Line Segment Intersection Research Articles

PEER-TO-PEER LOCALIZATION IN URBAN AND INDOOR ENVIRONMENTS

This paper presents a novel peer-to-peer or mobile- to-mobile localization scheme for general indoor and outdoor environments. In this scheme, two mobile nodes at arbitrary locations are able to locate each other without the need of Line-of-Sight (LOS) path between the two mobile device, and without the need for any reference devices such as GPS or land base beacons. Existing peer- to-peer localization techniques make use of Time of Arrival (TOA) and Angle of Arrival (AOA) of LOS and single bounce scattering paths to derive line of possible mobile device positions (LPMDs). The intersections of LPMDs are then used to estimate the unknown mobile device position | referred to as the Line Segment Intersection. However, in a heavy multipath environment with many multiple- bounce scattering paths, existing techniques require weighting factors and threshold values which are speciflcally chosen for that particular environment in order to select the LPMDs that correspond to LOS and single-bounce scattering paths for localization. Large localization error will occur if multiple-bounce scattering paths' LPMDs are mistakenly used for intersections. In addition, existing techniques also do not work well in a multipath environment with high level of TOA and AOA noises especially when the angles between LPMDs are small. The accuracy of the Line Segment Intersection also deteriorates as the distance traveled by multipath signals become comparable to each other. This renders the weighting and threshold values inefiective. This paper presents a novel Gaussian weighting process to remove the abovementioned limitations. The Gaussian weighting process also dramatically improves the accuracy of the localization. Experimental coupled with simulation results show that our proposed localization scheme outperforms existing Peer-to-peer localization technique by a signiflcant margin of up to 83% and 54%

Optimal in-place and cache-oblivious algorithms for 3-d convex hulls and 2-d segment intersection

We describe the first optimal randomized in-place algorithm for the basic 3-d convex hull problem (and, in particular, for 2-d Voronoi diagrams). The algorithm runs in O ( n log n ) expected time using only O ( 1 ) extra space; this improves the previous O ( n log 3 n ) bound by Brönnimann, Chan, and Chen (2004) [10]. The same approach leads to an optimal randomized in-place algorithm for the 2-d line segment intersection problem, with O ( n log n + K ) expected running time for output size K, improving the previous O ( n log 2 n + K ) bound by Vahrenhold (2007) [42]. As a bonus, we also point out a simplification of a known optimal cache-oblivious (non-in-place) algorithm by Kumar and Ramos (2002) [33] for 3-d convex hulls, and observe its applicability to 2-d segment intersection, extending a recent result for red/blue segment intersection by Arge, Mølhave, and Zeh (2008) [3]. Our results are all obtained by standard random sampling techniques, with some interesting twists.

A parallel line segment intersection strategy based on uniform grids

The line segment intersection problem is one of the basic problems in computational geometry and has been widely used in spatial analysis in Geographic Information Systems (GIS). Lots of traditional algorithms study the problem in a serial environment. However, in GIS, a spatial object is much more complicated and is considered to be always composed of multiple line segments, and one line segment connects another line segment at its endpoint. On the other hand, along with the advances made in computer hardware, more and more personal computers have multiple cores or CPUs equipped. Thus, to make full use of the increasing computing resources, parallel technique is applied as one of the most available methods. Apparently, the traditional algorithms should be improved to take advantage of the technologies. Under these circumstances, based on the modified uniform grid algorithm, which is adapted to dealing with spatial objects in GIS, this paper proposes a parallel strategy in a shared memory architecture. Also, experimental results are given in the final part of this paper to demonstrate the efficiency this strategy brings.

On the density of iterated line segment intersections

Given S 1 , a finite set of points in the plane, we define a sequence of point sets S i as follows: With S i already determined, let L i be the set of all the line segments connecting pairs of points of ⋃ j = 1 i S j , and let S i + 1 be the set of intersection points of those line segments in L i , which cross but do not overlap. We show that with the exception of some starting configurations the set of all crossing points ⋃ i = 1 ∞ S i is dense in a particular subset of the plane with nonempty interior. This region is the intersection of all closed half planes which contain all but at most one point from S 1 .

External-Memory Algorithms for Processing Line Segments in Geographic Information Systems

In the design of algorithms for large-scale applications it is essential to consider the problem of minimizing I/O communication. Geographical information systems (GIS) are good examples of such large-scale applications as they frequently handle huge amounts of spatial data. In this paper we develop efficient external-memory algorithms for a number of important problems involving line segments in the plane, including trapezoid decomposition, batched planar point location, triangulation, red--blue line segment intersection reporting, and general line segment intersection reporting. In GIS systems the first three problems are useful for rendering and modeling, and the latter two are frequently used for overlaying maps and extracting information from them.

Space-efficient geometric divide-and-conquer algorithms

We develop a number of space-efficient tools including an approach to simulate divide-and-conquer space-efficiently, stably selecting and unselecting a subset from a sorted set, and computing the kth smallest element in one dimension from a multi-dimensional set that is sorted in another dimension. We then apply these tools to solve several geometric problems that have solutions using some form of divide-and-conquer. Specifically, we present a deterministic algorithm running in O ( n log n ) time using O ( 1 ) extra memory given inputs of size n for the closest pair problem and a randomized solution running in O ( n log n ) expected time and using O ( 1 ) extra space for the bichromatic closest pair problem. For the orthogonal line segment intersection problem, we solve the problem in O ( n log n + k ) time using O ( 1 ) extra space where n is the number of horizontal and vertical line segments and k is the number of intersections.

Effects of Surface Markings on Judgments of Motion Direction

The effects of surface markings on perceived motion direction were examined for a rotating sphere in a structure-from-motion display. The markings were dot patterns representing separate line segments or intersecting line segments (crosses) covering the surface of the sphere. The orientation of the surface markings and their intersection angles affected the perceived direction of motion, suggesting that the markings were not interpreted as geodesics or planar cuts on the surface. The perceived direction of motion was biased towards the mean orientation of the markings over the visible area of the surface. A similar bias was observed for translating planar stimuli covered with crosses, suggesting that the bias is not specific to curved surfaces or motion in depth. The deviation between the simulated motion direction and the external horizontal and vertical axes also affected the perceived motion direction. These results suggest that the average orientation of surface contours with respect to an external reference frame influences the perceived direction of motion.

Raster milling tool‐path generation from STL files

PurposeThis paper presents an offset‐based tool path generation method for STL format three‐dimensional (3D) models. The created tool‐paths can be effectively used to near‐net‐shaped parts, in particular those created using rapid prototyping.Design/methodology/approachThe STL model is first offset by the distance of the selected cutter radius using a unique 3D offset method. The intersections between the top facing triangles of the offset model and tool‐path drive planes are calculated. The intersection line segments are sorted, trimmed and linked to generate continuous top envelope curves, which represent interference‐free tool paths.FindingsThe developed offset‐based algorithm can rapidly and successfully generate interference‐free tool paths as continuous lines, instead of a collection of discrete tool location points. The strategy of using adaptive step‐over distances based on local geometrical information can significantly increase machining efficiency.Research limitations/implicationsThe current tool path generation method only works for ball‐end mills. The entire surface of the STL model is treated as a single composite surface to be machined using raster milling. To improve machining efficiency, an automatic surface splitting algorithm could be developed to divide the model into several regions based on the characteristics of a group of triangular facets, and then machine these identified regions using different strategies and cutters.Originality/valueThe offset‐based tool‐path generation algorithm from STL models is a unique and novel development, which is useful in the rapid prototyping and computer‐aided machining areas.

A computer aided tool selection system for 3D die/mould-cavity NC machining using both a heuristic and analytical approach

The selection of cutting tools in die/mould-cavity NC machining affects not only the machining time and costs, but also the process quality. A systematic methodology for cutting tool selection, including tool type and size, for each operation, i.e. roughing, semi-roughing and finishing, is proposed in this article. First, a heuristic-based tool type selection method is discussed. Second, a tool combination optimization approach for roughing is presented and an optimization mathematical model is built. The related algorithms for realizing this approach, including acquiring loop from intersection line segments, identification of inner and outer loops, computation of key distance and merging of hunting layers, are presented in detail. Third, the methods for semi-roughing and finishing tool selection are also outlined. In order to validate the proposed methodology, a system called CATS has been developed in the UG/Open API environments and two examples of cavity with islands are included to demonstrate its feasibility. The work in this research facilitates both the elimination of the experience factors of human tool selection, and the seamless integration of CAD/CAM systems.

Line Segment Intersection Testing

A method for accurately determining whether two given line segments intersect is presented. This method uses the standard floating-point arithmetic that conforms to IEEE 754 standard. If three or four ending points of the two given line segments are on a same vertical or horizontal line, the intersection testing result is obtained directly. Otherwise, the ending points and their connections are mapped onto a 3×3 grid, and the intersection testing falls into one of the five testing classes. The intersection testing method is based on our method for floating-point dot product summation, whose error bound is 1ulp. Our method does not have the limitation in the method of Gavrilova and Rokne (2000) that the product of two floating-point numbers is calculated by a twice higher precision floating-point arithmetic than that of the multipliers. Furthermore, this method requires less than one-fifth of the running time used by the method of Gavrilova and Rokne (2000), and our new method for calculating the sign of a sum of n floating-point numbers requires less than one-fifteenth of the running time used by ESSA.

Generalized triangulations and diagonal-free subsets of stack polyominoes

For n ⩾ 3 , let Ω n be the set of line segments between vertices in a convex n-gon. For j ⩾ 1 , a j- crossing is a set of j distinct and mutually intersecting line segments from Ω n such that all 2 j endpoints are distinct. For k ⩾ 1 , let Δ n , k be the simplicial complex of subsets of Ω n not containing any ( k + 1 ) -crossing. For example, Δ n , 1 has one maximal set for each triangulation of the n-gon. Dress, Koolen, and Moulton were able to prove that all maximal sets in Δ n , k have the same number k ( 2 n - 2 k - 1 ) of line segments. We demonstrate that the number of such maximal sets is counted by a k × k determinant of Catalan numbers. By the work of Desainte-Catherine and Viennot, this determinant is known to count quite a few other objects including fans of non-crossing Dyck paths. We generalize our result to a larger class of simplicial complexes including some of the complexes appearing in the work of Herzog and Trung on determinantal ideals.

Parallel algorithms for identifying convex and non-convex basis polygons in an image

In this paper, we propose two novel parallel algorithms for identifying all the basis polygons in an image formed by n straight line segments each of which is represented by its two end points. The first algorithm is designed to tackle the simple situation where all basis polygons are convex. The second one deals with the general situation when the basis polygons can be both convex and non-convex. These algorithms are based on an idea of traversal along the periphery of the basis polygons in a well-defined manner so that each of these needs only O( n) time using an n × n processor array. Simulation results on various test input sets of intersecting line segments have also been found satisfactory.

Spanning Trees Crossing Few Barriers

We consider the problem of finding low-cost spanning trees for sets of $n$ points in the plane, where the cost of a spanning tree is defined as the total number of intersections of tree edges with a given set of $m$ barriers. We obtain the following results: (i) if the barriers are possibly intersecting line segments, then there is always a spanning tree of cost $O(\min(m^2,m\sqrt{n}))$; (ii) if the barriers are disjoint line segments, then there is always a spanning tree of cost $O(m)$; (iii) ] if the barriers are disjoint convex objects, then there is always a spanning tree of cost $O(n+m)$. All our bounds are worst-case optimal, up to multiplicative constants.

Cache Oblivious Distribution Sweeping

<p>We adapt the distribution sweeping method to the cache oblivious model. Distribution sweeping is the name used for a general approach for divide-and-conquer algorithms where the combination of solved subproblems can be viewed as a merging process of streams. We demonstrate by a series of algorithms for specific problems the feasibility of the method in a cache oblivious setting. The problems all come from computational geometry, and are: orthogonal line segment intersection reporting, the all nearest neighbors problem, the 3D maxima problem, computing the measure of a set of axis-parallel rectangles, computing the visibility of a set of line segments from a point, batched orthogonal range queries, and reporting pairwise intersections of axis-parallel rectangles. Our basic building block is a simplified version of the cache oblivious sorting algorithm Funnelsort of Frigo et al., which is of independent interest.</p><p> </p><p>Full text: http://dx.doi.org/10.1007/3-540-45465-9_37</p>

Reliable line segment intersection testing

The main result of this paper is a new algorithm that tests whether two line segments in the plane intersect. If the segments are defined using the coordinates of the endpoints in single-precision floating-point arithmetic, then the result of the test is exact. The equations of the segments are given in parametric form using the endpoint coordinates, and an equation whose solution would provide the coordinates of the intersection is developed. Interval arithmetic is then used to compute an inclusion of the coordinates of the intersection point. This inclusion is often sufficient to decide the intersection test. When it is not, a method for determining the exact sign of a sum is applied to the equations at an earlier stage of the solution process. Experimental results are presented that show that the number of intersections that cannot be resolved using interval tools is fairly large in close to degenerate configurations of the segments and that the algorithm is significantly faster than an algorithm implemented using exact arithmetic.

Efficient algorithms for line and curve segment intersection using restricted predicates

We consider whether restricted sets of geometric predicates support efficient algorithms to solve line and curve segment intersection problems in the plane. Our restrictions are based on the notion of algebraic degree, proposed by Preparata and others as a way to guide the search for efficient algorithms that can be implemented in more realistic computational models than the Real RAM. Suppose that n (pseudo-)segments have k intersections at which they cross. We show that intersection algorithms for monotone curves that use only comparisons and above/below tests for endpoints, and intersection tests, must take at least Ω(n k ) time. There are optimal O( nlog n+ k) algorithms that use a higher-degree test comparing x coordinates of an endpoint and intersection point; for line segments we show that this test can be simulated using CCW() tests with a logarithmic loss of efficiency. We also give an optimal O( nlog n+ k) algorithms for red/blue line and pseudo-segment intersection, in which the segments are colored red and blue so that there are no red/red or blue/blue crossings.

Efficient algorithms for line and curve segment intersection using restricted predicates

Efficient algorithms for line and curve segment intersection using restricted predicates

Improvement on the Plane Sweep Method for Finding All Intersections of Line Segments

Improvement on the Plane Sweep Method for Finding All Intersections of Line Segments

Robust Computation of Intersection Graph between Two Solids

We propose a new robust algorithm for Boolean operations on solid models. The algorithm produces a consistent intersection graph between two input solids whose geometrical data are represented in floating point numbers. In order to prevent numerical calculation errors and inaccuracy of input data from causing inconsistency of the output, we put higher priority on symbolical connectivity of the edge‐face intersection points than their numerical nearness. Each edge‐face intersection point is symbolically represented using face names, which generate connectivity relations between the intersection points and the intersection line segments. The symbols with the same connectivity are made into clusters. The intersection line segments connected together at their end clusters form the intersection graph of two solids. Inconsistency of the connectivity of the clusters is detected and the intersection graph is corrected automatically. We describe the algorithm in detail for polyhedral solids, discuss extension to curves solids, and show its effectiveness by some examples of Boolean operations for two solids whose faces intersect at a very small angle.

Robust Computation of Intersection Graph between Two Solids

We propose a new robust algorithm for Boolean operations on solid models. The algorithm produces a consistent intersection graph between two input solids whose geometrical data are represented in floating point numbers. In order to prevent numerical calculation errors and inaccuracy of input data from causing inconsistency of the output, we put higher priority on symbolical connectivity of the edge‐face intersection points than their numerical nearness. Each edge‐face intersection point is symbolically represented using face names, which generate connectivity relations between the intersection points and the intersection line segments. The symbols with the same connectivity are made into clusters. The intersection line segments connected together at their end clusters form the intersection graph of two solids. Inconsistency of the connectivity of the clusters is detected and the intersection graph is corrected automatically. We describe the algorithm in detail for polyhedral solids, discuss extension to curves solids, and show its effectiveness by some examples of Boolean operations for two solids whose faces intersect at a very small angle.