![WEAZER0](https://upload.wikimedia.org/wikipedia/commons/6/64/WEAZER0.jpg)
In
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 from related areas of geometric modeling and computer graphi ...
and
computer-aided design, boundary representation (often abbreviated B-rep or BREP) is a method for representing a
3D shape
In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), i ...
by defining the limits of its
volume
Volume is a measure of occupied 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). Th ...
. A
solid
Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structural ...
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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
components (
faces
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
,
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 ...
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
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 ...
; 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
In computer graphics, the winged edge data structure is a way to represent polygon meshes in computer memory. It is a type of boundary representation and describes both the geometry and topology of a model. Three types of records are used: vert ...
links'' or ''half-edges'') which are used to create the edge circuits.
Vs 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-sectional profile by pushing material through a die of the desired cross-section. Its two main advantages over other manufacturing processes are its ability to create very complex ...
(or sweeping),
chamfer, 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 College town, university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cam ...
(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
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 products. ...
, 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
In 3D computer graphics and 3D modeling, modeling, volumetric meshes are a polygonal representation of the interior volume of an object. Unlike polygon meshes, which represent only the surface as polygons, volumetric meshes also discretize the in ...
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 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
Step(s) or STEP may refer to:
Common meanings
* Stairs#Step, Steps, making a staircase
* Walking
* Dance move
* Military step, or march
** Marching
Arts Films and television
* Steps (TV series), ''Steps'' (TV series), Hong Kong
* Step (film), ' ...
, 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 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
See also
*
B-spline
*
Bezier curve
*
Bezier surface
*
Combinatorial maps A combinatorial map is a combinatorial representation of a graph on an orientable surface. A combinatorial map may also be called a combinatorial embedding, a rotation system, an orientable ribbon graph, a fat graph, or a cyclic graph. More gener ...
*
Constructive solid geometry (CSG)
*
Coons surface
*
Function representation Function Representation (FRep or F-Rep) 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 ...
*
Geometric modeling kernel
A geometric modeling kernel is a solid modeling software component used in computer-aided design (CAD) packages. Available modelling kernels include:
* ACIS is developed and licensed by Spatial Corporation of Dassault Systèmes.
*SMLib is develop ...
*
NURBS
Non-uniform rational basis spline (NURBS) is a mathematical model using B-spline, basis splines (B-splines) that is commonly used in computer graphics for representing curves and Surface (mathematics), surfaces. It offers great flexibility and pr ...
*
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 from related areas of geometric modeling and computer graphi ...
*
Spline
*
Winged edge
In computer graphics, the winged edge data structure is a way to represent polygon meshes in computer memory. It is a type of boundary representation and describes both the geometry and topology of a model. Three types of records are used: vert ...
Further reading
Further information about Boundary Representation can be found in a variety of papers and the following books:
*
*
*
* {{Cite book , title=Geometric Modeling: The Mathematics of Shapes , first=Nikolay , last=Golovanov , isbn=978-1497473195 , year=2014 , publisher=
CreateSpace Independent Publishing Platform
On-Demand Publishing, LLC, doing business as CreateSpace, is a self-publishing service owned by Amazon. The company was founded in 2000 in South Carolina as BookSurge and was acquired by Amazon in 2005.
History
CreateSpace publishes books conta ...
External links
OpenCascade - an open source library for BRep solid modeling Actual example of source boundary representation in STEP formatSimpleGeo - a B-rep/CSG hybrid modeling system for Monte Carlo particle transport simulationsFreeREP - Opensource geometry kernel developmentBrep - A boundary representation based modeler
Computer 3D graphics