Constructive Solid Geometry Representation Research Articles

A Survey of Methods for Converting Unstructured Data to CSG Models

The goal of this document is to survey existing methods for recovering or extracting CSG (Constructive Solid Geometry) representations from unstructured data such as 3D point-clouds or polygon meshes. We review and discuss related topics such as the segmentation and fitting of the input data. We cover techniques from solid modeling for the conversion of a polyhedron to a CSG expression and for the conversion of a B-rep to a CSG expression. We look at approaches coming from program synthesis, evolutionary techniques (such as genetic programming or genetic algorithm), and deep learning. Finally, we conclude our survey with a discussion of techniques for the generation of computer programs involving higher-level constructs, representations, and operations for representing solids.

Hardware-Accelerated Ray Tracing of CAD-Based Geometry for Monte Carlo Radiation Transport

Monte Carlo radiation transport (MCRT) methods have been used to simulate radiation environments for many decades by tracking individual particles through a model to accumulate statistical information. MCRT geometry is historically formed using the constructive solid geometry (CSG). Recently, significant work has been performed to support simulations using computer-aided design (CAD)-based tessellated surfaces to support highly complex geometries. Ray tracing acceleration data structures from the rendering and visualization community are applied to accelerate particle tracking in CAD-based models. Despite these efforts, CSG representations provide the superior performance in surface intersection operations during particle flight. Concurrently, pseudo Monte Carlo methods have become prevalent in rendering applications to support more realistic models for scattering media, motivating innovations that are advantageous for MCRT simulations. The authors’ work extends these innovations by employing Intel’s Embree ray tracing kernel within a geometry toolkit for Monte Carlo to improve the simulation performance using CAD-based models by factors of 1.5 to 2.

Robust event-driven particle tracking in complex geometries

Particle tracking, that is, the repeated localization of particles within a grid by means of tracking the particles’ trajectories, is routinely applied in particle-based schemes where the domain is described by an unstructured polyhedral grid. A range of tracking algorithms are available in the literature, which are inherently similar to algorithmic approaches common both in event-driven particle dynamics (EDPD) and ray-tracing methods. We propose a reformulation of existing particle tracking algorithms in the context of EDPD. On the one hand, this resolves inconsistencies in the mapping between particle positions and grid cells triggered, e.g., by imperfect grids. More importantly, it allows the specification of solid objects via constructive solid geometry (CSG), a standard technique for the modeling of solids in computer-aided design. While usually considered contrary approaches, our description of the computational domain as the combination of a bounding volume defined by an unstructured grid and solids modeled via CSG embedded into this volume can be highly advantageous. The two different approaches of modeling the computational domain complement each other perfectly, as the CSG representation is not only efficient in terms of memory and computing time, but also avoids the challenges of generating finely resolved unstructured grids in the presence of complicated boundaries. These benefits, as well as the positive impact of several algorithmic optimizations of the extended tracking algorithm, are exemplified via a particle-based simulation of a gas flow through a highly porous medium.

Structural topology optimization using multi-objective genetic algorithm with constructive solid geometry representation

This paper presents a constructive solid geometry based representation scheme for structural topology optimization. The proposed scheme encodes the topology using position of few joints and width of segments connecting them. Union of overlapping rectangular primitives is calculated using constructive solid geometry technique to obtain the topology. A valid topology in the design domain is ensured by representing the topology as a connected simple graph of nodes. A graph repair operator is applied to ensure a physically meaningful connected structure. The algorithm is integrated with single and multi-objective genetic algorithm and its performance is compared with those of other methods like SIMP. The multi-objective analysis provides the trade-off front between compliance and material availability, unveiling common design principles among optimized solutions. The proposed method is generic and can be easily extended to any two or three-dimensional topology optimization problem by using different shape primitives.

Simulation of a robot machining system based on heterogeneous-resolution representation

ABSTRACTCollision avoidance is a frequently encountered problem in machining processes, especially in robot-based machining. In a robot machining system, collision may occur between the robot arm, the tool, tool holder and the work piece together with its fixtures. Therefore, a precise collision detection algorithm is critical to ensure that the tool path is collision free, particularly in the area where the tool has contact with the work piece. To verify tool paths, a simulation system is developed. In the proposed system, the robotic system is modeled as a Constructive Solid Geometry (CSG) tree where the Denavit-Hartenberg (D-H) notation is applied to represent the robot arm transformation matrix. A combined CSG representation and STereoLithography (STL) representation, the so called heterogeneous-resolution method is proposed to represent the work piece. By heterogeneous-resolution, the area of the work piece near the current machining contact point is represented by triangular facets at a controlled a...

OpenMC: A state-of-the-art Monte Carlo code for research and development

This paper gives an overview of OpenMC, an open source Monte Carlo particle transport code recently developed at the Massachusetts Institute of Technology. OpenMC uses continuous-energy cross sections and a constructive solid geometry representation, enabling high-fidelity modeling of nuclear reactors and other systems. Modern, portable input/output file formats are used in OpenMC: XML for input, and HDF5 for output. High performance parallel algorithms in OpenMC have demonstrated near-linear scaling to over 100,000 processors on modern supercomputers. Other topics discussed in this paper include plotting, CMFD acceleration, variance reduction, eigenvalue calculations, and software development processes.

Reconstructing the World’s Museums

Virtual exploration tools for large indoor environments (e.g. museums) have so far been limited to either blueprint-style 2D maps that lack photo-realistic views of scenes, or ground-level image-to-image transitions, which are immersive but ill-suited for navigation. On the other hand, photorealistic aerial maps would be a useful navigational guide for large indoor environments, but it is impossible to directly acquire photographs covering a large indoor environment from aerial viewpoints. This paper presents a 3D reconstruction and visualization system for automatically producing clean and well-regularized texture-mapped 3D models for large indoor scenes, from ground-level photographs and 3D laser points. The key component is a new algorithm called “inverse constructive solid geometry (CSG)” for reconstructing a scene with a CSG representation consisting of volumetric primitives, which imposes powerful regularization constraints. We also propose several novel techniques to adjust the 3D model to make it suitable for rendering the 3D maps from aerial viewpoints. The visualization system enables users to easily browse a large-scale indoor environment from a bird’s-eye view, locate specific room interiors, fly into a place of interest, view immersive ground-level panorama views, and zoom out again, all with seamless 3D transitions. We demonstrate our system on various museums, including the Metropolitan Museum of Art in New York City—one of the largest art galleries in the world.

Reasoning about Solids Using Constraint Logic Programming

The embedding of constraint satisfaction on the domain of discourse into a rule-based programming paradigm like logic programming provides a powerful reasoning tool. We present an application in spatial reasoning that uses this combination to produce a clear, concise, yet very expressive system through its ability to manipulate partial information. Three-dimensional solid objects in constructive solid geometry representation are manipulated, and their spatial relationship with one another, points, or regions is reasoned about. The language used to develop this application is QUAD-CLP(ℜ), an experimental constraint logic programming language of our own design, which is equipped with a solver for quadratic and linear arithmetic constraints over the reals.

Computing occlusion-free viewpoints

This paper presents methods to compute the locus of all viewpoints from which features on known polyhedral objects can be viewed in their entirety without being occluded by anything in the environment. Convex and concave polyhedra with or without holes and the viewing model of perspective projection are employed in this work. Initially, properties of the occlusion-free and occluded loci of viewpoints are determined. Based on these properties, two methods to construct these loci together with their complexity analysis are presented. In the first method, a boundary representation of the occlusion-free locus is obtained. In the second method, the locus of occluded viewpoints is expressed in terms of a constructive solid geometry representation that consists of a union of component solids. Implementation results and comparison of the two methods are given.

Automatic construction of CSG solids from a single isometric drawing

This paper presents an algorithm for constructing a 3D workpiece in CSG (constructive solid geometry) representation from a single 2D isometric drawing. Currently, the concerned workpieces are limited to simple rectilinear polyhedra where any two faces or edges are either parallel or perpendicular to each other, however, the shape of constructed workpiece can further be modified by the application of wireframe-based approach of variational geometry. Compared to previous relevant work, the proposed algorithm is distinguished by its capability to deal with degenerate 2D line drawings; that is, problems where two distinct edges are projected on to one line can be solved.

Domain Extension of Isothetic Polyhedra with Minimal CSG Representation

AbstractWe 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.

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.

Three-dimensional shape optimization using fully automatic mesh generation

Introduction T HE technique for associating design variables with mesh data is the most crucial factor in three-dimensional shape optimization. Previously, work in three-dimensional shape optimization involved specifying design variables by associating parameters directly with grid points on an existing mesh. For realistic problems this can be a very tedious (and errorprone) process. In the past, shape optimization capabilities have been developed based on a variety of design/analysis capabilities ranging from associating parameters with a mesh created manually to associating parameters with control points of a mapped mesh generator. Special techniques have also been developed to properly move internal grid points during sensitivity calculations. More recently, a capability based on constructive solid geometry (CSG) has been developed, but CSG representations are not particularly suitable for design optimization. A new optimization approach is demonstrated in this Note that uses a new, fully automatic mesh generation capability. The design model is developed based on design-oriented geometric primitives that represent recognizable features of a part and can be assembled into complete solid models that are defined in terms of a small set of design parameters.

Fast display of solid objects

With the advance of hypermedia systems there is a need for high performance workstations to allow the engineer to manipulate and display solid models at interactive speeds. This paper deals with the development of the model structure and describes an algorithm which projects solid geometry onto the monitor. This algorithm will display both a boundary representation and a constructive solid geometry representation of the solid on the screen. Image display will be fast, since the subimages are calculated simultaneously.

Efficient CSG Representations of Two-Dimensional Solids

Good methods are known for converting a Constructive Solid Geometry (CSG) representation of a solid into a boundary representation (b-rep) of the solid, but not for performing the inverse conversion, b-rep→CSG, which is the subject of this paper. Important applications of b-rep→CSG conversion arise in solid modeling, image processing, and elsewhere. The problem can be divided into two tasks: (1) finding a set of halfspaces that is necessary and sufficient (but not unique) to represent a given solid, and (2) constructing an efficient CSG representation using those halfspaces. This paper solves the problem for curved planar solids, i.e., r-sets in E2, with or without holes, whose boundary is given by a collection of edges. The edges may be subsets of straight lines or convex curves (i.e., curves which intersect any line in at most two points). We prove a number of results and describe algorithms that have been fully implemented for solids bounded by line segments and circular arcs. Empirical results show that the computed CSG representations are superior to those produced by earlier algorithms, and produce superior three-dimensional CSG representations for mechanical parts defined by contour sweeping. A companion paper generalizes the results to higher dimensional solids.

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.

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 E 2, In E 3, an experimental system currently converts natural-quadric B-reps in PARASOLID to efficient CSG representations in PADL-2.

End Milling Force Algorithms for CAD Systems

This paper presents an efficient milling mechanics simulation system for use with solid modeller based tool path generation algorithms. The technique provides computationally efficient cutting force predictions for end milling of general part shapes. First, the part is described using the Constructive Solid Geometry representation scheme. Along the cutter path the tool-workpiece interaction is represented using immersion are segments obtained by intersecting the cutter with each part feature. Next, a cutting mechanics model is used to predict the instantaneous, average and peak forces, and maximum cutter deflections left on the finished part surface. By separating the cutter constants from the part geometry, the model allows automatic feed rate scheduling to satisfy end milling constraints. Verification of the method by both simulation and experiment is included.

Semantic CSG trees for finite element analysis

Solid modelling plays an important role in CAD applications, especially in finite element analysis. Constructive solid geometry (CSG) is one of the most popular methods used for representing solids. It preserves the three-dimensional structure of the object, and also provides a hierarchical representation which is useful in representing complex objects. Yet the representation is only syntactic. Finite element analysis needs additional information about the objects, such as topological and physical properties. This paper incorporates these properties in a CSG representation of solids and provides semantic constraints for manipulating these properties. Such a CSG tree is called a semantic CSG tree. Preserving the semantic information at the nodes of a CSG tree also guides the designer to reuse the analysis of the substructures of that tree.

Feature reasoning for automatic robotic assembly and machining in polyhedral representation†

SUMMARY An automatic planning system for robotic assembly must have knowledge of the workspace. In particular it must be capable of reasoning about the shape of objects within the workspace. This requires a knowledge of object features and their behaviour in assembly situations. In this paper, a system is presented which extracts features relevant to assembly and machining from a boundary model derived from a Constructive Solid Geometry representation of a complex shape. In particular, finding concavities plays an important role in identifying possible mating features for assembly. These relevant features can then be used for assembly planning based on finding compatible sets of mating features. In this work the CAD data is extracted from a boundary model represented in an ASCII file and converted into the data-structures of the AI language, Prolog.