Efficient algorithms for line and curve segment intersection using restricted predicates

Based on the unique geometric features of the ruled surface, a new efficient algorithm for ruled surface/plane intersection is proposed. The ruled surface is firstly dispersed into a set of line segments, and the ruled surface/plane intersection is transformed to the intersection of a group line segments with a plane. Then a set of ordered intersection points can be obtained by the proposed line/plane intersection algorithm. According to the serial number of every intersection point, all the intersection points are grouped and reorganized. Each group point corresponds to an intersection curve. All the intersection curves can be reconstructed by curve interpolation. Compared with the traditional tracing method, the proposed algorithm can avoid complex calculation including initial points searching and intersection points sorting, which is more efficient and stable.

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

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.

Purpose While existing 2D LiDAR algorithms can support mobile robots to perform localization and mapping in industrial scenarios, they often consume substantial computational resources, a critical limitation for lightweight robotic platforms. To enhance computational efficiency while maintaining localization accuracy, this paper aims to propose an efficient 2D LiDAR odometry algorithm based on the constraint of the intersection of lines and coordinate axes (LIA-LO). Design/methodology/approach The proposed LIA-LO algorithm consists of three key components: (1) in the body coordinate system, a seed region growing strategy is used to extract straight line segments from the environmental point cloud and fit their general expression parameters; (2) the intersection points between these line segments and the body coordinate axes are used as features to determine the matching relationship of line segments between two consecutive frames; and (3) the Gauss–Newton equation is applied to minimize the point-to-line distance residual, thereby solving the robot’s pose. Findings Experimental results demonstrate that the proposed algorithm achieves positioning accuracy comparable to established methods such as Hector-SLAM and PLICP, while taking less than 15 ms to compute on low-cost devices. Originality/value The novelty of this research lies in the introduction of line-axis intersection constraints as features for 2D LiDAR odometry. This approach, combined with a specialized line segment extraction and matching strategy, enables efficient and accurate pose estimation, offering a valuable contribution to lightweight SLAM systems, particularly in indoor environments where geometric structure information can be leveraged.

An improved method of rendering oil extraction contour under constrained conditions

Purpose/Objective(s): TTFields are FDA-approved for the treatment of recurrent and newly diagnosed Glioblastoma (GBM) in the supratentorial brain. The EF-14 trial showed that combining TTFields with adjuvant chemo-radiation leads to a significant increase in overall survival compared to adjuvant chemo-radiation alone. High-grade gliomas also occur in the infratentorium and brainstem of pediatric and adult patients. Brainstem gliomas have a median survival of 9 to 11 months, necessitating the exploration of novel treatment combinations, such as delivering TTFields to the infratentorial brain. We recently showed in a simulation-based study that TTFields can be successfully delivered to the infratentorium. Effective delivery was achieved by placement of the arrays on the vertex, bilateral posterolateral occiput, and superior-posterior neck; the array placement results in TTFields at therapeutic intensities throughout the brainstem and cerebellum of realistic computational head models. The previous simulation-based study was performed using realistic computational models of healthy individuals; it did not provide detailed insight into field distributions within tumor tissue located in the infratentorium. Here, we aim to expand the results of this previous study, by simulating the delivery of TTFields to the infratentorium of adult and pediatric patients with high-grade brainstem gliomas. Materials/Methods: A realistic computational model was created using MRI data of adult and pediatric patients with high-grade brainstem gliomas. Transducer arrays were placed on the model and the delivery of TTFields to the infratentorial brain was then simulated using a commercial numerical solver. The electric field distribution in the supratentorium, infratentorium, and within the tumor were derived from the simulations, and the dose in the tumor analyzed. Results: The distribution of calculated isofield lines demonstrates effective delivery of TTFields to infratentorial brain tumors. Conclusion: Our results provides rationale for clinically investigating the utility of TTFields in treating infratentorial high-grade gliomas. Citation Format: Marigdalia K. Ramirez-Fort, Brittany Cross, Ariel Naveh, Shearwood McClelland III, Melissa Mendez, Roberto Santiago, Sean S. Mahase, Jaime Matta, Ze9ev Bomzon, Christopher S. Lange. Simulating delivery of TTFields to the infratentorium in patients with brainstem gliomas [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2019; 2019 Mar 29-Apr 3; Atlanta, GA. Philadelphia (PA): AACR; Cancer Res 2019;79(13 Suppl):Abstract nr 4430.

Purpose/Objective(s): TTFields are FDA-approved for the treatment of recurrent and newly diagnosed Glioblastoma (GBM) in the supratentorial brain. The EF-14 trial showed that combining TTFields with adjuvant chemo-radiation leads to a significant increase in overall survival compared to adjuvant chemo-radiation alone. High-grade gliomas also occur in the infratentorium and brainstem of pediatric and adult patients. Brainstem gliomas have a median survival of 9 to 11 months, necessitating the exploration of novel treatment combinations, such as delivering TTFields to the infratentorial brain. We recently showed in a simulation-based study that TTFields can be successfully delivered to the infratentorium. Effective delivery was achieved by placement of the arrays on the vertex, bilateral posterolateral occiput, and superior-posterior neck; the array placement results in TTFields at therapeutic intensities throughout the brainstem and cerebellum of realistic computational head models. The previous simulation-based study was performed using realistic computational models of healthy individuals; it did not provide detailed insight into field distributions within tumor tissue located in the infratentorium. Here, we aim to expand the results of this previous study, by simulating the delivery of TTFields to the infratentorium of adult and pediatric patients with high-grade brainstem gliomas. Materials/Methods: A realistic computational model was created using MRI data of adult and pediatric patients with high-grade brainstem gliomas. Transducer arrays were placed on the model and the delivery of TTFields to the infratentorial brain was then simulated using a commercial numerical solver. The electric field distribution in the supratentorium, infratentorium, and within the tumor were derived from the simulations, and the dose in the tumor analyzed. Results: The distribution of calculated isofield lines demonstrates effective delivery of TTFields to infratentorial brain tumors. Conclusion: Our results provides rationale for clinically investigating the utility of TTFields in treating infratentorial high-grade gliomas. Citation Format: Marigdalia K. Ramirez-Fort, Brittany Cross, Ariel Naveh, Shearwood McClelland III, Melissa Mendez, Roberto Santiago, Sean S. Mahase, Jaime Matta, Ze'ev Bomzon, Christopher S. Lange. Simulating delivery of TTFields to the infratentorium in patients with brainstem gliomas [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2019; 2019 Mar 29-Apr 3; Atlanta, GA. Philadelphia (PA): AACR; Cancer Res 2019;79(13 Suppl):Abstract nr 4430.

The map construction problem (MCP) is defined as a spatial data integration problem relating to the integration of region data into map data. Although a purely geometric integration of regions into a map is known and is efficient, algorithms preserving thematic data of regions are much more difficult. A naive approach to the MCP runs in O((nmlg nm)2 + k) time for m regions composed with n line segments on average with k line segment intersections. A newO((n+k)(lg n+m+lgm2)) algorithm is presented to solve the MCP. The algorithm has been implemented and experiments show that it is significantly faster than the naive approach.

It may be possible to answer the question of whether or not a document was signed in blank, or after typing, by a microscopic examination of the line intersections. By removing the typewriter carbon at the point of the intersection, it can be clearly seen whether there is ink under the carbon or if the ink was removed with the carbon. A technique for determining the sequence of typewriting and most types of inked entries was developed. The results may be recorded photographically and subsequently used as a demonstrative exhibit in presenting the evidence during trial.

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 log3 n) bound by Bronnimann, Chan, and Chen [SoCG'04]. 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 log2 n + K) bound by Vahrenhold [WADS'05]. As a bonus, we also point out a simplification of a known optimal cache-oblivious (non-in-place) algorithm by Kumar and Ramos (2002) 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, Molhave, and Zeh [ESA'08]. Our results are all obtained by standard random sampling techniques, with some interesting twists.

The design-for-assembly technique requires realistic physically based simulation algorithms and in particular efficient geometric collision detection routines. Instead of approximating mechanical parts by large polygonal models, we work with the much smaller original CAD-data directly, thus avoiding precision and tolerance problems. We present a generic algorithm, which can decide whether two solids intersect or not. We identify classes of objects for which this algorithm can be efficiently specialized, and describe in detail how this specialization is done. These classes are objects that are bounded by quadric surface patches and conic arcs, objects that are bounded by natural quadric patches, torus patches, line segments and circular arcs, and objects that are bounded by quadric surface patches, segments of quadric intersection curves and segments of cubic spline curves. We show that all necessary geometric predicates can be evaluated by finding the roots of univariate polynomials of degree at most 4 for the first two classes, and at most 8 for the third class. In order to speed up the intersection tests we use bounding volume hierarchies. With the help of numerical optimization techniques we succeed in calculating smallest enclosing spheres and bounding boxes for a given set of surface patches fulfilling the properties mentioned above.

The straight line segment is the basic graphic element, and its generation algorithm has a great influence on the generation speed of complex graphics. In the past, the efficiency of integer operation is higher than that of floating-point operation, and the complexity of DDA algorithm is mainly floating-point addition and rounding operation, and the complexity of Bresenham algorithm is mainly integer operation through further improvement. Therefore, it is clearly explained in the existing computer graphics textbooks that the efficiency of Bresenham algorithm is higher than that of DDA algorithm. Through experiments, endpoints of 1,000 to 10,000 straight line segments are randomly generated, and the runtime of Bresenham algorithm and DDA algorithm is compared. It is found that the efficiency of DDA algorithm is higher than that of Bresenham algorithm. Then, by combining the comparison of the efficiency of modern CPU for addition, multiplication and division of integer and floating point with the improvement of floating-point computing hardware and instruction set in modern CPU, the computer's floating-point computing capabilities have been greatly improved, thus making the DDA algorithm more efficient than the Bresenham algorithm.

This chapter explores some basic elements of geometry and analytic geometry that are frequently encountered in computer graphics. For completeness, I have included a short review of important elements of Euclidean geometry with which you should be familiar. Perhaps the most important topics that you should try to understand concern the definitions of straight lines in space, 3D planes, and how points of intersection are computed. Another useful topic is the role of parameters in describing lines and line segments, and their intersection.

Dynamic analysis of many mechanical systems is often involved with contacts among bodies. This paper presents a relative contact formulation for multibody dynamics in the context of the compliance contact model. Many conventional collision detection algorithms are based on the absolute coordinate system. This paper proposes to use the relative coordinate system in detecting a contact. A contact reference frame is defined on the defense body of a contact pair. Since all geometric variables necessary to detect a contact are measured relative to the contact reference frame attached to the defense body, the variables for a defense body are constant, which significantly reduces computation time. Therefore, the contact frame plays a key role in developing an efficient contact search algorithm. Contour of a defense body is approximated by many piecewise straight lines, while contour of a hitting body is represented by hitting nodes along its boundary. Bounding boxes containing each body of a contact pair are defined at a pre-search stage to eliminate the exhaustive contact inspection process when two bodies are in a distance. Domain of the bounding box for a defense body is divided into many sectors each of which has a list of line segments lying inside or on the sector boundary. Post-search for a contact is processed in the sequence of broad and narrow phases. In the broad phase, the bounding boxes of a contact pair are inspected for a contact. If two boxes are in a contact, each node on the hitting boundary is inspected to find out to which sector the node belongs. Since each domain sector of the defense body has a list of line segments, each node on the hitting boundary is tested for a contact only with the line segments in the list. In the narrow phase, actual contact calculation is carried out to find the contact penetration used in calculating the contact force. Since the searching algorithm is coupled with the stepping algorithm of the numerical integration, a strategy for deciding an integration stepsize is proposed. One numerical example is presented to demonstrate the validity of the proposed method.

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.