Performance of scanline algorithms for direct display of CSG

Direct display algorithms display a CSG model without first converting the model into a boundary representation. Three such algorithms are described. All three are based on the scanline display algorithm, and are able to handle both polygonal and quadratic faces.The first algorithm is based on Atherton's recursive subdivision scanline algorithm, the second is a combination of a scanline and a ray casting algorithm, and the third is a scanline version of the Trickle algorithm. A multiprocessor system in which these algorithms can be incorporated is also described.The performances of the algorithms are compared. It turns out that the algorithms efficiently display CSG models on general-purpose architectures. A comparison is also made between the performances for polygon-approximated models and exact models for objects bounded by quadratic faces, such as spherical, cylindrical and conical faces, to get an indication of how many polygons can at most be used to approximate quadratic faces and still have better performance.     

Depth-Buffering Display Techniques for Constructive Solid Geometry

Solid modelers based on constructive solid geometry (CSG) typically generate shaded displays directly from CSG by using ray-casting techniques, which do not require informatin on the faces, edges, and vertices that bound a solid. This article describes an alternative-a simple new algorithm based on a depth-buffering or z-buffering approach. The z-buffer display algorithm operates directly on CSG, does not require explicit boundary data, and is easier to implement than ray casting. Ray-casting and z-beffering algorithms have comparable performances, but z-buffering is often faster for objects with complex surfaces, because it avoids expensive curve/surface intersection calculations. Because of their simplicity, depth-buffering algorithms for CSG are well-suited to hardware implementations, and may lead to machines simpler than those now being built for ray casting.     

CST: constructive solid trimming for rendering BReps and CSG

Abstract-To eliminate the need to evaluate the intersection curves in explicit representations of surface cutouts or of trimmed faces in BReps of CSG solids, we advocate using Constructive Solid Trimming (CST). A CST face is the intersection of a surface with a Blist representation of a trimming CSG volume. We propose a new GPU-based CSG rendering algorithm that trims the boundary of each primitive using a Blist of its active zone. This approach is faster than the previously reported Blister approach, eliminates occasional speckles of wrongly colored pixels, and provides additional capabilities: painting on surfaces, rendering semitransparent CSG models, and highlighting selected features in the BReps of CSG models.     

基于交线存储的快速实体边界生成算法

随着ＣＡＤ技术的日益普及，越来越复杂的设计对象进入系统，这就不可避免地导致了系统时空优化问题。目前实体造型中的主要表示方法，即边界模型及构造模型，由于各自特写的结构，无法较好地满足实际工程的需要。本文通过分析实体边界构造过程，提出了一个基个交线存储的实体存储方式及边界重构算法，该方法具有实体数据存储空间小，边界生成速度快等特点，较好地解决了系统存在的时空优化问题。     

Deferred boundary evaluation of complex CSG models

In this paper we present a deferred method for evaluating a complete CSG tree based on triangulated solids. It allows the exact evaluation of the surface of the entire model in a single step, using regularized Boolean classifications. The overall performance with this approach is better than with the classical method, which incrementally evaluates a CSG tree with single Boolean operations. The deferred algorithm does not use any intermediate result for the nodes of the CSG tree. It uses a very simple data structure and an octree that speeds up spatial queries for the entire CSG tree. The algorithm intensively uses multitasking and is ready for working with very complex CSG expressions, including the application of an out-of-core based approach.     

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.     

An efficient algorithm for ray casting of CSG animation frames

This paper presents a new algorithm to generate ray-cast CSG animation frames. We consider sequences of frames where only the objects can move; in this way, we take advantage of the high screen area coherence of this kind of animation. A new definition of bounding box allows us to reduce the number of pixels to be computed for the frames after the first. We associate a CSG subtree and two new flags, denoting if the box has changed in the current frame and if it will change in the next frame, with each box. We show with three examples the advantages of our technique when compared with an algorithm which entirely renders each frame of an animation. Intersections with CSG objects may be reduced to about one-fifth, while the rendering may be computed up to four times faster for the test sequences. © 1998 John Wiley & Sons, Ltd.     

A new space subdivision method for ray tracing CSG modelled scenes

A new algorithm for space tracing with CSG modelled scenes is presented. Space is divided in an irregular fashion to fit the objects as closely as possible. For this reason, primitive minimal bounding boxes are computed. Space subdivision is achieved in two steps: partitioning in projection plane and depth partitioning. A set of 3D regions named cells are then created. A Boolean CSG tree is distributed into the cell structure to form in each cell the minimal boolean CSG tree using the relevant primitives. The searching process for the next cell along the ray path is performed by using a local data structure associated with each cell. The goal of this structure is to link the cells together. An improvement, named mailbox, for all space tracing algorithms is detailed. Results are presented for two scenes to compare this new algorithm with Roth's algorithm.     

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.     

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.     

明暗效果快速多变的CSG实本显示技术

CSG(Constructive Solid Geometry)是实体造型中一种重要表示方法.传统的CSG实 体显示方法不仅费时,而且难以控制明暗效果.本文提出的方法以一种新的CSG体素表面光 强生成模型为基础,结台深度缓冲技术,能快速有效地显示CSG实体,方便灵活地控制明暗效 果.     

Domain Extension of Isothetic Polyhedra with Minimal CSG Representation

We consider the problem of converting boundary representations of isothetic polyhedra into constructive solid geometry (CSG) representations. The CSG representation is a boolean formula based on the half-spaces supporting the faces of the polyhedron. This boolean formula exhibits two important features: no term is complemented (it is monotone) and each supporting half-space appears in the formula once and only once. It is known that such formulas do not always exist for general polyhedra in the three-dimensional space. In this work first we give a procedure that extends the domain of polyhedra for which such a nice representation can be computed. Then we prove that not all cyclic isothetic polyhedra have a CSG representation of the style given above.     

Interactive vizualization of constructive solid geometry scenes on graphic processors

A ray-tracing algorithm for interactive visualization of very large and structurally complicated scenes presented in the constructive solid geometry (CSG) form is suggested. The algorithm is capable of visualizing such scenes in real time by using a graphic processor. As primitives, classical shapes and objects represented in an analytical form (in particular, second-order surfaces and implicit functions) are used. Unlike other similar algorithms, our algorithm produces the final image in a single pass and has no constraints on the maximum number of primitives and on the CSG tree depth. The key feature of the algorithm is a method for optimizing CSG models, which converts the input tree to an equivalent spatially coherent and well-balanced form (a completely balanced equivalent tree may not exist). The performance of visualization after applying the optimization technique is shown to depend on only the computational resource of the GPU (in contrast to multi-pass algorithms whose performance is restricted by memory capacity). It has been shown experimentally that our algorithm is capable of rendering CSG models consisting of more than a million CSG primitives with the tree depth up to 24.     

Efficient slicing procedure based on adaptive layer depth normal image

In layered modeling for rapid prototyping of products, compromising slicing accuracy and time is a critical issue. Based on adaptive Layer Depth Normal Image (LDNI), this paper proposes an efficient algorithm to achieve this compromise for complex Constructive Solid Geometry (CSG) models. First, each primitive at the tree leaf is converted into an adaptive LDNI solid whose Boolean operation can be performed efficiently. Then, a layered model is constructed directly from the Booleaned LDNI solid since it is actually a set of a layered and ordered point cloud. In addition to speed, efficient use of memory is also taken into account in design of the LDNI-based slicing algorithm. The capability and efficiency of this slicing algorithm are demonstrated by examples.     

基于图卷积网络的BREP→CSG转换方法及其应用研究

边界表示法(BREP)和构造实体表示法(CSG)是应用最广泛的2种实体表示法,在粒子输运计算辅助建模等领域对BREP→CSG自动转换算法有迫切的需求,但目前最常用的基于分割的BREP→CSG转换算法存在"计算量大、CSG表达过于复杂"等不足.观察到"拓扑相似的BREP模型的CSG表达结构类似",因此提出建立包含(BRE...     

Creation and boundary evaluation of CSG-models

This paper deals with one particular type of solid model representation. Constructive solid geometry (CSG) describes solids in terms of primitive shapes combined by Boolean operators. The goal of this paper is to present an algorithm suitable for the visualization of a given CSG-object for verification purposes. To accomplish this, the CSG representation is to be converted into a polygonal boundary representation (B-Rep), which allows the easy display of the solid's visible edges and surfaces. An algorithm performing this conversion is called a boundary evaluator. This paper describes the fundamental concepts leading to this new algorithm and presents some examples.The boundary evaluator described here follows the following steps: (1) A polygonal mesh is created that covers the primitive shapes. (2) Interfering polygons are repertedly subdivided until the resulting mesh contains the intersection curves between the primitives. (3) The location and validity of all polygons must be determined since only valid polygons reflect the solid's final shape. (4) Adjoining coplanar polygons are collected and merged by deleting their common edges. The final result of this procedure is a consistent, nonredundant polygon mesh describing the given solid model. Now the object may be displayed by simple perspective plots, by isometric projections, by hidden line drawings, or through the use of hidden surface removal techniques as color-shaded pictures. The algorithm presented here has been successfully used to create a variety of renderings of various CSG-models. The interpretation, display, and validation of computer-generated CSG models is essential for the successful use of such models in the engineering design process.     

Stewart平台机构仿真研究中的实体模型

本文根据构造的实体几何造型的思想建立了Stewart 平台机构的实体模型,并采用画家算法,结合Stewart平台机构的结构特点,提出了针对该 机构的消隐算法．另外,作为一种并联机器人机构,Stewart平台机构在运动过程中也较容 易出现干涉现象,本文对干涉情况也作了讨论．上述这些算法用于Stewart平台机构运动学 仿真软件的开发实践中,取得了较好的效果．     

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.     

三维布局中八叉树节点的快速分解算法

本文在对八叉树子节点的顶点类型及构成三维实体精确ＣＳＧ模型的半空间的类型进行系统地分类的基础上，结合不同类型半空间的性质给出了一个三维实体的精确ＣＳＧ模型转换到八叉树模型的综合算法。通过对若干个不同的实体进行实际计算，结果表明该算法能够满足三维实体自动化布局的需要。