Algorithm For Boolean Operations Research Articles

Interactive and Robust Mesh Booleans

Boolean operations are among the most used paradigms to create and edit digital shapes. Despite being conceptually simple, the computation of mesh Booleans is notoriously challenging. Main issues come from numerical approximations that make the detection and processing of intersection points inconsistent and unreliable, exposing implementations based on floating point arithmetic to many kinds of degeneracy and failure. Numerical methods based on rational numbers or exact geometric predicates have the needed robustness guarantees, that are achieved at the cost of increased computation times that, as of today, has always restricted the use of robust mesh Booleans to offline applications. We introduce an algorithm for Boolean operations with robustness guarantees that is capable of operating at interactive frame rates on meshes with up to 200K triangles. We evaluate our tool thoroughly, considering not only interactive applications but also batch processing of large collections of meshes, processing of huge meshes containing millions of elements and variadic Booleans of hundreds of shapes altogether. In all these experiments, we consistently outperform prior robust floating point methods by at least one order of magnitude.

Voxel-Based Geometry Reconstruction for Repairing and Remanufacturing of Metallic Components Via Additive Manufacturing

In the component repair process using additive manufacturing (AM) technique, reconstructing the repair volume is essential to generate the repair tool path and guide the AM system to deposit correct geometry on the worn parts. In this study, a novel repair volume reconstruction methodology based on voxel modeling was introduced. At first, the stereolithography (STL) models of the nominal and damaged components were acquired either from computer-aided design (CAD) modeling or through robot-aided 3D scanning. The pre-machining approach was introduced to guarantee the defective zone is accessible to repair tools. The damaged models were aligned with the nominal models by best-fitting their common convex-hull centroids. After that, these STL models were converted to voxel models using the Marching Cube algorithm. The accuracy of the transferred voxel models was investigated by comparing them with the exact geometry. Boolean operations algorithm based on constructive solid geometry was proposed to extract the repair volume by comparing the nominal and damaged voxel models. The Boolean operations can output both the repair volume that is missing from the damaged parts which should be restored by AM and the excess geometry which should be removed by subtractive machining. The proposed approach was implemented on three CAD models and three scanned models. The repair volumes for these damaged parts were successfully reconstructed. Three components were repaired via the directed energy deposition AM process. Finally, the bonding performance between the deposited layers and base parts was evaluated via tensile testing.

Optimal Reliable Point-in-Polygon Test and Differential Coding Boolean Operations on Polygons

This paper provides a full theoretical and experimental analysis of a serial algorithm for the point-in-polygon test, which requires less running time than previous algorithms and can handle all degenerate cases. The serial algorithm can quickly determine whether a point is inside or outside a polygon and accurately determine the contours of input polygon. It describes all degenerate cases and simultaneously provides a corresponding solution to each degenerate case to ensure the stability and reliability. This also creates the prerequisites and basis for our novel boolean operations algorithm that inherits all the benefits of the serial algorithm. Using geometric probability and straight-line equation F ( P ) = ( y i − y i + 1 ) ( x p − x i ) − ( y i − y p ) ( x i + 1 − x i ) , it optimizes our two algorithms that avoid the division operation and do not need to compute any intersection points. Our algorithms are applicable to any polygon that may be self-intersecting or with holes nested to any level of depth. They do not have to sort the vertices clockwise or counterclockwise beforehand. Consequently, they process all edges one by one in any order for input polygons. This allows a parallel implementation of each algorithm to be made very easily. We also prove several theorems guaranteeing the correctness of algorithms. To speed up the operations, we assign each vector a number code and derive two iterative formulas using differential calculus. However, the experimental results as well as the theoretical proof show that our serial algorithm for the point-in-polygon test is optimal and the time complexities of all algorithms are linear. Our methods can be extended to three-dimensional space, in particular, they can be applied to 3D printing to improve its performance.

Robust Boolean operations algorithm on regularized triangular mesh and implementation

Boolean operations are an essential tool for creating complex entities in many fields. In order to implement complex entity modeling, we proposed a method based on robust Boolean operations that focused on the robustness of geometric calculations caused by computational errors and data error. This method used a uniform logical judgment to analyze the specific conditions of the intersection of vertices or edges in advance, and avoided the inconsistency between logical judgment results and geometric relations. We correspondingly obtained the positions of the two triangles, the validity of the intersection and the intersection edges from tetrahedral-volume calculations, the triangle-area calculations, and the topology information to mark the triangles instead of intersecting lines tracking and the judgment of the triangles inside the entities. Finally, the experimental results indicate that this method realized the three-dimensional modeling of any complex geological body.

Fourth- and Higher-order Interface Tracking Via Mapping and Adjusting Regular Semianalytic sets Represented by Cubic Splines

This work is a further development and the culmination along our research line of interface tracking in two dimensions [Zhang and Liu, J. Comput. Phys., 27 (2008), pp. 4063--4088; Zhang, SIAM J. Numer. Anal., 51 (2013), pp. 2822--2850; Zhang and Fogelson, SIAM J. Sci. Comput., 36 (2014), pp. A2369--A2400; Zhang and Fogelson, SIAM J. Numer. Anal., 54 (2016), pp. 530--560]. Previously, we have proposed an analytic framework for explicit interface tracking and a fourth-order interface tracking method. In this paper, we continue our approach to propose a new method, called the cubic MARS method, that features (1) a solid theoretical foundation consisting of well developed concepts from distinct branches of mathematics, (2) a representation of the interface with cubic splines, (3) a novel algorithm for Boolean operations on linear/curvilinear polygons, and (4) fourth-, sixth-, and eighth-order accuracy via a refined control of the relation between the Eulerian grid size $h$ and a Lagrangian length scale $h_L$ of interface markers. Volume conservation of the cubic MARS method also has convergence rates at 4, 6, and 8. A simple analysis and numerical results from several benchmark problems show that the cubic MARS method is superior to existing interface tracking methods in terms of accuracy and efficiency. In particular, for the single-vortex test and the deformation test, errors of an eighth-order cubic MARS method are close to machine precision on a 128-by-128 Eulerian grid! Currently the new method does not tackle topological changes of the interface, but its theoretical foundation provides a solid footing for future treatments of such phenomena.

A GPU-Based Rasterization Algorithm for Boolean Operations on Polygons

A GPU-Based Rasterization Algorithm for Boolean Operations on Polygons

Boolean operations on rasterized shapes represented by chain codes using space filling curves

This paper introduces a new algorithm for Boolean operations on rasterized geometric shapes that are represented with chain codes. The algorithm works in three steps. Firstly, the chain code symbols are transformed in the Hilbert space, where the overlaid chain code symbols are recognised. After that, a suitable starting cell is determined. Finally, the walk-about through the sequence of the initial chain code symbols is performed to obtain the sequence of chain code symbols representing the shape of the required Boolean operation. The algorithm is demonstrated on Freeman chain code in four directions. The time and space complexity of the proposed algorithm is linear, which was proven theoretically and confirmed by experiments.

On the nesting of a contour dataset and its use in repair

ABSTRACTA contour dataset is a common source of data in biomedical imaging and GIS. In practice, many contour datasets violate key assumptions: contours self-intersect or intersect other contours, which can disrupt the performance of algorithms. We call such data dirty. Avoiding this issue, algorithms in the literature assume clean data. This paper considers the issue of dirty datasets, in the context of nesting.Contours are often nested inside other contours, and the nesting level of a contour is important, as it affects the position of the inside of the shape. This paper analyzes the nesting of a dirty dataset, then shows how to use this nesting analysis to repair the data. A theme is the power of image space algorithms in the presence of noise. Image space algorithms for polygon area and Boolean operations of polygons are provided.

Two-Dimensional Boundary Recovery Procedure for Non-Manifold and Intersecting Boundaries

To ensure all boundary edges of each region are retained in the final mesh during mesh generation, a procedure suitable for non-manifold with intersecting boundaries is proposed to recover the non-manifold and intersecting boundaries. In the procedure, Steiner points are inserted directly on the interFigure of missing boundaries and mesh edges from a triangulation. Based on the boundary recovery procedure, Delaunay triangulation algorithms for the domain with interior constraints and for the non-manifold are proposed, and the latter can be used as a new effective polygon Boolean operations algorithm.

Fast Filter-Based Boolean Matchers

<?Pub Dtl?>Boolean matching is one of the fundamental and time-consuming procedures in field-programmable gate array (FPGA) synthesis. The SAT-based Boolean matchers (BMs) are not scalable while other Boolean matchers based on complicated Boolean logic operation algorithms are not flexible for complex PLBs. Recently, a scalable Boolean matcher (F-BM) based on the Bloom filter has been proposed for both scalability and flexibility. However, it requires large amount of memory space which can be a bottleneck for traditional personal computers. To tackle that problem, this letter proposes a novel Boolean matcher with much less memory requirement. Compared with F-BM, the proposed Boolean matcher has achieved an average of 5% better result with 2000x smaller storage and only 1.6x more runtime when applying to the same application. The significant reduction of storage requirements makes the proposed Boolean matcher able to handle more complicated PLB structures with larger input sizes.

A simple algorithm for Boolean operations on polygons

In this paper a simple and efficient algorithm for computing Boolean operations on polygons is presented. The algorithm works with almost any kind of input polygons: concave polygons, polygons with holes, several contours and self-intersecting edges. Important topological information, as the holes of the result polygon, is computed.

Collision free region determination by modified polygonal Boolean operations

Cutting and packing problems are found in numerous industries such as garment, wood and shipbuilding. The collision free region concept is presented, as it represents all the translations possible for an item to be inserted into a container with already placed items. The often adopted nofit polygon concept and its analogous concept inner fit polygon are used to determine the collision free region. Boolean operations involving nofit polygons and inner fit polygons are used to determine the collision free region. New robust non-regularized Boolean operations algorithm is proposed to determine the collision free region. The algorithm is capable of dealing with degenerated boundaries. This capability is important because degenerated boundaries often represent local optimal placements. A parallelized version of the algorithm is also proposed and tests are performed in order to determine the execution times of both the serial and parallel versions of the algorithm.

An algorithm for discrete booleans with applications to finite element modeling of complex systems

In this paper we describe a robust algorithm for three-dimensional boolean operations between boundary representation objects whose geometry is given by discrete (faceted) data. The algorithm presents a new approach for computing the intersection graph which is critical for robustness. It uses elementary computational-geometry operations such as, facet–segment intersection, point containment in simplices and edge recovery in a plane, to produce high-level boolean operations including union, intersection, difference as well as the imprint of the boundary of one object onto another. We also demonstrate the extension and application of the algorithm to mesh-based volumes. We show the robustness and efficacy of our algorithm by employing it to model complex three-dimensional finite element mesh models such as a complete ship where some of the model components are defined in a CAD-based system while others come from legacy mesh-based facetized representations. Use of our algorithm has enabled automation of modeling of very complex configurations reducing the turnaround time for analyses-ready numerical representations from several months to hours or less.

Boolean Operations Algorithm on Triangle Meshes for Modeling Bone Scaffold

Boolean operation is the key technology for modeling the bone scaffold. This paper proposes a Boolean operation algorithm based on triangle mesh model. It firstly voxelizes the mesh models based on project vector and octree, and classifies the vertices into inside, outside and surface type according to the position relationship between the vertex and the other mesh model’s voxel. Then the triangles can be easily classified based on the vertices class. Finally the Boolean model can be composed by the corresponding triangles of the Boolean operation. In order to obtain the intersection features well, it detects the intersection lines and gets the intersection polygons further, and then triangulates the polygon using the ear clipping method. This Boolean operation algorithm has been applied to the bone scaffold modeling and got good performance．

A vector-based algorithm to generate and update multiplicatively weighted Voronoi diagrams for points, polylines, and polygons

A Voronoi diagram is a basic data structure in geometry with many applications. Existing research studies have focused on ordinary Voronoi diagrams, and some vector-based algorithms have been developed to generate multiplicatively weighted Voronoi diagrams (MWVDs) for points. An algorithm to construct MWVDs for points, polylines, and polygons is raster-based and has drawbacks. We propose a vector-based algorithm to generate and update MWVDs for points, polylines, and polygons. The MWVDs of two sites characterize the geometric features of multiplicatively weighted Voronoi regions, and the algorithm for Boolean operations on conic polygons is the computational preliminary to the MWVDs of N sites. The proposed algorithm is vector-based and can deal with sites with mixed spatial extents and various weights. We implement the algorithm in C# and present several examples of generating and updating MWVDs.

Boolean operations on multi-region solids for mesh generation

An algorithm for Boolean operations on non-manifold models is proposed to allow the treatment of solids with multiple regions (internal interfaces) and degenerate portions (shells and wires), in the context of mesh generation. In a solid modeler, one of the most powerful tools to create three-dimensional objects with any level of geometric complexity is the Boolean set operators. They are intuitive and popular ways to combine solids, based on the operations applied to point sets. To assure that the resulting objects have the same dimension as the original objects, without loose or dangling parts, a regularization process is usually applied after a Boolean operation. In practice, the regularization is performed classifying the topological elements and removing internal or lower-dimensional structures. However, in many engineering applications, the adopted geometric model may contain idealized internal parts, as in the case of multi-region models, or lower-dimensional parts, as in the case of solids that contain dangling slabs that are represented as zero-thickness surfaces or wireframes in the model. Therefore, the aim of this work is the development of a generic algorithm that allows the application of the Boolean set operations in a geometric modeling environment applied to finite and boundary element mesh generation. This environment adopts a non-manifold boundary representation that considers an undefined number of topological entities (group concept), and works with objects of different dimensions and with objects not necessarily plane or polyhedral (parametric curved surfaces). Numerical examples are presented to illustrate the proposed methodology.

Boolean Operations on Conic Polygons

An algorithm for Boolean operations on conic polygons is proposed. Conic polygons are polygons consisting of conic segments or bounded conics with directions. Preliminaries of Boolean operations on general polygons are presented. In our algorithm, the intersection points and the topological relationships between two conic polygons are computed. Boundaries are obtained by tracking path and selecting uncrossed boundaries following rule tables to build resulting conic polygons. We define a set of rules for the intersection, union, and subtraction operations on conic polygons. The algorithm considers degeneration cases such as homology, complement, interior, and exterior. The algorithm is also evaluated and implemented.

Boolean Set Operations with Cubic Algebraic Patches

In this paper we discuss a symbolic-numeric algorithm for Boolean operations, closed in the algebra of curved polyhedra whose boundary is triangulated with algebraic patches (A-patches). This approach uses a linear polyhedron as a first approximation of both the arguments and the result. On each triangle of a boundary representation of such linear approximation, a piecewise cubic algebraic interpolant is built, using a C(1)-continuous prism algebraic patch (prism A-patch) that interpolates the three triangle vertices, with given normal vectors. The boundary representation only stores the vertices of the initial triangulation and their external vertex normals. In order to represent also flat and/or sharp local features, the corresponding normal-per-face and/or normal-per-edge may be also given, respectively. The topology is described by storing, for each curved triangle, the two triples of pointers to incident vertices and to adjacent triangles. For each triangle, a scaffolding prism is built, produced by its extreme vertices and normals, which provides a containment volume for the curved interpolating A-patch. When looking for the result of a regularized Boolean operation, the 0-set of a tri-variate polynomial within each such prism is generated, and intersected with the analogous 0-sets of the other curved polyhedron, when two prisms have non-empty intersection. The intersection curves of the boundaries are traced and used to decompose each boundary into the 3 standard classes of subpatches, denoted in, out and on. While tracing the intersection curves, the locally refined triangulation of intersecting patches is produced, and added to the boundary representation.

Boolean operations on 3D selective Nef complexes: Data structure, algorithms, optimized implementation and experiments

Nef polyhedra in d-dimensional space are the closure of half-spaces under boolean set operations. In consequence, they can represent non-manifold situations, open and closed sets, mixed-dimensional complexes, and they are closed under all boolean and topological operations, such as complement and boundary. They were introduced by W. Nef in his seminal 1978 book on polyhedra. The generality of Nef complexes is essential for some applications. In this paper, we present a new data structure for the boundary representation of three-dimensional Nef polyhedra and efficient algorithms for boolean operations. We use exact arithmetic to avoid well-known problems with floating-point arithmetic and handle all degeneracies. Furthermore, we present important optimizations for the algorithms, and evaluate this optimized implementation with extensive experiments. The experiments supplement the theoretical runtime analysis and illustrate the effectiveness of our optimizations. We compare our implementation with the Acis CAD kernel. Acis is mostly faster, by a factor up to six. There are examples on which Acis fails. The implementation was released as Open Source in the Computational Geometry Algorithm Library ( Cgal) release 3.1 in December 2004.

A topologically robust algorithm for Boolean operations on polyhedral shapes using approximate arithmetic

We present a topologically robust algorithm for Boolean operations on polyhedral boundary models. The algorithm can be proved always to generate a result with valid connectivity if the input shape representations have valid connectivity, irrespective of the type of arithmetic used or the extent of numerical errors in the computations or input data. The main part of the algorithm is based on a series of interdependent operations. The relationship between these operations ensures a consistency in the intermediate results that guarantees correct connectivity in the final result. Either a triangle mesh or polygon mesh can be used. Although the basic algorithm may generate geometric artifacts, principally gaps and slivers, a data smoothing post-process can be applied to the result to remove such artifacts, thereby making the combined process a practical and reliable way of performing Boolean operations.