solid modeling Solid modeling (or solid modelling) is a consistent set of principles for mathematical and computer modeling of three-dimensional shapes '' (solids)''. Solid modeling is distinguished within the broader related areas of geometric modeling and ...

and

computer-aided design Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve c ...

, boundary representation (often abbreviated B-rep or BREP) is a method for representing a 3D shape by defining the limits of its

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

. A

solid Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...

is represented as a collection of connected

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

elements, which define the boundary between interior and exterior points.

Overview

A boundary representation of a model comprises

topological Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...

components ( faces,

edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...

s and vertices) and the connections between them, along with geometric definitions for those components (surfaces, curves and points, respectively). A face is a bounded portion of a

; an edge is a bounded piece of a curve and a vertex lies at a point. Other elements are the ''shell'' (a set of connected faces), the ''loop'' (a circuit of edges bounding a face) and ''loop-edge links'' (also known as '' winged edge links'' or ''half-edges'') which are used to create the edge circuits.

Compared to constructive solid geometry

Compared to the constructive solid geometry (CSG) representation, which uses only primitive objects and Boolean operations to combine them, boundary representation is more flexible and has a much richer operation set. In addition to the Boolean operations, B-rep has

extrusion Extrusion is a process used to create objects of a fixed cross section (geometry), cross-sectional profile by pushing material through a Die (manufacturing), die of the desired cross-section. Its two main advantages over other manufacturing pro ...

(or sweeping),

chamfer A chamfer ( ) is a transitional edge between two faces of an object. Sometimes defined as a form of bevel, it is often created at a 45° angle between two adjoining right-angled faces. Chamfers are frequently used in machining, carpentry, fur ...

, blending, drafting, shelling, tweaking and other operations which make use of these.

History

The basic method for BREP was developed independently in the early 1970s by both Ian C. Braid in

Cambridge Cambridge ( ) is a List of cities in the United Kingdom, city and non-metropolitan district in the county of Cambridgeshire, England. It is the county town of Cambridgeshire and is located on the River Cam, north of London. As of the 2021 Unit ...

(for CAD) and Bruce G. Baumgart at

Stanford Leland Stanford Junior University, commonly referred to as Stanford University, is a private research university in Stanford, California, United States. It was founded in 1885 by railroad magnate Leland Stanford (the eighth governor of and th ...

(for

computer vision Computer vision tasks include methods for image sensor, acquiring, Image processing, processing, Image analysis, analyzing, and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical ...

). Braid continued his work with the research solid modeller BUILD which was the forerunner of many research and commercial solid modelling systems. Braid worked on the commercial systems

ROMULUS Romulus (, ) was the legendary founder and first king of Rome. Various traditions attribute the establishment of many of Rome's oldest legal, political, religious, and social institutions to Romulus and his contemporaries. Although many of th ...

, the forerunner of

Parasolid Parasolid is a geometric modeling kernel originally developed by Shape Data Limited, now owned and developed by Siemens Digital Industries Software. It can be licensed by other companies for use in their 3D computer graphics software product ...

, and on

ACIS The 3D ACIS Modeler (ACIS) is a geometric modeling kernel developed by Spatial Corporation (formerly Spatial Technology), part of Dassault Systèmes. ACIS is used by software developers in industries such as computer-aided design, computer-ai ...

. Parasolid and ACIS are the basis for many of today's commercial CAD systems. Following Braid's work for solids, a Swedish team led by Professor Torsten Kjellberg, developed the philosophy and methods for working with hybrid models, wire-frames, sheet objects and volumetric models during the early 1980s. In Finland, Martti Mäntylä produced a solid modelling system called GWB. In the USA Eastman and Weiler were also working on Boundary Representation and in Japan Professor Fumihiko Kimura and his team at Tokyo University also produced their own B-rep modelling system. Initially CSG was used by several commercial systems because it was easier to implement. The advent of reliable commercial B-rep kernel systems like Parasolid and ACIS, mentioned above, as well as OpenCASCADE and C3D that were later developed, has led to widespread adoption of B-rep for CAD. Boundary representation is essentially a local representation connecting faces, edges and vertices. An extension of this was to group sub-elements of the shape into logical units called ''geometric features'', or simply ''features''. Pioneering work was done by Kyprianou in Cambridge also using the BUILD system and continued and extended by Jared and others. Features are the basis of many other developments, allowing high-level "geometric reasoning" about shape for comparison, process-planning, manufacturing, etc. Boundary representation has also been extended to allow special, non-solid model types called non-manifold models. As described by Braid, normal solids found in nature have the property that, at every point on the boundary, a small enough sphere around the point is divided into two pieces, one inside and one outside the object. Non-manifold models break this rule. An important sub-class of non-manifold models are sheet objects which are used to represent thin-plate objects and integrate surface modelling into a solid modelling environment.

Standardization

Standardization for boundary representation took time to develop. In a meeting organized by the Computer-Aided Manufacturing International (CAM-I) in 1979 the

IGES The Initial Graphics Exchange Specification (IGES) is a vendor-neutral List of file formats, file format that allows the CAD data exchange, digital exchange of information among computer-aided design (CAD) systems. It is an ASCII-based textual for ...

format was discussed for solid model transfer. IGES was not, then, suitable. Another complication was the coexistence of, then, two major representations, CSG and Boundary Representation, although use of CSG in commercial systems started to decline later. Further developments within CAM-I led to the Experimental Boundary Format, known as XBF, which was proposed to IGES as a possibility for extension to cover Boundary Representation models. However, this was not taken up. Towards the end of the 1980s a project called CAD*I developed a standard representation which then became one of the bases for the development of the STEP solid model format, the first widely accepted data exchange format for Boundary Representation. In the world of data-exchange, STEP, the Standard for the Exchange of Product Model data also defines some data models for boundary representations in a neutral form which can be mapped to specific data structures. The common generic topological and geometric models are defined in ISO 10303-42 Geometric and topological representation. The following Application Integrated Resources (AICs) define boundary models that are constraints of the generic geometric and topological capabilities: * ISO 10303-511 ''Topologically bounded surface'', definition of an advanced face, that is a bounded surface where the surface is of type elementary (plane, cylindrical, conical, spherical or toroidal), or a swept surface, or

B-spline In numerical analysis, a B-spline (short for basis spline) is a type of Spline (mathematics), spline function designed to have minimal Support (mathematics), support (overlap) for a given Degree of a polynomial, degree, smoothness, and set of bre ...

surface. The boundaries are defined by lines, conics, polylines, surface curves, or b spline curves * ISO 10303-514 ''Advanced boundary representation'', a solid defining a volume with possible voids that is composed by advanced faces * ISO 10303-509 ''Manifold surface'', a non intersecting area in 3D that is composed by advanced faces * ISO 10303-521 ''Manifold subsurface'', a sub-area out of a manifold surface * ISO 10303-508 ''Non-manifold surface'', any kind of advanced face arrangement * ISO 10303-513 ''Elementary boundary representation'' similar to ISO 10303-514, but restricted to the elementary surfaces only * ISO 10303-512 ''Faceted boundary representation'' a simplified surface model constructed by planar surfaces only

References

External links

OpenCascade - an open source library for BRep solid modeling

Actual example of source boundary representation in STEP format

SimpleGeo - a B-rep/CSG hybrid modeling system for Monte Carlo particle transport simulations

FreeREP - Opensource geometry kernel development

Brep - A boundary representation based modeler
3D computer graphics Computer-related introductions in the 1970s