Construction and optimization of CSG representations

Boundary representations (B-reps) and constructive solid geometry (CSG) are widely used representation schemes for solids. While the problem of computing a B-rep from a CSG representation is relatively well understood, the inverse problem of B-rep to CSG conversion has not been addressed in general. The ability to perform B-rep to CSG conversion has important implications for the architecture of solid modelling systems and, in addition, is of considerable theoretical interest.

The paper presents a general approach to B-rep to CSG conversion based on a partition of Euclidean space by surfaces induced from a B-rep, and on the well known fact that closed regular sets and regularized set operations form a Boolean algebra. It is shown that the conversion problem is well defined, and that the solution results in a CSG representation that is unique for a fixed set of halfspaces that serve as a ‘basis’ for the representation. The ‘basis’ set contains halfspaces induced from a B-rep plus additional non-unique separating halfspaces.

An important characteristic of B-rep to CSG conversion is the size of a resulting CSG representation. We consider minimization of CSG representations in some detail and suggest new minimization techniques.

While many important geometric and combinatorial issues remain open, a companion paper shows that the proposed approach to B-rep to CSG conversion and minimization is effective in E², In E³, an experimental system currently converts natural-quadric B-reps in PARASOLID to efficient CSG representations in PADL-2.     

Techniques for reducing Boolean evaluation time in CSG scan-line algorithms

Atherton's scan-line hidden surface algorithm for Boolean combinations of plane-faced primitives is intended for fast image generation of complex models.

An important step in the algorithm is the Boolean evaluation at the required positions on each scan-line. This step, however, can be very time consuming if performed in a straightforward way.

Some techniques are presented here that considerably reduce the time needed for the Boolean evaluation. These include a fast bottom-up evaluation technique and partial backface elimination.     

An algorithm for visible-line and visible-surface display of CSG models

An algorithm is presented for display of Constructive Solid Geometry (CSG) models, in which Boolean evaluation of a model is done during image generation only for the visible parts of the model.The algorithm is based on Atherton's CSG scan-line algorithm. It involves, however, dividing the image plane into strips of varying width, inside which areas are determined where only one face is visible. This may involve subdivision of an area into smaller areas.Two versions of the algorithm are presented: an efficient visible-line version for a raster display, and a visible-surface version, which turns out to be an improved variant for simple models of Atherton's algorithm.Sample images and CPU times for some models are given to show the efficiency of the algorithm.     

Solving CSG equations for checking equivalency between two different geometric models

For two given parametric constructive solid geometry (CSG) models, the problem of determining their parameter domains in which the two models are equivalent is addressed. Here, two CSG models are equivalent if they represent the exact same region in R³, although their constituent features and feature attributes may differ. In this paper, an approach for solving the problem in a limited scope is proposed, in which a CSG model is polyhedral, its parametric form is explicit and its feature orientations are fixed. The solution includes the equivalent parameter domain for each model and parameter mapping that associates these two models on their equivalent parameter domains. One application of this research is to identify the equivalent parameter domains of two parametric part models, respectively, in the design option space and in the capability envelope set of a parametric machining process in order to facilitate the interoperation between part design and process planning through the generated parameter mapping between these two different kinds of models.     

An efficient algorithm for finding the CSG representation of a simple polygon

Modeling two-dimensional and three-dimensional objects is an important theme in computer graphics. Two main types of models are used in both cases: boundary representations, which represent the surface of an object explicitly but represent its interior only implicitly, and constructive solid geometry representations, which model a complex object, surface and interior together, as a boolean combination of simpler objects. Because neither representation is good for all applications, conversion between the two is often necessary.We consider the problem of converting boundary representations of polyhedral objects into constructive solid geometry (CSG) representations. The CSG representations for a polyhedronP are based on the half-spaces supporting the faces ofP. For certain kinds of polyhedra this problem is equivalent to the corresponding problem for simple polygons in the plane. We give a new proof that the interior of each simple polygon can be represented by a monotone boolean formula based on the half-planes supporting the sides of the polygon and using each such half-plane only once. Our main contribution is an efficient and practicalO(n logn) algorithm for doing this boundary-to-CSG conversion for a simple polygon ofn sides. We also prove that such nice formulae do not always exist for general polyhedra in three dimensions.The first author would like to acknowledge the support of the National Science Foundation under Grants CCR87-00917 and CCR90-02352. The fourth author was supported in part by a National Science Foundation Graduate Fellowship. This work was begun while the first author was visiting the DEC Systems Research Center.     

Hybrid Booleans

In this paper, we present a novel method to compute Boolean operations on polygonal meshes. Given a Boolean expression over an arbitrary number of input meshes we reliably and efficiently compute an output mesh which faithfully preserves the existing sharp features and precisely reconstructs the new features appearing along the intersections of the input meshes. The term "hybrid" applies to our method in two ways: First, our algorithm operates on a hybrid data structure which stores the original input polygons (surface data) in an adaptively refined octree (volume data). By this we combine the robustness of volumetric techniques with the accuracy of surface-oriented techniques. Second, we generate a new triangulation only in a close vicinity around the intersections of the input meshes and thus preserve as much of the original mesh structure as possible (hybrid mesh). Since the actual processing of the Boolean operation is confined to a very small region around the intersections of the input meshes, we can achieve very high adaptive refinement resolutions and hence very high precision. We demonstrate our method on a number of challenging examples.     

Identifying rectangles in laser range data for urban scene reconstruction

In urban scenes, many of the surfaces are planar and bounded by simple shapes. In a laser scan of such a scene, these simple shapes can still be identified. We present a one-parameter algorithm that can identify point sets on a plane for which a rectangle is a fitting boundary. These rectangles have a guaranteed density: no large part of the rectangle is empty of points. We prove that our algorithm identifies all angles for which a rectangle fits the point set of size n in O(nlogn) time. We evaluate our method experimentally on 13 urban data sets and we compare the rectangles found by our algorithm to the α‐shape as a surface boundary.     

Asymptotic speed-ups in constructive solid geometry

We convert constructive solid geometry input to explicit representations of polygons, polyhedra, or more generallyd-dimensional polyhedra, in time and space 0(n^d), improving a previous0(n^d logn) time bound. We then show that any Boolean formula can be preprocessed in time0(n log n/log logn) and linear space so that the value of the formula can be maintained, as variables are changed one by one, in time O(log n/log logn) per change; this speeds up certain output-sensitive algorithms for constructive solid geometry.     

Efficient editing of solid models by exploiting structural and spatial locality

Computer aided design systems based on solid modellers must provide fast visual feedback to users when objects are edited. This implies that boundary representations must be updated rapidly, because displays typically are generated in current-generation modellers from face, edge and vertex data.This paper describes algorithms for updating a boundary representation when an object's constructive solid geometry (CSG) representation is edited. The algorithms exploit the structural (representational) locality inherent in most object modifications by taking advantage of previously computed boundary representations for (sub-) objects that are not affected by the editing operations. They also exploit spatial locality by re-computing boundaries only within the spatial region where changes can occur. The algorithms are efficient, and are guaranteed to produce valid solids because they are based on CSG.     

Topologically exact evaluation of polyhedra defined in CSG with loose primitives

Floating point round-off causes erroneous and inconsistent decisions in geometric modelling algorithms. These errors lead to the generation of topologically invalid boundary models for CSG objects and significantly reduce the reliability of CAD applications. Previously known methods that guarantee topological consistency by relying on arbitrary precision rational arithmetic or on symbol-manipulation techniques are too expensive for practical purposes. This paper presents a new solution which takes as input a “fixed precision” regularized Boolean combination of linear half-spaces and produces a polyhedral boundary model that has the exact topology of the corresponding solid. Each half-space is represented by four homogeneous coefficients infixed precision format (L_a bits for the three direction cosines and L_d bits for the constant term, i.e. the distance from the origin). Exact answers to all topological and ordering questions are computed using a fixed length, 3 L_a+ L_d+ 2 bits, integer format. This new guaranteed tight limit on the number of bits necessary for performing intermediate calculations is achieved by expressing all of the topological decisions based on geometric computations in terms of the signs of 4 by 4 determinants of the input coefficients. The coordinates of intersection vertices are not required for making the correct topological decisions and hence vertices and lines are represented implicitly in terms of planes.     

Interpolatory,solid subdivision of unstructured hexahedral meshes

This paper presents a new, volumetric subdivision scheme for interpolation of arbitrary hexahedral meshes. To date, nearly every existing volumetric subdivision scheme is approximating, i.e., with each application of the subdivision algorithm, the geometry shrinks away from its control mesh. Often, an approximating algorithm is undesirable and inappropriate, producing unsatisfactory results for certain applications in solid modeling and engineering design (e.g., finite element meshing). We address this lack of smooth, interpolatory subdivision algorithms by devising a new scheme founded upon the concept of tri-cubic Lagrange interpolating polynomials. We show that our algorithm is a natural generalization of the butterfly subdivision surface scheme to a tri-variate, volumetric setting.     

Approximate motion planning and the complexity of the boundary of the union of simple geometric figures

We study rigid motions of a rectangle amidst polygonal obstacles. The best known algorithms for this problem have a running time of (n ²), wheren is the number of obstacle corners. We introduce thetightness of a motion-planning problem as a measure of the difficulty of a planning problem in an intuitive sense and describe an algorithm with a running time ofO((a/b · 1/_crit + 1)n(logn)²), wherea b are the lengths of the sides of a rectangle and _crit is the tightness of the problem. We show further that the complexity (= number of vertices) of the boundary ofn bow ties (see Figure 1) isO(n). Similar results for the union of other simple geometric figures such as triangles and wedges are also presented.This work was supported partially by the DFG Schwerpunkt Datenstrukturen und Algorithmen, Grants Me 620/6 and Al 253/1, and by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM).     

基于自适应延迟切割的三角网格布尔运算优化

规则化的布尔运算被广泛应用在三维建模系统中.近年来,随着图形硬件的发展,基于三角网格的规则化布尔算法由于输出结果能直接被图形硬件处理,表现出了明显的优势.但是传统的算法由于采用CSG树局部评估策略,使得面片在相交测试中反复被切割,并且由于面片分类在切割后的模型之间直接进行,导致算法无法在保证鲁棒性的同时实现高性能.为了避免这些问题,本文呈现了一种CSG树全局评估算法来统一执行单次和连续布尔运算.算法由两部分组成：自适应的延迟切割和全局化面片分类.在自适应的延迟切割阶段,算法通过仔细处理多个三角面片相交导致的各种情况使得延迟切割被扩展到整个CSG树来避免由于面片的反复切割带来的数值误差累积并利用自适应的八叉树使得相交测试能在线性时间内完成.在全局化面片分类阶段,算法通过分治法使得分类始终在切割后的面片和原始输入模型之间进行来保证分类的精度;通过结合组分类策略和自适应的八叉树来进一步优化了分类性能。实验结果表明,本文提出的算法无论是在执行单次或连续布尔运算时都能在保证鲁棒性同时性能优于其他的算法,因此本文算法可广泛应用于交互式建模系统中,如数字雕刻、计算机辅助设计和制造（CAD/CAM）等.     

Integration of CAD and boundary element analysis through subdivision methods

Subdivision methods have been mainly used in computer graphics. This paper extends their applications to mechanical design and boundary element analysis (BEA), and fulfills the seamless integration of CAD and BEA in the model and representation.Traditionally, geometric design and BEA are treated as separate modules requiring different representations and models, which include continuous parametric models and discrete models. Due to the incompatibility of the involved representations and models, the post-processing in geometric design or the pre-processing in BEA is essential. The transition from geometric design to BEA requires substantial effort and errors are inevitably introduced during the transition. In this paper, a framework of realizing the integration of CAD and BEA was first presented based on subdivision methods. A common model or a unified representation for geometric design and BEA was created with subdivision surfaces. For general 3D structures, automatic mesh generation for geometric design and BEA was fulfilled through subdivision methods. The seamless integration improves the accuracy of numerical analysis and shortens the cycle of geometric design and BEA.     

A decision-support method for multi-parameter editing of parametric CAD models

Parametric modeling is a computer-aided design (CAD) paradigm where a design can be created by defining geometric constraints with parameters. In design change as well as design optimization, the design is often edited by modifying the values of relevant parameters. Without guidance on allowable parameter ranges that can guarantee the intrinsic solvability of the geometric constraint system, the user could assign improper values to the model’s parameters, which would further lead to a failure in model updating. However, current commercial CAD systems provide little support on such guidance. Existing methods, though able to compute allowable ranges for individual parameters, face difficulties in handling multi-parameter situations. To solve this problem, a systematic method is proposed, supporting decision-making in multi-parameter model editing by computing allowable parameter ranges. In the method, a set of variable parameters from the geometric constraint system are first selected by the user; these variable parameters are to be sequentially edited within several editing operations. Before each editing operation, 1D allowable ranges of the variable parameters are computed. By editing parameter values within the provided ranges, the solvability of the geometric constraint system can be guaranteed. The effectiveness and efficiency of the proposed approach is verified by several experimental results.     

Multi-objective optimization of expensive electromagnetic simulation models

Vast majority of practical engineering design problems require simultaneous handling of several criteria. For the sake of simplicity and through a priori preference articulation one can turn many design tasks into single-objective problems that can be handled using conventional numerical optimization routines. However, in some situations, acquiring comprehensive knowledge about the system at hand, in particular, about possible trade-offs between conflicting objectives may be necessary. This calls for multi-objective optimization that aims at identifying a set of alternative, Pareto-optimal designs. The most popular solution approaches include population-based metaheuristics. Unfortunately, such methods are not practical for problems involving expensive computational models. This is particularly the case for microwave and antenna engineering where design reliability requires utilization of CPU-intensive electromagnetic (EM) analysis. In this work, we discuss methodologies for expedited multi-objective design optimization of expensive EM simulation models. The solution approaches that we present here rely on surrogate-based optimization (SBO) paradigm, where the design speedup is obtained by shifting the optimization burden into a cheap replacement model (the surrogate). The latter is utilized for generating the initial approximation of the Pareto front representation as well as further front refinement (to elevate it to the high-fidelity EM simulation model level). We demonstrate several application case studies, including a wideband matching transformer, a dielectric resonator antenna and an ultra-wideband monopole antenna. Dimensionality of the design spaces in the considered examples vary from six to fifteen, and the design optimization cost is about one hundred of high-fidelity EM simulations of the respective structure, which is extremely low given the problem complexity.     

Multi-objective optimization design of a complex building based on an artificial neural network and performance evaluation of algorithms

While optimization studies focusing on real-world buildings are somewhat limited, many building optimization studies to date have used simple hypothetical buildings for the following three reasons: (1) the shape and form of real buildings are complex and difficult to mathematically describe; (2) computer models built based on real buildings are computationally expensive, which makes the optimization process time-consuming and impractical and (3) although algorithm performance is crucial for achieving effective building performance optimization (BPO), there is a lack of agreement regarding the proper selection of optimization algorithms and algorithm control parameters. This study applied BPO to the design of a newly built complex building. A number of design variables, including the shape of the building’s eaves, were optimized to improve building energy efficiency and indoor thermal comfort. Instead of using a detailed simulation model, a surrogate model developed by an artificial neural network (ANN) was used to reduce the computing time. In this study, the performance of four multi-objective algorithms was evaluated by using the proposed performance evaluation criteria to select the best algorithm and parameter values for population size and number of generations. The performance evaluation results of the algorithms implied that NSGA-II (with a population size and number of generations of 40 and 45, respectively) performed the best in the case study. The final optimal solution significantly improves building performance, demonstrating the success of the BPO technique in solving complex building design problems. In addition, the findings on the performance evaluation of the algorithms provide guidance for users regarding the selection of suitable algorithms and parameter settings based on the most important performance criteria.