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Efficient Booleans Algorithms for Triangulated Meshes of Geometric Modeling

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Efficient Booleans Algorithms for Triangulated Meshes of Geometric Modeling COMPUTER-AIDED DESIGN & APPLICATIONS, 2016 VOL. 13, NO. 4, 419–430 http://dx.doi.org/10.1080/16864360.2015.1131530 Efficient Booleans algorithms for triangulated meshes of geometric modeling Xiaotong Jiang1 , Qingjin Peng2 , Xiaosheng Cheng1 ,NingDai1 , Cheng Cheng1 and Dawei Li1 1Nanjing University of Aeronautics and Astronautics, China; 2University of Manitoba, Canada ABSTRACT KEYWORDS Boolean operation of geometric models is an essential element in computational geometry. An effi- Octree; floating point cient approach is developed in this research to perform Boolean operation for triangulated meshes arithmetic errors; singularity represented by B-rep. This approach is much fast and robust than many existing methods. The Octree of intersections; sub-meshes technique is adapted to facilitate the division of the common space of two meshes in order to reduce classify; fast and robust the time of Octree’s construction and intersection detection. Floating point arithmetic errors and sin- gularity of intersections are then analyzed to guarantee the unique intersection between a segment and a face, and the continuity of intersections. A novel technique based on intersecting triangles is finally proposed to create required sub-meshes based on the type of Boolean operations. Some experimental results and comparisons with other methods are presented to prove that the proposed method is fast and robust. 1. Introduction improve the stability of the exact arithmetic. To avoid Boolean operations of geometric models are important the complex exact computation and effectively deal with processes in computational geometry [7,15,16]. A most degenerated intersections, approximate arithmetic meth- commonly used method in Boolean operations to dis- ods and volumetric methods were suggested. For approx- play geometric models is the boundary representation imate arithmetic methods, Smith and Dodgson [32] (B-Rep) based on parametric surfaces. Booleans oper- presented a topologically robust algorithm. Ming et al. ations on B-Rep models were introduced in the 1980 s [6] proposed an approach to compute the approximate [24, 30]. With the development of reverse engineering Boolean operations of two free-form polygonal solids and 3D printing technologies, the use of discrete sur- efficiently with the help of Layered Depth Images. The faces, especially the triangulated meshes, is increasing. volumetric method converts mesh surfaces into the vol- Triangulatedmodelsareusedinmanyfields,suchas umetric representation. Booleans can then be operated architectural design, industrial design, and CAE. Numer- based on volumetric data [10, 27]. Because of the lim- ous Boolean algorithms have been proposed to resolve ited precision of the representation, the loss of geomet- triangulated meshes in geometric modeling. However, ric features is unavoidable. Marching Cubes algorithm the complex problem challenges performance of these is usually used to extract features from the volumetric methods in efficiency and robustness. result [22]. Pavic et al. [29]presentedahybridmethodto Boolean algorithms can be classified into three broad combine the volumetric technique with surface-oriented categories: exact arithmetic, approximate arithmetic and techniques. The method can obtain an output meth volumetric methods [6]. The exact arithmetic operates to preserve the existing sharp features and reconstruct Boolean operations directly over geometric elements, new features appearing along intersections of the input such as faces, edges and vertices [2, 30]. However, they meshes. This work was for the exact Booleans using B- often suffer the problem in searching stable solutions. Rep triangle meshes which are exactly edge-based data Various methods have been proposed to improve the structures [25]. robustness of the exact arithmetic. Boolean operations Robustness and efficiency of exact Booleans on were implemented by Keyser [21] in a curved domain for triangulated meshes have proven a challenging and com- an exact arithmetic. An interval computation is adopted plicated task. There are two main procedures of the trian- by Hu [17, 18]tooperateBooleansonsolidmodelswith gulated meshes in Boolean algorithms: the intersection spline surfaces. The BSP (binary space partition) tech- detection between two meshes and Boolean operations. nique was proposed by Bernstein and Campen [3, 4]to Intersection detection and Boolean operations are crucial CONTACT Jiang Xiaotong [email protected] © 2016 CAD Solutions, LLC, http://www.cadanda.com 420 X. JIANG ET AL. to the performance of the algorithm. This paper presents betweenasegmentandamesh.Thisalsoinfluencesthe a fast and stable intersection detection technique and continuity of intersections. Boolean operations to achieve excellent performance. For Boolean operations, a key technique is the point- For the intersection detection between two meshes, in-solid test to classify required sub-meshes. Several many techniques have been proposed to accelerate the approaches can be used to perform the point-in-solid speed of intersection detections. A BSP-based method test. Feito et al. [28] used the simplicial covering of the was used to accelerate the spatial queries in the pro- solids.Thismethodisveryfastwhenitisimplementedon cess [3]. The method is stable and can operate com- GPU, however it needs a large number of tests to classify plex meshes, however it encounters memory issues when the newly created sub-mesh. A method of Jordan Curve mesheshavealargenumberoffaces[9]. A technique was TheoremwasadoptedbyVeblen[36], the method uses a adopted by Campen [4] to combine an adaptive Octree ray-surface intersection test in order to obtain the par- with the nested binary space partition, which can achieve ity of the number of intersections. The method is fast, the high performance and less memory consumption. however it needs the Octree calculated in advance, the Jing et al. [20] adopted an oriented bounding box (OBB) method may fail when it deals with the open mesh. Guo tree method. The OBB tree is fast to obtain the intersec- et al. [13] used a loop detection to decide the candidate tion lines once it is built. However, the building time of point whether inside or outside the solid. However, these the structure is slow and is inconvenient to use. Feito et al. methods are only designed for solid models, and require [9] used the Octree method which requires the less stor- meshes no boundary edges. Jacobson et al. [19]used age and can run in a multithreaded environment. The a generalized winding numbers method for the inside- Octree method has the good performance to deal with outside segmentation to handle the open mesh situation. complex meshes. Some methods were proposed to cal- However, there is still not a unified rule of Booleans for culate the intersection line of a pair of triangles. Moller open meshes. As shown in Fig. 1,usingthecommercial [26] presented a method to compute whether or not two CAD software Geomagic Studio 12, the Booleans differ- triangles intersect. Tropp et al. [35] developed an opti- ence operation is applied on two open meshes. If an open mummethodtosaveabout20%timeofthemathematical mesh’s normal is flipped, the result will be difference. operations. When calculating intersection lines of two meshes, there are two factors that influence efficiency and 2. Method stability of the algorithm. The first is that each intersected edge of one mesh will produce two points with another In this paper, a fast and robust Boolean algorithm for mesh. The two points have the same coordinate value the triangulated mesh is presented. It involves two states theoretically. The floating point arithmetic errors may as shown in Fig. 2: intersection detection between two cause the incontinuity of intersections. The second is that meshes and Boolean operations. For the intersection there may be singularity when judging the intersection detection between two meshes, the Octree technique is (a) (b) Figure 1. Booleans difference results using Geomagic Studio 12. Figure 2. Flow of the approach. COMPUTER-AIDED DESIGN & APPLICATIONS 421 adapted to facilitate the division of the common space of 2.1.1. Building Octree of the common space two meshes in order to reduce the time of Octree’s con- Many techniques have been used to accelerate the inter- struction and intersection detection, floating point arith- section detection of two meshes, such as methods of BSP meticerrorsandsingularityofintersectionsareanalyzed (binary space partitions) [3, 4], OBB tree [20], Spatial to find intersections. In this way, the unique intersec- hashing [8, 33, 38], and Octree [9]. The Octree technique tion between a segment and a mesh can be obtained to is adopted in this research. guarantee the continuity of intersections. For Boolean For two intersected meshes, the size of the common operations, a fast and stable technique is presented to spaceissmall.Octreecanbebuilttodividethecommon realize union, intersection and difference operations. The space to accelerate the speed of intersection detection and method is simple as it finds only points that belong to reduce memory utilization. The common space can be these intersected triangles to create required sub-meshes calculated as follows: based on the type of Booleans. The method is suitable for Given two triangulated meshes SA and SB as shown in both closed and open meshes. Fig. 3.LetBoxA and BoxB are the smallest axis-aligned In summary, the main contributions of this work are bounding boxes (AABBs) of SA and SB. BoxA and BoxB
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