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Triangle 1.6 (A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator)
Triangle
A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator.
Version 1.6
Copyright 1993, 1995, 1997, 1998, 2002, 2005 Jonathan Richard Shewchuk
2360 Woolsey #H / Berkeley, California 94705-1927
Bugs/comments to jrs@cs.berkeley.edu
Created as part of the Quake project (tools for earthquake simulation).
Supported in part by NSF Grant CMS-9318163 and an NSERC 1967 Scholarship.
There is no warranty whatsoever. Use at your own risk.
This executable is compiled for double precision arithmetic.
Triangle generates exact Delaunay triangulations, constrained Delaunay
triangulations, conforming Delaunay triangulations, Voronoi diagrams, and
high-quality triangular meshes. The latter can be generated with no small
or large angles, and are thus suitable for finite element analysis. If no
command line switch is specified, your .node input file is read, and the
Delaunay triangulation is returned in .node and .ele output files. The
command syntax is:
triangle [-prq__a__uAcDjevngBPNEIOXzo_YS__iFlsCQVh] input_file
Underscores indicate that numbers may optionally follow certain switches.
Do not leave any space between a switch and its numeric parameter.
input_file must be a file with extension .node, or extension .poly if the
-p switch is used. If -r is used, you must supply .node and .ele files,
and possibly a .poly file and an .area file as well. The formats of these
files are described below.
Command Line Switches:
-p Reads a Planar Straight Line Graph (.poly file), which can specify
vertices, segments, holes, regional attributes, and regional area
constraints. Generates a constrained Delaunay triangulation (CDT)
fitting the input; or, if -s, -q, -a, or -u is used, a conforming
constrained Delaunay triangulation (CCDT). If you want a truly
Delaunay (not just constrained Delaunay) triangulation, use -D as
well. When -p is not used, Triangle reads a .node file by default.
-r Refines a previously generated mesh. The mesh is read from a .node
file and an .ele file. If -p is also used, a .poly file is read
and used to constrain segments in the mesh. If -a is also used
(with no number following), an .area file is read and used to
impose area constraints on the mesh. Further details on refinement
appear below.
-q Quality mesh generation by Delaunay refinement (a hybrid of Paul
Chew‘s and Jim Ruppert‘s algorithms). Adds vertices to the mesh to
ensure that all angles are between 20 and 140 degrees. An
alternative bound on the minimum angle, replacing 20 degrees, may
be specified after the `q‘. The specified angle may include a
decimal point, but not exponential notation. Note that a bound of
theta degrees on the smallest angle also implies a bound of
(180 - 2 theta) on the largest angle. If the minimum angle is 28.6
degrees or smaller, Triangle is mathematically guaranteed to
terminate (assuming infinite precision arithmetic--Triangle may
fail to terminate if you run out of precision). In practice,
Triangle often succeeds for minimum angles up to 34 degrees. For
some meshes, however, you might need to reduce the minimum angle to
avoid problems associated with insufficient floating-point
precision.
-a Imposes a maximum triangle area. If a number follows the `a‘, no
triangle is generated whose area is larger than that number. If no
number is specified, an .area file (if -r is used) or .poly file
(if -r is not used) specifies a set of maximum area constraints.
An .area file contains a separate area constraint for each
triangle, and is useful for refining a finite element mesh based on
a posteriori error estimates. A .poly file can optionally contain
an area constraint for each segment-bounded region, thereby
controlling triangle densities in a first triangulation of a PSLG.
You can impose both a fixed area constraint and a varying area
constraint by invoking the -a switch twice, once with and once
without a number following. Each area specified may include a
decimal point.
-u Imposes a user-defined constraint on triangle size. There are two
ways to use this feature. One is to edit the triunsuitable()
procedure in triangle.c to encode any constraint you like, then
recompile Triangle. The other is to compile triangle.c with the
EXTERNAL_TEST symbol set (compiler switch -DEXTERNAL_TEST), then
link Triangle with a separate object file that implements
triunsuitable(). In either case, the -u switch causes the user-
defined test to be applied to every triangle.
-A Assigns an additional floating-point attribute to each triangle
that identifies what segment-bounded region each triangle belongs
to. Attributes are assigned to regions by the .poly file. If a
region is not explicitly marked by the .poly file, triangles in
that region are assigned an attribute of zero. The -A switch has
an effect only when the -p switch is used and the -r switch is not.
-c Creates segments on the convex hull of the triangulation. If you
are triangulating a vertex set, this switch causes a .poly file to
be written, containing all edges of the convex hull. If you are
triangulating a PSLG, this switch specifies that the whole convex
hull of the PSLG should be triangulated, regardless of what
segments the PSLG has. If you do not use this switch when
triangulating a PSLG, Triangle assumes that you have identified the
region to be triangulated by surrounding it with segments of the
input PSLG. Beware: if you are not careful, this switch can cause
the introduction of an extremely thin angle between a PSLG segment
and a convex hull segment, which can cause overrefinement (and
possibly failure if Triangle runs out of precision). If you are
refining a mesh, the -c switch works differently: it causes a
.poly file to be written containing the boundary edges of the mesh
(useful if no .poly file was read).
-D Conforming Delaunay triangulation: use this switch if you want to
ensure that all the triangles in the mesh are Delaunay, and not
merely constrained Delaunay; or if you want to ensure that all the
Voronoi vertices lie within the triangulation. (Some finite volume
methods have this requirement.) This switch invokes Ruppert‘s
original algorithm, which splits every subsegment whose diametral
circle is encroached. It usually increases the number of vertices
and triangles.
-j Jettisons vertices that are not part of the final triangulation
from the output .node file. By default, Triangle copies all
vertices in the input .node file to the output .node file, in the
same order, so their indices do not change. The -j switch prevents
duplicated input vertices, or vertices `eaten‘ by holes, from
appearing in the output .node file. Thus, if two input vertices
have exactly the same coordinates, only the first appears in the
output. If any vertices are jettisoned, the vertex numbering in
the output .node file differs from that of the input .node file.
-e Outputs (to an .edge file) a list of edges of the triangulation.
-v Outputs the Voronoi diagram associated with the triangulation.
Does not attempt to detect degeneracies, so some Voronoi vertices
may be duplicated. See the discussion of Voronoi diagrams below.
-n Outputs (to a .neigh file) a list of triangles neighboring each
triangle.
-g Outputs the mesh to an Object File Format (.off) file, suitable for
viewing with the Geometry Center‘s Geomview package.
-B No boundary markers in the output .node, .poly, and .edge output
files. See the detailed discussion of boundary markers below.
-P No output .poly file. Saves disk space, but you lose the ability
to maintain constraining segments on later refinements of the mesh.
-N No output .node file.
-E No output .ele file.
-I No iteration numbers. Suppresses the output of .node and .poly
files, so your input files won‘t be overwritten. (If your input is
a .poly file only, a .node file is written.) Cannot be used with
the -r switch, because that would overwrite your input .ele file.
Shouldn‘t be used with the -q, -a, -u, or -s switch if you are
using a .node file for input, because no .node file is written, so
there is no record of any added Steiner points.
-O No holes. Ignores the holes in the .poly file.
-X No exact arithmetic. Normally, Triangle uses exact floating-point
arithmetic for certain tests if it thinks the inexact tests are not
accurate enough. Exact arithmetic ensures the robustness of the
triangulation algorithms, despite floating-point roundoff error.
Disabling exact arithmetic with the -X switch causes a small
improvement in speed and creates the possibility that Triangle will
fail to produce a valid mesh. Not recommended.
-z Numbers all items starting from zero (rather than one). Note that
this switch is normally overridden by the value used to number the
first vertex of the input .node or .poly file. However, this
switch is useful when calling Triangle from another program.
-o2 Generates second-order subparametric elements with six nodes each.
-Y No new vertices on the boundary. This switch is useful when the
mesh boundary must be preserved so that it conforms to some
adjacent mesh. Be forewarned that you will probably sacrifice much
of the quality of the mesh; Triangle will try, but the resulting
mesh may contain poorly shaped triangles. Works well if all the
boundary vertices are closely spaced. Specify this switch twice
(`-YY‘) to prevent all segment splitting, including internal
boundaries.
-S Specifies the maximum number of Steiner points (vertices that are
not in the input, but are added to meet the constraints on minimum
angle and maximum area). The default is to allow an unlimited
number. If you specify this switch with no number after it,
the limit is set to zero. Triangle always adds vertices at segment
intersections, even if it needs to use more vertices than the limit
you set. When Triangle inserts segments by splitting (-s), it
always adds enough vertices to ensure that all the segments of the
PLSG are recovered, ignoring the limit if necessary.
-i Uses an incremental rather than a divide-and-conquer algorithm to
construct a Delaunay triangulation. Try it if the divide-and-
conquer algorithm fails.
-F Uses Steven Fortune‘s sweepline algorithm to construct a Delaunay
triangulation. Warning: does not use exact arithmetic for all
calculations. An exact result is not guaranteed.
-l Uses only vertical cuts in the divide-and-conquer algorithm. By
default, Triangle alternates between vertical and horizontal cuts,
which usually improve the speed except with vertex sets that are
small or short and wide. This switch is primarily of theoretical
interest.
-s Specifies that segments should be forced into the triangulation by
recursively splitting them at their midpoints, rather than by
generating a constrained Delaunay triangulation. Segment splitting
is true to Ruppert‘s original algorithm, but can create needlessly
small triangles. This switch is primarily of theoretical interest.
-C Check the consistency of the final mesh. Uses exact arithmetic for
checking, even if the -X switch is used. Useful if you suspect
Triangle is buggy.
-Q Quiet: Suppresses all explanation of what Triangle is doing,
unless an error occurs.
-V Verbose: Gives detailed information about what Triangle is doing.
Add more `V‘s for increasing amount of detail. `-V‘ is most
useful; itgives information on algorithmic progress and much more
detailed statistics. `-VV‘ gives vertex-by-vertex details, and
prints so much that Triangle runs much more slowly. `-VVVV‘ gives
information only a debugger could love.
-h Help: Displays these instructions.
Definitions:
A Delaunay triangulation of a vertex set is a triangulation whose
vertices are the vertex set, that covers the convex hull of the vertex
set. A Delaunay triangulation has the property that no vertex lies
inside the circumscribing circle (circle that passes through all three
vertices) of any triangle in the triangulation.
A Voronoi diagram of a vertex set is a subdivision of the plane into
polygonal cells (some of which may be unbounded, meaning infinitely
large), where each cell is the set of points in the plane that are closer
to some input vertex than to any other input vertex. The Voronoi diagram
is a geometric dual of the Delaunay triangulation.
A Planar Straight Line Graph (PSLG) is a set of vertices and segments.
Segments are simply edges, whose endpoints are all vertices in the PSLG.
Segments may intersect each other only at their endpoints. The file
format for PSLGs (.poly files) is described below.
A constrained Delaunay triangulation (CDT) of a PSLG is similar to a
Delaunay triangulation, but each PSLG segment is present as a single edge
of the CDT. (A constrained Delaunay triangulation is not truly a
Delaunay triangulation, because some of its triangles might not be
Delaunay.) By definition, a CDT does not have any vertices other than
those specified in the input PSLG. Depending on context, a CDT might
cover the convex hull of the PSLG, or it might cover only a segment-
bounded region (e.g. a polygon).
A conforming Delaunay triangulation of a PSLG is a triangulation in which
each triangle is truly Delaunay, and each PSLG segment is represented by
a linear contiguous sequence of edges of the triangulation. New vertices
(not part of the PSLG) may appear, and each input segment may have been
subdivided into shorter edges (subsegments) by these additional vertices.
The new vertices are frequently necessary to maintain the Delaunay
property while ensuring that every segment is represented.
A conforming constrained Delaunay triangulation (CCDT) of a PSLG is a
triangulation of a PSLG whose triangles are constrained Delaunay. New
vertices may appear, and input segments may be subdivided into
subsegments, but not to guarantee that segments are respected; rather, to
improve the quality of the triangles. The high-quality meshes produced
by the -q switch are usually CCDTs, but can be made conforming Delaunay
with the -D switch.
File Formats:
All files may contain comments prefixed by the character ‘#‘. Vertices,
triangles, edges, holes, and maximum area constraints must be numbered
consecutively, starting from either 1 or 0. Whichever you choose, all
input files must be consistent; if the vertices are numbered from 1, so
must be all other objects. Triangle automatically detects your choice
while reading the .node (or .poly) file. (When calling Triangle from
another program, use the -z switch if you wish to number objects from
zero.) Examples of these file formats are given below.
.node files:
First line: <# of vertices> <dimension (must be 2)> <# of attributes>
<# of boundary markers (0 or 1)>
Remaining lines: <vertex #> <x> <y> [attributes] [boundary marker]
The attributes, which are typically floating-point values of physical
quantities (such as mass or conductivity) associated with the nodes of
a finite element mesh, are copied unchanged to the output mesh. If -q,
-a, -u, -D, or -s is selected, each new Steiner point added to the mesh
has attributes assigned to it by linear interpolation.
If the fourth entry of the first line is `1‘, the last column of the
remainder of the file is assumed to contain boundary markers. Boundary
markers are used to identify boundary vertices and vertices resting on
PSLG segments; a complete description appears in a section below. The
.node file produced by Triangle contains boundary markers in the last
column unless they are suppressed by the -B switch.
.ele files:
First line: <# of triangles> <nodes per triangle> <# of attributes>
Remaining lines: <triangle #> <node> <node> <node> ... [attributes]
Nodes are indices into the corresponding .node file. The first three
nodes are the corner vertices, and are listed in counterclockwise order
around each triangle. (The remaining nodes, if any, depend on the type
of finite element used.)
The attributes are just like those of .node files. Because there is no
simple mapping from input to output triangles, Triangle attempts to
interpolate attributes, and may cause a lot of diffusion of attributes
among nearby triangles as the triangulation is refined. Attributes do
not diffuse across segments, so attributes used to identify
segment-bounded regions remain intact.
In .ele files produced by Triangle, each triangular element has three
nodes (vertices) unless the -o2 switch is used, in which case
subparametric quadratic elements with six nodes each are generated.
The first three nodes are the corners in counterclockwise order, and
the fourth, fifth, and sixth nodes lie on the midpoints of the edges
opposite the first, second, and third vertices, respectively.
.poly files:
First line: <# of vertices> <dimension (must be 2)> <# of attributes>
<# of boundary markers (0 or 1)>
Following lines: <vertex #> <x> <y> [attributes] [boundary marker]
One line: <# of segments> <# of boundary markers (0 or 1)>
Following lines: <segment #> <endpoint> <endpoint> [boundary marker]
One line: <# of holes>
Following lines: <hole #> <x> <y>
Optional line: <# of regional attributes and/or area constraints>
Optional following lines: <region #> <x> <y> <attribute> <max area>
A .poly file represents a PSLG, as well as some additional information.
The first section lists all the vertices, and is identical to the
format of .node files. <# of vertices> may be set to zero to indicate
that the vertices are listed in a separate .node file; .poly files
produced by Triangle always have this format. A vertex set represented
this way has the advantage that it may easily be triangulated with or
without segments (depending on whether the -p switch is invoked).
The second section lists the segments. Segments are edges whose
presence in the triangulation is enforced. (Depending on the choice of
switches, segment might be subdivided into smaller edges). Each
segment is specified by listing the indices of its two endpoints. This
means that you must include its endpoints in the vertex list. Each
segment, like each point, may have a boundary marker.
If -q, -a, -u, and -s are not selected, Triangle produces a constrained
Delaunay triangulation (CDT), in which each segment appears as a single
edge in the triangulation. If -q, -a, -u, or -s is selected, Triangle
produces a conforming constrained Delaunay triangulation (CCDT), in
which segments may be subdivided into smaller edges. If -D is
selected, Triangle produces a conforming Delaunay triangulation, so
that every triangle is Delaunay, and not just constrained Delaunay.
The third section lists holes (and concavities, if -c is selected) in
the triangulation. Holes are specified by identifying a point inside
each hole. After the triangulation is formed, Triangle creates holes
by eating triangles, spreading out from each hole point until its
progress is blocked by segments in the PSLG. You must be careful to
enclose each hole in segments, or your whole triangulation might be
eaten away. If the two triangles abutting a segment are eaten, the
segment itself is also eaten. Do not place a hole directly on a
segment; if you do, Triangle chooses one side of the segment
arbitrarily.
The optional fourth section lists regional attributes (to be assigned
to all triangles in a region) and regional constraints on the maximum
triangle area. Triangle reads this section only if the -A switch is
used or the -a switch is used without a number following it, and the -r
switch is not used. Regional attributes and area constraints are
propagated in the same manner as holes: you specify a point for each
attribute and/or constraint, and the attribute and/or constraint
affects the whole region (bounded by segments) containing the point.
If two values are written on a line after the x and y coordinate, the
first such value is assumed to be a regional attribute (but is only
applied if the -A switch is selected), and the second value is assumed
to be a regional area constraint (but is only applied if the -a switch
is selected). You may specify just one value after the coordinates,
which can serve as both an attribute and an area constraint, depending
on the choice of switches. If you are using the -A and -a switches
simultaneously and wish to assign an attribute to some region without
imposing an area constraint, use a negative maximum area.
When a triangulation is created from a .poly file, you must either
enclose the entire region to be triangulated in PSLG segments, or
use the -c switch, which automatically creates extra segments that
enclose the convex hull of the PSLG. If you do not use the -c switch,
Triangle eats all triangles that are not enclosed by segments; if you
are not careful, your whole triangulation may be eaten away. If you do
use the -c switch, you can still produce concavities by the appropriate
placement of holes just inside the boundary of the convex hull.
An ideal PSLG has no intersecting segments, nor any vertices that lie
upon segments (except, of course, the endpoints of each segment). You
aren‘t required to make your .poly files ideal, but you should be aware
of what can go wrong. Segment intersections are relatively safe--
Triangle calculates the intersection points for you and adds them to
the triangulation--as long as your machine‘s floating-point precision
doesn‘t become a problem. You are tempting the fates if you have three
segments that cross at the same location, and expect Triangle to figure
out where the intersection point is. Thanks to floating-point roundoff
error, Triangle will probably decide that the three segments intersect
at three different points, and you will find a minuscule triangle in
your output--unless Triangle tries to refine the tiny triangle, uses
up the last bit of machine precision, and fails to terminate at all.
You‘re better off putting the intersection point in the input files,
and manually breaking up each segment into two. Similarly, if you
place a vertex at the middle of a segment, and hope that Triangle will
break up the segment at that vertex, you might get lucky. On the other
hand, Triangle might decide that the vertex doesn‘t lie precisely on
the segment, and you‘ll have a needle-sharp triangle in your output--or
a lot of tiny triangles if you‘re generating a quality mesh.
When Triangle reads a .poly file, it also writes a .poly file, which
includes all the subsegments--the edges that are parts of input
segments. If the -c switch is used, the output .poly file also
includes all of the edges on the convex hull. Hence, the output .poly
file is useful for finding edges associated with input segments and for
setting boundary conditions in finite element simulations. Moreover,
you will need the output .poly file if you plan to refine the output
mesh, and don‘t want segments to be missing in later triangulations.
.area files:
First line: <# of triangles>
Following lines: <triangle #> <maximum area>
An .area file associates with each triangle a maximum area that is used
for mesh refinement. As with other file formats, every triangle must
be represented, and the triangles must be numbered consecutively. A
triangle may be left unconstrained by assigning it a negative maximum
area.
.edge files:
First line: <# of edges> <# of boundary markers (0 or 1)>
Following lines: <edge #> <endpoint> <endpoint> [boundary marker]
Endpoints are indices into the corresponding .node file. Triangle can
produce .edge files (use the -e switch), but cannot read them. The
optional column of boundary markers is suppressed by the -B switch.
In Voronoi diagrams, one also finds a special kind of edge that is an
infinite ray with only one endpoint. For these edges, a different
format is used:
<edge #> <endpoint> -1 <direction x> <direction y>
The `direction‘ is a floating-point vector that indicates the direction
of the infinite ray.
.neigh files:
First line: <# of triangles> <# of neighbors per triangle (always 3)>
Following lines: <triangle #> <neighbor> <neighbor> <neighbor>
Neighbors are indices into the corresponding .ele file. An index of -1
indicates no neighbor (because the triangle is on an exterior
boundary). The first neighbor of triangle i is opposite the first
corner of triangle i, and so on.
Triangle can produce .neigh files (use the -n switch), but cannot read
them.
Boundary Markers:
Boundary markers are tags used mainly to identify which output vertices
and edges are associated with which PSLG segment, and to identify which
vertices and edges occur on a boundary of the triangulation. A common
use is to determine where boundary conditions should be applied to a
finite element mesh. You can prevent boundary markers from being written
into files produced by Triangle by using the -B switch.
The boundary marker associated with each segment in an output .poly file
and each edge in an output .edge file is chosen as follows:
- If an output edge is part or all of a PSLG segment with a nonzero
boundary marker, then the edge is assigned the same marker.
- Otherwise, if the edge lies on a boundary of the triangulation
(even the boundary of a hole), then the edge is assigned the marker
one (1).
- Otherwise, the edge is assigned the marker zero (0).
The boundary marker associated with each vertex in an output .node file
is chosen as follows:
- If a vertex is assigned a nonzero boundary marker in the input file,
then it is assigned the same marker in the output .node file.
- Otherwise, if the vertex lies on a PSLG segment (even if it is an
endpoint of the segment) with a nonzero boundary marker, then the
vertex is assigned the same marker. If the vertex lies on several
such segments, one of the markers is chosen arbitrarily.
- Otherwise, if the vertex occurs on a boundary of the triangulation,
then the vertex is assigned the marker one (1).
- Otherwise, the vertex is assigned the marker zero (0).
If you want Triangle to determine for you which vertices and edges are on
the boundary, assign them the boundary marker zero (or use no markers at
all) in your input files. In the output files, all boundary vertices,
edges, and segments will be assigned the value one.
Triangulation Iteration Numbers:
Because Triangle can read and refine its own triangulations, input
and output files have iteration numbers. For instance, Triangle might
read the files mesh.3.node, mesh.3.ele, and mesh.3.poly, refine the
triangulation, and output the files mesh.4.node, mesh.4.ele, and
mesh.4.poly. Files with no iteration number are treated as if
their iteration number is zero; hence, Triangle might read the file
points.node, triangulate it, and produce the files points.1.node and
points.1.ele.
Iteration numbers allow you to create a sequence of successively finer
meshes suitable for multigrid methods. They also allow you to produce a
sequence of meshes using error estimate-driven mesh refinement.
If you‘re not using refinement or quality meshing, and you don‘t like
iteration numbers, use the -I switch to disable them. This switch also
disables output of .node and .poly files to prevent your input files from
being overwritten. (If the input is a .poly file that contains its own
points, a .node file is written. This can be quite convenient for
computing CDTs or quality meshes.)
Examples of How to Use Triangle:
`triangle dots‘ reads vertices from dots.node, and writes their Delaunay
triangulation to dots.1.node and dots.1.ele. (dots.1.node is identical
to dots.node.) `triangle -I dots‘ writes the triangulation to dots.ele
instead. (No additional .node file is needed, so none is written.)
`triangle -pe object.1‘ reads a PSLG from object.1.poly (and possibly
object.1.node, if the vertices are omitted from object.1.poly) and writes
its constrained Delaunay triangulation to object.2.node and object.2.ele.
The segments are copied to object.2.poly, and all edges are written to
object.2.edge.
`triangle -pq31.5a.1 object‘ reads a PSLG from object.poly (and possibly
object.node), generates a mesh whose angles are all between 31.5 and 117
degrees and whose triangles all have areas of 0.1 or less, and writes the
mesh to object.1.node and object.1.ele. Each segment may be broken up
into multiple subsegments; these are written to object.1.poly.
Here is a sample file `box.poly‘ describing a square with a square hole:
# A box with eight vertices in 2D, no attributes, one boundary marker.
8 2 0 1
# Outer box has these vertices:
1 0 0 0
2 0 3 0
3 3 0 0
4 3 3 33 # A special marker for this vertex.
# Inner square has these vertices:
5 1 1 0
6 1 2 0
7 2 1 0
8 2 2 0
# Five segments with boundary markers.
5 1
1 1 2 5 # Left side of outer box.
# Square hole has these segments:
2 5 7 0
3 7 8 0
4 8 6 10
5 6 5 0
# One hole in the middle of the inner square.
1
1 1.5 1.5
Note that some segments are missing from the outer square, so you must
use the `-c‘ switch. After `triangle -pqc box.poly‘, here is the output
file `box.1.node‘, with twelve vertices. The last four vertices were
added to meet the angle constraint. Vertices 1, 2, and 9 have markers
from segment 1. Vertices 6 and 8 have markers from segment 4. All the
other vertices but 4 have been marked to indicate that they lie on a
boundary.
12 2 0 1
1 0 0 5
2 0 3 5
3 3 0 1
4 3 3 33
5 1 1 1
6 1 2 10
7 2 1 1
8 2 2 10
9 0 1.5 5
10 1.5 0 1
11 3 1.5 1
12 1.5 3 1
# Generated by triangle -pqc box.poly
Here is the output file `box.1.ele‘, with twelve triangles.
12 3 0
1 5 6 9
2 10 3 7
3 6 8 12
4 9 1 5
5 6 2 9
6 7 3 11
7 11 4 8
8 7 5 10
9 12 2 6
10 8 7 11
11 5 1 10
12 8 4 12
# Generated by triangle -pqc box.poly
Here is the output file `box.1.poly‘. Note that segments have been added
to represent the convex hull, and some segments have been subdivided by
newly added vertices. Note also that <# of vertices> is set to zero to
indicate that the vertices should be read from the .node file.
0 2 0 1
12 1
1 1 9 5
2 5 7 1
3 8 7 1
4 6 8 10
5 5 6 1
6 3 10 1
7 4 11 1
8 2 12 1
9 9 2 5
10 10 1 1
11 11 3 1
12 12 4 1
1
1 1.5 1.5
# Generated by triangle -pqc box.poly
Refinement and Area Constraints:
The -r switch causes a mesh (.node and .ele files) to be read and
refined. If the -p switch is also used, a .poly file is read and used to
specify edges that are constrained and cannot be eliminated (although
they can be subdivided into smaller edges) by the refinement process.
When you refine a mesh, you generally want to impose tighter constraints.
One way to accomplish this is to use -q with a larger angle, or -a
followed by a smaller area than you used to generate the mesh you are
refining. Another way to do this is to create an .area file, which
specifies a maximum area for each triangle, and use the -a switch
(without a number following). Each triangle‘s area constraint is applied
to that triangle. Area constraints tend to diffuse as the mesh is
refined, so if there are large variations in area constraint between
adjacent triangles, you may not get the results you want. In that case,
consider instead using the -u switch and writing a C procedure that
determines which triangles are too large.
If you are refining a mesh composed of linear (three-node) elements, the
output mesh contains all the nodes present in the input mesh, in the same
order, with new nodes added at the end of the .node file. However, the
refinement is not hierarchical: there is no guarantee that each output
element is contained in a single input element. Often, an output element
can overlap two or three input elements, and some input edges are not
present in the output mesh. Hence, a sequence of refined meshes forms a
hierarchy of nodes, but not a hierarchy of elements. If you refine a
mesh of higher-order elements, the hierarchical property applies only to
the nodes at the corners of an element; the midpoint nodes on each edge
are discarded before the mesh is refined.
Maximum area constraints in .poly files operate differently from those in
.area files. A maximum area in a .poly file applies to the whole
(segment-bounded) region in which a point falls, whereas a maximum area
in an .area file applies to only one triangle. Area constraints in .poly
files are used only when a mesh is first generated, whereas area
constraints in .area files are used only to refine an existing mesh, and
are typically based on a posteriori error estimates resulting from a
finite element simulation on that mesh.
`triangle -rq25 object.1‘ reads object.1.node and object.1.ele, then
refines the triangulation to enforce a 25 degree minimum angle, and then
writes the refined triangulation to object.2.node and object.2.ele.
`triangle -rpaa6.2 z.3‘ reads z.3.node, z.3.ele, z.3.poly, and z.3.area.
After reconstructing the mesh and its subsegments, Triangle refines the
mesh so that no triangle has area greater than 6.2, and furthermore the
triangles satisfy the maximum area constraints in z.3.area. No angle
bound is imposed at all. The output is written to z.4.node, z.4.ele, and
z.4.poly.
The sequence `triangle -qa1 x‘, `triangle -rqa.3 x.1‘, `triangle -rqa.1
x.2‘ creates a sequence of successively finer meshes x.1, x.2, and x.3,
suitable for multigrid.
Convex Hulls and Mesh Boundaries:
If the input is a vertex set (not a PSLG), Triangle produces its convex
hull as a by-product in the output .poly file if you use the -c switch.
There are faster algorithms for finding a two-dimensional convex hull
than triangulation, of course, but this one comes for free.
If the input is an unconstrained mesh (you are using the -r switch but
not the -p switch), Triangle produces a list of its boundary edges
(including hole boundaries) as a by-product when you use the -c switch.
If you also use the -p switch, the output .poly file contains all the
segments from the input .poly file as well.
Voronoi Diagrams:
The -v switch produces a Voronoi diagram, in files suffixed .v.node and
.v.edge. For example, `triangle -v points‘ reads points.node, produces
its Delaunay triangulation in points.1.node and points.1.ele, and
produces its Voronoi diagram in points.1.v.node and points.1.v.edge. The
.v.node file contains a list of all Voronoi vertices, and the .v.edge
file contains a list of all Voronoi edges, some of which may be infinite
rays. (The choice of filenames makes it easy to run the set of Voronoi
vertices through Triangle, if so desired.)
This implementation does not use exact arithmetic to compute the Voronoi
vertices, and does not check whether neighboring vertices are identical.
Be forewarned that if the Delaunay triangulation is degenerate or
near-degenerate, the Voronoi diagram may have duplicate vertices or
crossing edges.
The result is a valid Voronoi diagram only if Triangle‘s output is a true
Delaunay triangulation. The Voronoi output is usually meaningless (and
may contain crossing edges and other pathology) if the output is a CDT or
CCDT, or if it has holes or concavities. If the triangulated domain is
convex and has no holes, you can use -D switch to force Triangle to
construct a conforming Delaunay triangulation instead of a CCDT, so the
Voronoi diagram will be valid.
Mesh Topology:
You may wish to know which triangles are adjacent to a certain Delaunay
edge in an .edge file, which Voronoi cells are adjacent to a certain
Voronoi edge in a .v.edge file, or which Voronoi cells are adjacent to
each other. All of this information can be found by cross-referencing
output files with the recollection that the Delaunay triangulation and
the Voronoi diagram are planar duals.
Specifically, edge i of an .edge file is the dual of Voronoi edge i of
the corresponding .v.edge file, and is rotated 90 degrees counterclock-
wise from the Voronoi edge. Triangle j of an .ele file is the dual of
vertex j of the corresponding .v.node file. Voronoi cell k is the dual
of vertex k of the corresponding .node file.
Hence, to find the triangles adjacent to a Delaunay edge, look at the
vertices of the corresponding Voronoi edge. If the endpoints of a
Voronoi edge are Voronoi vertices 2 and 6 respectively, then triangles 2
and 6 adjoin the left and right sides of the corresponding Delaunay edge,
respectively. To find the Voronoi cells adjacent to a Voronoi edge, look
at the endpoints of the corresponding Delaunay edge. If the endpoints of
a Delaunay edge are input vertices 7 and 12, then Voronoi cells 7 and 12
adjoin the right and left sides of the corresponding Voronoi edge,
respectively. To find which Voronoi cells are adjacent to each other,
just read the list of Delaunay edges.
Triangle does not write a list of the edges adjoining each Voronoi cell,
but you can reconstructed it straightforwardly. For instance, to find
all the edges of Voronoi cell 1, search the output .edge file for every
edge that has input vertex 1 as an endpoint. The corresponding dual
edges in the output .v.edge file form the boundary of Voronoi cell 1.
For each Voronoi vertex, the .neigh file gives a list of the three
Voronoi vertices attached to it. You might find this more convenient
than the .v.edge file.
Quadratic Elements:
Triangle generates meshes with subparametric quadratic elements if the
-o2 switch is specified. Quadratic elements have six nodes per element,
rather than three. `Subparametric‘ means that the edges of the triangles
are always straight, so that subparametric quadratic elements are
geometrically identical to linear elements, even though they can be used
with quadratic interpolating functions. The three extra nodes of an
element fall at the midpoints of the three edges, with the fourth, fifth,
and sixth nodes appearing opposite the first, second, and third corners
respectively.
Domains with Small Angles:
If two input segments adjoin each other at a small angle, clearly the -q
switch cannot remove the small angle. Moreover, Triangle may have no
choice but to generate additional triangles whose smallest angles are
smaller than the specified bound. However, these triangles only appear
between input segments separated by small angles. Moreover, if you
request a minimum angle of theta degrees, Triangle will generally produce
no angle larger than 180 - 2 theta, even if it is forced to compromise on
the minimum angle.
Statistics:
After generating a mesh, Triangle prints a count of entities in the
output mesh, including the number of vertices, triangles, edges, exterior
boundary edges (i.e. subsegments on the boundary of the triangulation,
including hole boundaries), interior boundary edges (i.e. subsegments of
input segments not on the boundary), and total subsegments. If you‘ve
forgotten the statistics for an existing mesh, run Triangle on that mesh
with the -rNEP switches to read the mesh and print the statistics without
writing any files. Use -rpNEP if you‘ve got a .poly file for the mesh.
The -V switch produces extended statistics, including a rough estimate
of memory use, the number of calls to geometric predicates, and
histograms of the angles and the aspect ratios of the triangles in the
mesh.
Exact Arithmetic:
Triangle uses adaptive exact arithmetic to perform what computational
geometers call the `orientation‘ and `incircle‘ tests. If the floating-
point arithmetic of your machine conforms to the IEEE 754 standard (as
most workstations do), and does not use extended precision internal
floating-point registers, then your output is guaranteed to be an
absolutely true Delaunay or constrained Delaunay triangulation, roundoff
error notwithstanding. The word `adaptive‘ implies that these arithmetic
routines compute the result only to the precision necessary to guarantee
correctness, so they are usually nearly as fast as their approximate
counterparts.
May CPUs, including Intel x86 processors, have extended precision
floating-point registers. These must be reconfigured so their precision
is reduced to memory precision. Triangle does this if it is compiled
correctly. See the makefile for details.
The exact tests can be disabled with the -X switch. On most inputs, this
switch reduces the computation time by about eight percent--it‘s not
worth the risk. There are rare difficult inputs (having many collinear
and cocircular vertices), however, for which the difference in speed
could be a factor of two. Be forewarned that these are precisely the
inputs most likely to cause errors if you use the -X switch. Hence, the
-X switch is not recommended.
Unfortunately, the exact tests don‘t solve every numerical problem.
Exact arithmetic is not used to compute the positions of new vertices,
because the bit complexity of vertex coordinates would grow without
bound. Hence, segment intersections aren‘t computed exactly; in very
unusual cases, roundoff error in computing an intersection point might
actually lead to an inverted triangle and an invalid triangulation.
(This is one reason to specify your own intersection points in your .poly
files.) Similarly, exact arithmetic is not used to compute the vertices
of the Voronoi diagram.
Another pair of problems not solved by the exact arithmetic routines is
underflow and overflow. If Triangle is compiled for double precision
arithmetic, I believe that Triangle‘s geometric predicates work correctly
if the exponent of every input coordinate falls in the range [-148, 201].
Underflow can silently prevent the orientation and incircle tests from
being performed exactly, while overflow typically causes a floating
exception.
Calling Triangle from Another Program:
Read the file triangle.h for details.
Troubleshooting:
Please read this section before mailing me bugs.
`My output mesh has no triangles!‘
If you‘re using a PSLG, you‘ve probably failed to specify a proper set
of bounding segments, or forgotten to use the -c switch. Or you may
have placed a hole badly, thereby eating all your triangles. To test
these possibilities, try again with the -c and -O switches.
Alternatively, all your input vertices may be collinear, in which case
you can hardly expect to triangulate them.
`Triangle doesn‘t terminate, or just crashes.‘
Bad things can happen when triangles get so small that the distance
between their vertices isn‘t much larger than the precision of your
machine‘s arithmetic. If you‘ve compiled Triangle for single-precision
arithmetic, you might do better by recompiling it for double-precision.
Then again, you might just have to settle for more lenient constraints
on the minimum angle and the maximum area than you had planned.
You can minimize precision problems by ensuring that the origin lies
inside your vertex set, or even inside the densest part of your
mesh. If you‘re triangulating an object whose x-coordinates all fall
between 6247133 and 6247134, you‘re not leaving much floating-point
precision for Triangle to work with.
Precision problems can occur covertly if the input PSLG contains two
segments that meet (or intersect) at an extremely small angle, or if
such an angle is introduced by the -c switch. If you don‘t realize
that a tiny angle is being formed, you might never discover why
Triangle is crashing. To check for this possibility, use the -S switch
(with an appropriate limit on the number of Steiner points, found by
trial-and-error) to stop Triangle early, and view the output .poly file
with Show Me (described below). Look carefully for regions where dense
clusters of vertices are forming and for small angles between segments.
Zoom in closely, as such segments might look like a single segment from
a distance.
If some of the input values are too large, Triangle may suffer a
floating exception due to overflow when attempting to perform an
orientation or incircle test. (Read the section on exact arithmetic
above.) Again, I recommend compiling Triangle for double (rather
than single) precision arithmetic.
Unexpected problems can arise if you use quality meshing (-q, -a, or
-u) with an input that is not segment-bounded--that is, if your input
is a vertex set, or you‘re using the -c switch. If the convex hull of
your input vertices has collinear vertices on its boundary, an input
vertex that you think lies on the convex hull might actually lie just
inside the convex hull. If so, the vertex and the nearby convex hull
edge form an extremely thin triangle. When Triangle tries to refine
the mesh to enforce angle and area constraints, Triangle might generate
extremely tiny triangles, or it might fail because of insufficient
floating-point precision.
`The numbering of the output vertices doesn‘t match the input vertices.‘
You may have had duplicate input vertices, or you may have eaten some
of your input vertices with a hole, or by placing them outside the area
enclosed by segments. In any case, you can solve the problem by not
using the -j switch.
`Triangle executes without incident, but when I look at the resulting
mesh, it has overlapping triangles or other geometric inconsistencies.‘
If you select the -X switch, Triangle occasionally makes mistakes due
to floating-point roundoff error. Although these errors are rare,
don‘t use the -X switch. If you still have problems, please report the
bug.
`Triangle executes without incident, but when I look at the resulting
Voronoi diagram, it has overlapping edges or other geometric
inconsistencies.‘
If your input is a PSLG (-p), you can only expect a meaningful Voronoi
diagram if the domain you are triangulating is convex and free of
holes, and you use the -D switch to construct a conforming Delaunay
triangulation (instead of a CDT or CCDT).
Strange things can happen if you‘ve taken liberties with your PSLG. Do
you have a vertex lying in the middle of a segment? Triangle sometimes
copes poorly with that sort of thing. Do you want to lay out a collinear
row of evenly spaced, segment-connected vertices? Have you simply
defined one long segment connecting the leftmost vertex to the rightmost
vertex, and a bunch of vertices lying along it? This method occasionally
works, especially with horizontal and vertical lines, but often it
doesn‘t, and you‘ll have to connect each adjacent pair of vertices with a
separate segment. If you don‘t like it, tough.
Furthermore, if you have segments that intersect other than at their
endpoints, try not to let the intersections fall extremely close to PSLG
vertices or each other.
If you have problems refining a triangulation not produced by Triangle:
Are you sure the triangulation is geometrically valid? Is it formatted
correctly for Triangle? Are the triangles all listed so the first three
vertices are their corners in counterclockwise order? Are all of the
triangles constrained Delaunay? Triangle‘s Delaunay refinement algorithm
assumes that it starts with a CDT.
Show Me:
Triangle comes with a separate program named `Show Me‘, whose primary
purpose is to draw meshes on your screen or in PostScript. Its secondary
purpose is to check the validity of your input files, and do so more
thoroughly than Triangle does. Unlike Triangle, Show Me requires that
you have the X Windows system. Sorry, Microsoft Windows users.
Triangle on the Web:
To see an illustrated version of these instructions, check out
http://www.cs.cmu.edu/~quake/triangle.html
A Brief Plea:
If you use Triangle, and especially if you use it to accomplish real
work, I would like very much to hear from you. A short letter or email
(to jrs@cs.berkeley.edu) describing how you use Triangle will mean a lot
to me. The more people I know are using this program, the more easily I
can justify spending time on improvements, which in turn will benefit
you. Also, I can put you on a list to receive email whenever a new
version of Triangle is available.
If you use a mesh generated by Triangle in a publication, please include
an acknowledgment as well. And please spell Triangle with a capital `T‘!
If you want to include a citation, use `Jonathan Richard Shewchuk,
``Triangle: Engineering a 2D Quality Mesh Generator and Delaunay
Triangulator,‘‘ in Applied Computational Geometry: Towards Geometric
Engineering (Ming C. Lin and Dinesh Manocha, editors), volume 1148 of
Lecture Notes in Computer Science, pages 203-222, Springer-Verlag,
Berlin, May 1996. (From the First ACM Workshop on Applied Computational
Geometry.)‘
Research credit:
Of course, I can take credit for only a fraction of the ideas that made
this mesh generator possible. Triangle owes its existence to the efforts
of many fine computational geometers and other researchers, including
Marshall Bern, L. Paul Chew, Kenneth L. Clarkson, Boris Delaunay, Rex A.
Dwyer, David Eppstein, Steven Fortune, Leonidas J. Guibas, Donald E.
Knuth, Charles L. Lawson, Der-Tsai Lee, Gary L. Miller, Ernst P. Mucke,
Steven E. Pav, Douglas M. Priest, Jim Ruppert, Isaac Saias, Bruce J.
Schachter, Micha Sharir, Peter W. Shor, Daniel D. Sleator, Jorge Stolfi,
Robert E. Tarjan, Alper Ungor, Christopher J. Van Wyk, Noel J.
Walkington, and Binhai Zhu. See the comments at the beginning of the
source code for references.
Triangle 1.6 (A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator)
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