<s>
In	O
solid	O
modeling	O
and	O
computer-aided	B-Application
design	I-Application
,	O
boundary	B-Algorithm
representation	I-Algorithm
(	O
often	O
abbreviated	O
B-rep	B-Algorithm
or	O
BREP	B-Algorithm
)	O
is	O
a	O
method	O
for	O
representing	O
a	O
3D	O
shape	O
by	O
defining	O
the	O
limits	O
of	O
its	O
volume	O
.	O
</s>
<s>
A	O
boundary	B-Algorithm
representation	I-Algorithm
of	O
a	O
model	O
comprises	O
topological	B-Architecture
components	O
(	O
faces	O
,	O
edges	O
and	O
vertices	O
)	O
and	O
the	O
connections	O
between	O
them	O
,	O
along	O
with	O
geometric	O
definitions	O
for	O
those	O
components	O
(	O
surfaces	O
,	O
curves	O
and	O
points	O
,	O
respectively	O
)	O
.	O
</s>
<s>
Other	O
elements	O
are	O
the	O
shell	O
(	O
a	O
set	O
of	O
connected	O
faces	O
)	O
,	O
the	O
loop	O
(	O
a	O
circuit	O
of	O
edges	O
bounding	O
a	O
face	O
)	O
and	O
loop-edge	O
links	O
(	O
also	O
known	O
as	O
winged	B-Algorithm
edge	I-Algorithm
links	O
or	O
half-edges	O
)	O
which	O
are	O
used	O
to	O
create	O
the	O
edge	O
circuits	O
.	O
</s>
<s>
Compared	O
to	O
the	O
constructive	O
solid	O
geometry	O
(	O
CSG	O
)	O
representation	O
,	O
which	O
uses	O
only	O
primitive	O
objects	O
and	O
Boolean	O
operations	O
to	O
combine	O
them	O
,	O
boundary	B-Algorithm
representation	I-Algorithm
is	O
more	O
flexible	O
and	O
has	O
a	O
much	O
richer	O
operation	O
set	O
.	O
</s>
<s>
In	O
addition	O
to	O
the	O
Boolean	O
operations	O
,	O
B-rep	B-Algorithm
has	O
extrusion	O
(	O
or	O
sweeping	O
)	O
,	O
chamfer	O
,	O
blending	O
,	O
drafting	O
,	O
shelling	O
,	O
tweaking	O
and	O
other	O
operations	O
which	O
make	O
use	O
of	O
these	O
.	O
</s>
<s>
The	O
basic	O
method	O
for	O
BREP	B-Algorithm
was	O
developed	O
independently	O
in	O
the	O
early	O
1970s	O
by	O
both	O
Ian	O
C	O
.	O
Braid	O
in	O
Cambridge	O
(	O
for	O
CAD	B-Application
)	O
and	O
Bruce	O
G	O
.	O
Baumgart	O
at	O
Stanford	O
(	O
for	O
computer	B-Application
vision	I-Application
)	O
.	O
</s>
<s>
Braid	O
worked	O
on	O
the	O
commercial	O
systems	O
ROMULUS	B-Application
,	O
the	O
forerunner	O
of	O
Parasolid	B-Application
,	O
and	O
on	O
ACIS	O
.	O
</s>
<s>
Parasolid	B-Application
and	O
ACIS	O
are	O
the	O
basis	O
for	O
many	O
of	O
today	O
's	O
commercial	O
CAD	B-Application
systems	O
.	O
</s>
<s>
Following	O
Braid	O
's	O
work	O
for	O
solids	O
,	O
a	O
Swedish	O
team	O
led	O
by	O
Professor	O
Torsten	O
Kjellberg	O
,	O
developed	O
the	O
philosophy	O
and	O
methods	O
for	O
working	O
with	O
hybrid	O
models	O
,	O
wire-frames	O
,	O
sheet	O
objects	O
and	O
volumetric	B-Algorithm
models	I-Algorithm
during	O
the	O
early	O
1980s	O
.	O
</s>
<s>
In	O
the	O
USA	O
Eastman	O
and	O
Weiler	O
were	O
also	O
working	O
on	O
Boundary	B-Algorithm
Representation	I-Algorithm
and	O
in	O
Japan	O
Professor	O
Fumihiko	O
Kimura	O
and	O
his	O
team	O
at	O
Tokyo	O
University	O
also	O
produced	O
their	O
own	O
B-rep	B-Algorithm
modelling	O
system	O
.	O
</s>
<s>
The	O
advent	O
of	O
reliable	O
commercial	O
B-rep	B-Algorithm
kernel	O
systems	O
like	O
Parasolid	B-Application
and	O
ACIS	O
,	O
mentioned	O
above	O
,	O
as	O
well	O
as	O
OpenCASCADE	O
and	O
C3D	B-Language
that	O
were	O
later	O
developed	O
,	O
has	O
led	O
to	O
widespread	O
adoption	O
of	O
B-rep	B-Algorithm
for	O
CAD	B-Application
.	O
</s>
<s>
Boundary	B-Algorithm
representation	I-Algorithm
is	O
essentially	O
a	O
local	O
representation	O
connecting	O
faces	O
,	O
edges	O
and	O
vertices	O
.	O
</s>
<s>
Boundary	B-Algorithm
representation	I-Algorithm
has	O
also	O
been	O
extended	O
to	O
allow	O
special	O
,	O
non-solid	O
model	O
types	O
called	O
non-manifold	B-Algorithm
models	I-Algorithm
.	O
</s>
<s>
Non-manifold	B-Algorithm
models	I-Algorithm
break	O
this	O
rule	O
.	O
</s>
<s>
An	O
important	O
sub-class	O
of	O
non-manifold	B-Algorithm
models	I-Algorithm
are	O
sheet	O
objects	O
which	O
are	O
used	O
to	O
represent	O
thin-plate	O
objects	O
and	O
integrate	O
surface	O
modelling	O
into	O
a	O
solid	O
modelling	O
environment	O
.	O
</s>
<s>
Standardization	O
for	O
boundary	B-Algorithm
representation	I-Algorithm
took	O
time	O
to	O
develop	O
.	O
</s>
<s>
Another	O
complication	O
was	O
the	O
coexistence	O
of	O
,	O
then	O
,	O
two	O
major	O
representations	O
,	O
CSG	O
and	O
Boundary	B-Algorithm
Representation	I-Algorithm
,	O
although	O
use	O
of	O
CSG	O
in	O
commercial	O
systems	O
started	O
to	O
decline	O
later	O
.	O
</s>
<s>
Further	O
developments	O
within	O
CAM-I	O
led	O
to	O
the	O
Experimental	O
Boundary	O
Format	O
,	O
known	O
as	O
XBF	O
,	O
which	O
was	O
proposed	O
to	O
IGES	O
as	O
a	O
possibility	O
for	O
extension	O
to	O
cover	O
Boundary	B-Algorithm
Representation	I-Algorithm
models	O
.	O
</s>
<s>
Towards	O
the	O
end	O
of	O
the	O
1980s	O
a	O
project	O
called	O
CAD*I	O
developed	O
a	O
standard	O
representation	O
which	O
then	O
became	O
one	O
of	O
the	O
bases	O
for	O
the	O
development	O
of	O
the	O
STEP	O
solid	O
model	O
format	O
,	O
the	O
first	O
widely	O
accepted	O
data	O
exchange	O
format	O
for	O
Boundary	B-Algorithm
Representation	I-Algorithm
.	O
</s>
<s>
In	O
the	O
world	O
of	O
data-exchange	O
,	O
STEP	O
,	O
the	O
Standard	O
for	O
the	O
Exchange	O
of	O
Product	O
Model	O
data	O
also	O
defines	O
some	O
data	O
models	O
for	O
boundary	B-Algorithm
representations	I-Algorithm
in	O
a	O
neutral	O
form	O
which	O
can	O
be	O
mapped	O
to	O
specific	O
data	O
structures	O
.	O
</s>
<s>
The	O
common	O
generic	O
topological	B-Architecture
and	O
geometric	O
models	O
are	O
defined	O
in	O
ISO	O
10303-42	O
Geometric	O
and	O
topological	B-Architecture
representation	O
.	O
</s>
<s>
The	O
following	O
Application	O
Integrated	O
Resources	O
(	O
AICs	O
)	O
define	O
boundary	O
models	O
that	O
are	O
constraints	O
of	O
the	O
generic	O
geometric	O
and	O
topological	B-Architecture
capabilities	O
:	O
</s>
<s>
ISO	O
10303-511	O
Topologically	O
bounded	O
surface	O
,	O
definition	O
of	O
an	O
advanced	O
face	O
,	O
that	O
is	O
a	O
bounded	O
surface	O
where	O
the	O
surface	O
is	O
of	O
type	O
elementary	O
(	O
plane	O
,	O
cylindrical	O
,	O
conical	O
,	O
spherical	O
or	O
toroidal	O
)	O
,	O
or	O
a	O
swept	O
surface	O
,	O
or	O
b	B-Algorithm
spline	I-Algorithm
surface	O
.	O
</s>
