Boundary representation
Encyclopedia
In solid modeling
and computer-aided design
, boundary representation—often abbreviated as B-rep or BREP—is a method for representing shapes using the limits. A solid is represented as a collection of connected surface elements, the boundary between solid and non-solid.
and geometry (surfaces, curves and points). The main topological items are: faces
, edge
s and vertices
. A face is a bounded portion of a surface
; 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. The edges are like the edges of a table, bounding a surface portion.
(for CAD) and Bruce G. Baumgart at Stanford (for computer vision
). 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
, the forerunner of Parasolid
, and on ACIS
. 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, Marti Mäntylä produced a solid modelling system called GWB. In the USA Eastman and Weiler were also working on Boundary Representation and in Japan Professor Kimura and his team at Tokyo University also produced their own B-rep modelling system.
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. This makes boundary representation a more appropriate choice for CAD systems. CSG was used initially 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, has led to widespread adoption of B-rep for CAD. As well as the Boolean operations, B-rep has extrusion
(or sweeping), chamfer, blending, drafting, shelling, tweaking and other operations which make use of these.
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.
, the Standard for the Exchange of Product Model data also defines some data models for boundary representations. 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:
Martti Mäntylä: "An Introduction to Solid Modeling", Computer Science Press, ISBN 0-88175-108-1, 1988.
H. Chiyokura: "Solid Modelling with DESIGNBASE", Addison-Wesley Publishing Company, ISBN 0-201-19245-4, 1988.
Ian Stroud: "Boundary Representation Modelling Techniques, Springer, ISBN 1-84628-312-4, 2006
Each edge is defined by vertices. Each vertex is defined by a point.
Orientation of a face: The orientation of a face can be represented in one of the two ways, associating a normal with it or associating an order in the list of edges that define the face.
Solid modeling
Solid modeling is a consistent set of principles for mathematical and computer modeling of three dimensional solids. Solid modeling is distinguished from related areas of Geometric modeling and Computer graphics by its emphasis on physical fidelity...
and computer-aided design
Computer-aided design
Computer-aided design , also known as computer-aided design and drafting , is the use of computer technology for the process of design and design-documentation. Computer Aided Drafting describes the process of drafting with a computer...
, boundary representation—often abbreviated as B-rep or BREP—is a method for representing shapes using the limits. A solid is represented as a collection of connected surface elements, the boundary between solid and non-solid.
Overview
Boundary representation models are composed of two parts: topologyTopology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and geometry (surfaces, curves and points). The main topological items are: faces
Face (geometry)
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
, edge
Edge (geometry)
In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
s and vertices
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
. A face is a bounded portion of a surface
Computer representation of surfaces
In technical applications of 3D computer graphics such as computer-aided design and computer-aided manufacturing, surfaces are one way of representing objects. The other ways are wireframe and solids...
; 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
Winged edge
The winged edge data structure is a data representation used to describe polygon models in computer graphics. It explicitly describes the geometry and topology of faces, edges, and vertices when three or more surfaces come together and meet at a common edge...
links or half-edges) which are used to create the edge circuits. The edges are like the edges of a table, bounding a surface portion.
History
The basic method for BREP was developed independently in the early 1970s by both Ian C. Braid in CambridgeUniversity of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
(for CAD) and Bruce G. Baumgart at Stanford (for computer vision
Computer vision
Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...
). 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 (b-rep solid modeler)
The Romulus b-rep solid modeler was released in 1982 by Ian Braid, Charles Lang and the Shape Data team in Cambridge, England. It was the first commercial solid modeling kernel designed for straightforward integration into CAD software...
, the forerunner of Parasolid
Parasolid
Parasolid is a geometric modeling kernel originally developed by ShapeData, now owned by Siemens PLM Software , that can be licensed by other companies for use in their 3D computer graphics software products....
, and on ACIS
ACIS
The 3D ACIS Modeler is a 3D modelling kernel owned by Spatial Corporation . ACIS is used by many software developers in industries such as computer-aided design , Computer-aided manufacturing , Computer-aided engineering , Architecture, engineering and construction , Coordinate-measuring machine...
. 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, Marti Mäntylä produced a solid modelling system called GWB. In the USA Eastman and Weiler were also working on Boundary Representation and in Japan Professor Kimura and his team at Tokyo University also produced their own B-rep modelling system.
Compared to the constructive solid geometry
Constructive solid geometry
Constructive solid geometry is a technique used in solid modeling. Constructive solid geometry allows a modeler to create a complex surface or object by using Boolean operators to combine objects...
(CSG) representation, which uses only primitive objects and Boolean operations
Boolean logic
Boolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of...
to combine them, boundary representation is more flexible and has a much richer operation set. This makes boundary representation a more appropriate choice for CAD systems. CSG was used initially 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, has led to widespread adoption of B-rep for CAD. As well as the Boolean operations, B-rep has extrusion
Extrusion
Extrusion is a process used to create objects of a fixed cross-sectional profile. A material is pushed or drawn through a die of the desired cross-section...
(or sweeping), chamfer, blending, drafting, shelling, tweaking and other operations which make use of these.
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
In the world of data-exchange, STEPISO 10303
ISO 10303 is an ISO standard for the computer-interpretable representation and exchange of product manufacturing information. Its official title is: Automation systems and integration — Product data representation and exchange...
, the Standard for the Exchange of Product Model data also defines some data models for boundary representations. 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 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
Further reading
Further information about Boundary Representation can be found in a variety of papers and the following books:Martti Mäntylä: "An Introduction to Solid Modeling", Computer Science Press, ISBN 0-88175-108-1, 1988.
H. Chiyokura: "Solid Modelling with DESIGNBASE", Addison-Wesley Publishing Company, ISBN 0-201-19245-4, 1988.
Ian Stroud: "Boundary Representation Modelling Techniques, Springer, ISBN 1-84628-312-4, 2006
Introduction
Let us consider BREP of a cube. It will be represented by six faces. Those faces are oriented. Each consist of four edges in a specific order. Each edge is defined by two vertices. Each vertex is defined by points.Details
In general a solid will consist of faces. A face will be defined by loops(closed curve). Each loop will be defined by one or more edges.Each edge is defined by vertices. Each vertex is defined by a point.
Orientation of a face: The orientation of a face can be represented in one of the two ways, associating a normal with it or associating an order in the list of edges that define the face.
See also
- Solid modelingSolid modelingSolid modeling is a consistent set of principles for mathematical and computer modeling of three dimensional solids. Solid modeling is distinguished from related areas of Geometric modeling and Computer graphics by its emphasis on physical fidelity...
- Constructive solid geometryConstructive solid geometryConstructive solid geometry is a technique used in solid modeling. Constructive solid geometry allows a modeler to create a complex surface or object by using Boolean operators to combine objects...
(CSG) - Winged edgeWinged edgeThe winged edge data structure is a data representation used to describe polygon models in computer graphics. It explicitly describes the geometry and topology of faces, edges, and vertices when three or more surfaces come together and meet at a common edge...
- Function representationFunction representationFunction Representation is used in solid modeling, volume modeling and computer graphics. FRep was introduced in "Function representation in geometric modeling: concepts, implementation and applications" as a uniform representation of multidimensional geometric objects...
- Combinatorial maps
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
- SMLib - A commercial solid modeling library supporting non-manifold breps, with source code available for licensees
- FreeREP - Opensource geometry kernel development
- Brep - A boundary representation based modeler