Triangle

一个二维高质量网格(mesh)生成器和Delaunay三角化工具.
PSLG(Planar Straight Line Graph)约束Delaunay三角网(CDT)与Delaunay三角网相似, 除了PSLG线段在CDT中表示为一条边.
.poly文件格式

    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>

附:

Triangle
一个二维高质量Mesh生成器和Delaunay三角化工具.

A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator.

Version 1.6

2360 Woolsey #H / Berkeley, California 94705-1927
错误/注释请发到[email protected]

Bugs/comments to [email protected]
Triangle作为Quake项目的一部分而创建(地震模拟工具)

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生成Delaunay三角形,约束Delaunay三角形,一致Delaunay三角形,Voronoi图, 以及高质量三角化网格.
最后一个可以不生成角度过大或过小,适合有限元分析的网格.
如果没有命令行参数,会读取.node输入文件, 进行Dealunay三角化, 并输出.node和.ele文件. 语法如下:

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

下划线表示在相应参数后的可选数字. 不要在参数和数字之间留空格. 输入文件必须是.node文件, 或者在-p开关下使用.poly文件.

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:

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 1.5 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 1.5 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 [email protected]) 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.

CDT

Triangle 1.6 (A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator)

Triangle