All bounded convex curves are rectifiable.