<s>
In	O
3D	O
computer	O
graphics	O
and	O
solid	O
modeling	O
,	O
a	O
polygon	B-Algorithm
mesh	I-Algorithm
is	O
a	O
collection	O
of	O
,	O
s	O
and	O
s	O
that	O
defines	O
the	O
shape	O
of	O
a	O
polyhedral	O
object	O
.	O
</s>
<s>
The	O
faces	O
usually	O
consist	O
of	O
triangles	O
(	O
triangle	B-Algorithm
mesh	I-Algorithm
)	O
,	O
quadrilaterals	O
(	O
quads	O
)	O
,	O
or	O
other	O
simple	O
convex	O
polygons	B-General_Concept
(	O
n-gons	O
)	O
,	O
since	O
this	O
simplifies	O
rendering	O
,	O
but	O
may	O
also	O
be	O
more	O
generally	O
composed	O
of	O
concave	O
polygons	B-General_Concept
,	O
or	O
even	O
polygons	B-General_Concept
with	O
holes	O
.	O
</s>
<s>
The	O
study	O
of	O
polygon	B-Algorithm
meshes	I-Algorithm
is	O
a	O
large	O
sub-field	O
of	O
computer	O
graphics	O
(	O
specifically	O
3D	O
computer	O
graphics	O
)	O
and	O
geometric	B-Application
modeling	I-Application
.	O
</s>
<s>
Different	O
representations	O
of	O
polygon	B-Algorithm
meshes	I-Algorithm
are	O
used	O
for	O
different	O
applications	O
and	O
goals	O
.	O
</s>
<s>
The	O
variety	O
of	O
operations	O
performed	O
on	O
meshes	B-Algorithm
may	O
include	O
:	O
Boolean	O
logic	O
(	O
Constructive	O
solid	O
geometry	O
)	O
,	O
smoothing	O
,	O
simplification	O
,	O
and	O
many	O
others	O
.	O
</s>
<s>
Algorithms	O
also	O
exist	O
for	O
ray	B-Algorithm
tracing	I-Algorithm
,	O
collision	O
detection	O
,	O
and	O
rigid-body	O
dynamics	O
with	O
polygon	B-Algorithm
meshes	I-Algorithm
.	O
</s>
<s>
If	O
the	O
mesh	B-Algorithm
's	O
edges	O
are	O
rendered	O
instead	O
of	O
the	O
faces	O
,	O
then	O
the	O
model	O
becomes	O
a	O
wireframe	B-Algorithm
model	I-Algorithm
.	O
</s>
<s>
Volumetric	B-Algorithm
meshes	I-Algorithm
are	O
distinct	O
from	O
polygon	B-Algorithm
meshes	I-Algorithm
in	O
that	O
they	O
explicitly	O
represent	O
both	O
the	O
surface	O
and	O
volume	O
of	O
a	O
structure	O
,	O
while	O
polygon	B-Algorithm
meshes	I-Algorithm
only	O
explicitly	O
represent	O
the	O
surface	O
(	O
the	O
volume	O
is	O
implicit	O
)	O
.	O
</s>
<s>
Several	O
methods	O
exist	O
for	O
mesh	B-Architecture
generation	I-Architecture
,	O
including	O
the	O
marching	B-Algorithm
cubes	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Objects	O
created	O
with	O
polygon	B-Algorithm
meshes	I-Algorithm
must	O
store	O
different	O
types	O
of	O
elements	O
.	O
</s>
<s>
These	O
include	O
vertices	O
,	O
edges	O
,	O
faces	O
,	O
polygons	B-General_Concept
and	O
surfaces	O
.	O
</s>
<s>
In	O
many	O
applications	O
,	O
only	O
vertices	O
,	O
edges	O
and	O
either	O
faces	O
or	O
polygons	B-General_Concept
are	O
stored	O
.	O
</s>
<s>
A	O
renderer	O
may	O
support	O
only	O
3-sided	O
faces	O
,	O
so	O
polygons	B-General_Concept
must	O
be	O
constructed	O
of	O
many	O
of	O
these	O
,	O
as	O
shown	O
above	O
.	O
</s>
<s>
However	O
,	O
many	O
renderers	O
either	O
support	O
quads	O
and	O
higher-sided	O
polygons	B-General_Concept
,	O
or	O
are	O
able	O
to	O
convert	O
polygons	B-General_Concept
to	O
triangles	O
on	O
the	O
fly	O
,	O
making	O
it	O
unnecessary	O
to	O
store	O
a	O
mesh	B-Algorithm
in	O
a	O
triangulated	B-Algorithm
form	O
.	O
</s>
<s>
Polygon	B-Algorithm
meshes	I-Algorithm
may	O
be	O
represented	O
in	O
a	O
variety	O
of	O
ways	O
,	O
using	O
different	O
methods	O
to	O
store	O
the	O
vertex	O
,	O
edge	O
and	O
face	O
data	O
.	O
</s>
<s>
For	O
example	O
,	O
it	O
is	O
easier	O
to	O
deal	O
with	O
triangles	O
than	O
general	O
polygons	B-General_Concept
,	O
especially	O
in	O
computational	O
geometry	O
.	O
</s>
<s>
For	O
certain	O
operations	O
it	O
is	O
necessary	O
to	O
have	O
a	O
fast	O
access	O
to	O
topological	O
information	O
such	O
as	O
edges	O
or	O
neighboring	O
faces	O
;	O
this	O
requires	O
more	O
complex	O
structures	O
such	O
as	O
the	O
winged-edge	B-Algorithm
representation	O
.	O
</s>
<s>
For	O
hardware	O
rendering	O
,	O
compact	O
,	O
simple	O
structures	O
are	O
needed	O
;	O
thus	O
the	O
corner-table	O
(	O
triangle	B-General_Concept
fan	I-General_Concept
)	O
is	O
commonly	O
incorporated	O
into	O
low-level	O
rendering	O
APIs	O
such	O
as	O
DirectX	B-Application
and	O
OpenGL	B-Application
.	O
</s>
<s>
Vertex-vertex	O
meshes	B-Algorithm
represent	O
an	O
object	O
as	O
a	O
set	O
of	O
vertices	O
connected	O
to	O
other	O
vertices	O
.	O
</s>
<s>
However	O
,	O
VV	O
meshes	B-Algorithm
benefit	O
from	O
small	O
storage	O
space	O
and	O
efficient	O
morphing	O
of	O
shape	O
.	O
</s>
<s>
The	O
above	O
figure	O
shows	O
a	O
four-sided	O
box	O
as	O
represented	O
by	O
a	O
VV	O
mesh	B-Algorithm
.	O
</s>
<s>
For	O
a	O
complete	O
description	O
of	O
VV	O
meshes	B-Algorithm
see	O
Smith	O
(	O
2006	O
)	O
.	O
</s>
<s>
Face-vertex	O
meshes	B-Algorithm
represent	O
an	O
object	O
as	O
a	O
set	O
of	O
faces	O
and	O
a	O
set	O
of	O
vertices	O
.	O
</s>
<s>
This	O
is	O
the	O
most	O
widely	O
used	O
mesh	B-Algorithm
representation	O
,	O
being	O
the	O
input	O
typically	O
accepted	O
by	O
modern	O
graphics	O
hardware	O
.	O
</s>
<s>
Face-vertex	O
meshes	B-Algorithm
improve	O
on	O
VV-mesh	O
for	O
modeling	O
in	O
that	O
they	O
allow	O
explicit	O
lookup	O
of	O
the	O
vertices	O
of	O
a	O
face	O
,	O
and	O
the	O
faces	O
surrounding	O
a	O
vertex	O
.	O
</s>
<s>
The	O
above	O
figure	O
shows	O
the	O
"	O
box-cylinder	O
"	O
example	O
as	O
an	O
FV	O
mesh	B-Algorithm
.	O
</s>
<s>
With	O
face-vertex	O
meshes	B-Algorithm
it	O
is	O
easy	O
to	O
find	O
the	O
vertices	O
of	O
a	O
face	O
.	O
</s>
<s>
Unlike	O
VV	O
meshes	B-Algorithm
,	O
both	O
faces	O
and	O
vertices	O
are	O
explicit	O
,	O
so	O
locating	O
neighboring	O
faces	O
and	O
vertices	O
is	O
constant	O
time	O
.	O
</s>
<s>
Other	O
dynamic	O
operations	O
,	O
such	O
as	O
splitting	O
or	O
merging	O
a	O
face	O
,	O
are	O
also	O
difficult	O
with	O
face-vertex	O
meshes	B-Algorithm
.	O
</s>
<s>
Introduced	O
by	O
Baumgart	O
in	O
1975	O
,	O
winged-edge	B-Algorithm
meshes	B-Algorithm
explicitly	O
represent	O
the	O
vertices	O
,	O
faces	O
,	O
and	O
edges	O
of	O
a	O
mesh	B-Algorithm
.	O
</s>
<s>
This	O
representation	O
is	O
widely	O
used	O
in	O
modeling	O
programs	O
to	O
provide	O
the	O
greatest	O
flexibility	O
in	O
dynamically	O
changing	O
the	O
mesh	B-Algorithm
geometry	O
,	O
because	O
split	O
and	O
merge	O
operations	O
can	O
be	O
done	O
quickly	O
.	O
</s>
<s>
A	O
good	O
discussion	O
of	O
implementation	O
issues	O
of	O
Winged-edge	B-Algorithm
meshes	B-Algorithm
may	O
be	O
found	O
in	O
the	O
book	O
Graphics	O
Gems	O
II	O
.	O
</s>
<s>
Winged-edge	B-Algorithm
meshes	B-Algorithm
address	O
the	O
issue	O
of	O
traversing	O
from	O
edge	O
to	O
edge	O
,	O
and	O
providing	O
an	O
ordered	O
set	O
of	O
faces	O
around	O
an	O
edge	O
.	O
</s>
<s>
To	O
simplify	O
this	O
,	O
winged-edge	B-Algorithm
meshes	B-Algorithm
provide	O
only	O
four	O
,	O
the	O
nearest	O
clockwise	O
and	O
counter-clockwise	O
edges	O
at	O
each	O
end	O
.	O
</s>
<s>
The	O
information	O
for	O
each	O
edge	O
therefore	O
resembles	O
a	O
butterfly	O
,	O
hence	O
"	O
winged-edge	B-Algorithm
"	O
meshes	B-Algorithm
.	O
</s>
<s>
The	O
above	O
figure	O
shows	O
the	O
"	O
box-cylinder	O
"	O
as	O
a	O
winged-edge	B-Algorithm
mesh	B-Algorithm
.	O
</s>
<s>
The	O
total	O
data	O
for	O
an	O
edge	O
consists	O
of	O
2	O
vertices	O
(	O
endpoints	O
)	O
,	O
2	O
faces	O
(	O
on	O
each	O
side	O
)	O
,	O
and	O
4	O
edges	O
(	O
winged-edge	B-Algorithm
)	O
.	O
</s>
<s>
Rendering	O
of	O
winged-edge	B-Algorithm
meshes	B-Algorithm
for	O
graphics	O
hardware	O
requires	O
generating	O
a	O
Face	O
index	O
list	O
.	O
</s>
<s>
Winged-edge	B-Algorithm
meshes	B-Algorithm
are	O
ideally	O
suited	O
for	O
dynamic	O
geometry	O
,	O
such	O
as	O
subdivision	O
surfaces	O
and	O
interactive	O
modeling	O
,	O
since	O
changes	O
to	O
the	O
mesh	B-Algorithm
can	O
occur	O
locally	O
.	O
</s>
<s>
Traversal	O
across	O
the	O
mesh	B-Algorithm
,	O
as	O
might	O
be	O
needed	O
for	O
collision	O
detection	O
,	O
can	O
be	O
accomplished	O
efficiently	O
.	O
</s>
<s>
Winged-edge	B-Algorithm
meshes	B-Algorithm
are	O
not	O
the	O
only	O
representation	O
which	O
allows	O
for	O
dynamic	O
changes	O
to	O
geometry	O
.	O
</s>
<s>
A	O
new	O
representation	O
which	O
combines	O
winged-edge	B-Algorithm
meshes	B-Algorithm
and	O
face-vertex	O
meshes	B-Algorithm
is	O
the	O
render	O
dynamic	O
mesh	B-Algorithm
,	O
which	O
explicitly	O
stores	O
both	O
,	O
the	O
vertices	O
of	O
a	O
face	O
and	O
faces	O
of	O
a	O
vertex	O
(	O
like	O
FV	O
meshes	B-Algorithm
)	O
,	O
and	O
the	O
faces	O
and	O
vertices	O
of	O
an	O
edge	O
(	O
like	O
winged-edge	B-Algorithm
)	O
.	O
</s>
<s>
Render	O
dynamic	O
meshes	B-Algorithm
require	O
slightly	O
less	O
storage	O
space	O
than	O
standard	O
winged-edge	B-Algorithm
meshes	B-Algorithm
,	O
and	O
can	O
be	O
directly	O
rendered	O
by	O
graphics	O
hardware	O
since	O
the	O
face	O
list	O
contains	O
an	O
index	O
of	O
vertices	O
.	O
</s>
<s>
RD	O
meshes	B-Algorithm
do	O
not	O
require	O
the	O
four	O
outgoing	O
edges	O
since	O
these	O
can	O
be	O
found	O
by	O
traversing	O
from	O
edge	O
to	O
face	O
,	O
then	O
face	O
to	O
neighboring	O
edge	O
.	O
</s>
<s>
RD	O
meshes	B-Algorithm
benefit	O
from	O
the	O
features	O
of	O
winged-edge	B-Algorithm
meshes	B-Algorithm
by	O
allowing	O
for	O
geometry	O
to	O
be	O
dynamically	O
updated	O
.	O
</s>
<s>
For	O
example	O
,	O
to	O
get	O
"	O
all	O
vertices	O
around	O
a	O
given	O
vertex	O
V	O
"	O
using	O
the	O
face-vertex	O
mesh	B-Algorithm
,	O
it	O
is	O
necessary	O
to	O
first	O
find	O
the	O
faces	O
around	O
the	O
given	O
vertex	O
V	O
using	O
the	O
vertex	O
list	O
.	O
</s>
<s>
Winged-edge	B-Algorithm
meshes	B-Algorithm
explicitly	O
store	O
nearly	O
all	O
information	O
,	O
and	O
other	O
operations	O
always	O
traverse	O
to	O
the	O
edge	O
first	O
to	O
get	O
additional	O
info	O
.	O
</s>
<s>
Vertex-vertex	O
meshes	B-Algorithm
are	O
the	O
only	O
representation	O
that	O
explicitly	O
stores	O
the	O
neighboring	O
vertices	O
of	O
a	O
given	O
vertex	O
.	O
</s>
<s>
As	O
the	O
mesh	B-Algorithm
representations	O
become	O
more	O
complex	O
(	O
from	O
left	O
to	O
right	O
in	O
the	O
summary	O
)	O
,	O
the	O
amount	O
of	O
information	O
explicitly	O
stored	O
increases	O
.	O
</s>
<s>
As	O
a	O
general	O
rule	O
,	O
face-vertex	O
meshes	B-Algorithm
are	O
used	O
whenever	O
an	O
object	O
must	O
be	O
rendered	O
on	O
graphics	O
hardware	O
that	O
does	O
not	O
change	O
geometry	O
(	O
connectivity	O
)	O
,	O
but	O
may	O
deform	O
or	O
morph	O
shape	O
(	O
vertex	O
positions	O
)	O
such	O
as	O
real-time	B-General_Concept
rendering	I-General_Concept
of	O
static	O
or	O
morphing	O
objects	O
.	O
</s>
<s>
Winged-edge	B-Algorithm
or	O
render	O
dynamic	O
meshes	B-Algorithm
are	O
used	O
when	O
the	O
geometry	O
changes	O
,	O
such	O
as	O
in	O
interactive	O
modeling	O
packages	O
or	O
for	O
computing	O
subdivision	O
surfaces	O
.	O
</s>
<s>
Vertex-vertex	O
meshes	B-Algorithm
are	O
ideal	O
for	O
efficient	O
,	O
complex	O
changes	O
in	O
geometry	O
or	O
topology	O
so	O
long	O
as	O
hardware	O
rendering	O
is	O
not	O
of	O
concern	O
.	O
</s>
<s>
There	O
exist	O
many	O
different	O
file	O
formats	O
for	O
storing	O
polygon	B-Algorithm
mesh	I-Algorithm
data	O
.	O
</s>
<s>
File	O
suffix	O
Format	O
name	O
Organization(s )	O
Program(s )	O
Description	O
.raw	O
Raw	O
mesh	B-Algorithm
Unknown	O
Various	O
Open	O
,	O
ASCII-only	O
format	O
.	O
</s>
<s>
Each	O
line	O
contains	O
3	O
vertices	O
,	O
separated	O
by	O
spaces	O
,	O
to	O
form	O
a	O
triangle	O
,	O
like	O
so	O
:	O
X1	O
Y1	O
Z1	O
X2	O
Y2	O
Z2	O
X3	O
Y3	O
Z3	O
.blend	O
Blender	B-Application
File	O
Format	O
Blender	B-License
Foundation	I-License
Blender	B-Application
3D	I-Application
Open	O
source	O
,	O
binary-only	O
format	O
.fbx	B-Application
Autodesk	O
FBX	B-Application
Format	O
Autodesk	O
Various	O
Proprietary	O
.	O
</s>
<s>
.3ds	O
3ds	B-Language
Max	I-Language
File	O
Autodesk	O
3ds	B-Language
Max	I-Language
A	O
common	O
but	O
outdated	O
format	O
with	O
hard	O
16-bit	O
limits	O
on	O
the	O
number	O
of	O
vertices	O
and	O
faces	O
.	O
</s>
<s>
.dae	O
Digital	O
Asset	O
Exchange	O
(	O
COLLADA	O
)	O
Sony	O
Computer	O
Entertainment	O
,	O
Khronos	B-Library
Group	I-Library
N/A	O
Stands	O
for	O
"	O
COLLAborative	O
Design	O
Activity	O
"	O
.	O
</s>
<s>
.dgn	B-Algorithm
MicroStation	B-Algorithm
File	O
Bentley	O
Systems	O
MicroStation	B-Algorithm
There	O
are	O
two	O
dgn	O
file	O
formats	O
:	O
pre-version	O
8	O
and	O
version	O
8	O
(	O
V8	O
)	O
.3dm	B-Algorithm
Rhino	O
File	O
Robert	B-Algorithm
McNeel	I-Algorithm
&	I-Algorithm
Associates	I-Algorithm
Rhinoceros	B-Algorithm
3D	I-Algorithm
.dxf	O
,	O
.dwg	O
Drawing	O
Exchange	O
Format	O
Autodesk	O
AutoCAD	B-Algorithm
.obj	O
Wavefront	O
OBJ	O
Wavefront	B-Application
Technologies	I-Application
Various	O
ASCII	O
format	O
describing	O
3D	O
geometry	O
.	O
</s>
<s>
.ply	O
Polygon	B-General_Concept
File	O
Format	O
Stanford	O
University	O
Various	O
Binary	O
and	O
ASCII	O
.pmd	B-Application
Polygon	B-General_Concept
Movie	O
Maker	O
data	O
Yu	O
Higuchi	O
MikuMikuDance	B-Application
Proprietary	O
binary	O
file	O
format	O
for	O
storing	O
humanoid	O
model	O
geometry	O
with	O
rigging	O
,	O
material	O
,	O
and	O
physics	O
information	O
.	O
</s>
<s>
.wrl	B-Application
Virtual	B-Application
Reality	I-Application
Modeling	I-Application
Language	I-Application
Web3D	O
Consortium	O
Web	O
Browsers	O
ISO	O
Standard	O
14772-1:1997	O
.wrz	B-Application
VRML	B-Application
Compressed	O
Web3D	O
Consortium	O
Web	O
Browsers	O
.x3d	B-Application
,	I-Application
.x3db	I-Application
,	I-Application
.x3dv	I-Application
Extensible	B-Application
3D	I-Application
Web3D	O
Consortium	O
Web	O
Browsers	O
XML-based	O
,	O
open	O
source	O
,	O
royalty-free	O
,	O
extensible	O
,	O
and	O
interoperable	O
;	O
also	O
supports	O
color	O
,	O
texture	O
,	O
and	O
scene	O
information	O
.	O
</s>
<s>
ISO	O
Standard	O
19775/19776/19777	O
.x3dz	B-Application
,	I-Application
.x3dbz	I-Application
,	I-Application
.x3dvz	I-Application
X3D	B-Application
Compressed	O
Binary	O
Web3D	O
Consortium	O
Web	O
Browsers	O
.c4d	B-Device
Cinema	B-Device
4D	I-Device
File	O
Maxon	O
CINEMA	B-Device
4D	I-Device
.lwo	B-Device
LightWave	B-Device
3D	I-Device
object	O
File	O
NewTek	O
LightWave	B-Device
3D	I-Device
.smbSCOREC	O
apfRPI	O
SCORECPUMIOpen	O
source	O
parallel	O
adaptive	O
unstructured	O
3D	B-Algorithm
meshes	I-Algorithm
for	O
PDE	O
based	O
simulation	O
workflows	O
.	O
</s>
<s>
.msh	O
Gmsh	B-Language
Mesh	B-Algorithm
GMsh	B-Language
Developers	O
GMsh	B-Language
Project	I-Language
Open	O
source	O
,	O
providing	O
an	O
ASCII	O
mesh	B-Algorithm
description	O
for	O
linear	O
and	O
polynomially	O
interpolated	O
elements	O
in	O
1	O
to	O
3	O
dimensions	O
.	O
</s>
<s>
.mesh	O
OGRE	O
XML	O
OGRE	O
Development	O
Team	O
OGRE	O
,	O
purebasic	O
Open	O
Source	O
.	O
</s>
<s>
Binary	O
(	O
.mesh	O
)	O
and	O
ASCII	O
(	O
.mesh.xml	O
)	O
format	O
available	O
.	O
</s>
<s>
.veg	O
Vega	O
FEM	O
tetrahedral	O
mesh	B-Algorithm
Jernej	O
Barbič	O
Vega	O
FEM	O
Open	O
Source	O
.	O
</s>
<s>
Stores	O
a	O
tetrahedral	O
mesh	B-Algorithm
and	O
its	O
material	O
properties	O
for	O
FEM	O
simulation	O
.	O
</s>
<s>
.z3d	O
Z3d	O
Oleg	O
Melashenko	O
Zanoza	O
Modeler	O
-	O
.vtk	B-Language
VTK	B-Language
mesh	B-Algorithm
VTK	B-Language
,	O
Kitware	O
VTK	B-Language
,	O
Paraview	B-Language
Open	O
,	O
ASCII	O
or	O
binary	O
format	O
that	O
contains	O
many	O
different	O
data	O
fields	O
,	O
including	O
point	O
data	O
,	O
cell	O
data	O
,	O
and	O
field	O
data	O
.	O
</s>
