+Triangle
+A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator.
+Version 1.3
+
+Copyright 1996 Jonathan Richard Shewchuk (bugs/comments to jrs@cs.cmu.edu)
+School of Computer Science / Carnegie Mellon University
+5000 Forbes Avenue / Pittsburgh, Pennsylvania 15213-3891
+Created as part of the Archimedes project (tools for parallel FEM).
+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, and quality conforming Delaunay triangulations. The latter
+can be generated with no small angles, and are thus suitable for finite
+element analysis. If no command line switches are specified, your .node
+input file will be read, and the Delaunay triangulation will be returned in
+.node and .ele output files. The command syntax is:
+
+triangle [-prq__a__AcevngBPNEIOXzo_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 .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
+ points, segments, holes, and regional attributes and area
+ constraints. Will generate a constrained Delaunay triangulation
+ fitting the input; or, if -s, -q, or -a is used, a conforming
+ Delaunay triangulation. If -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 edges in the mesh. Further details on
+ refinement are given below.
+ -q Quality mesh generation by Jim Ruppert's Delaunay refinement
+ algorithm. Adds points to the mesh to ensure that no angles
+ smaller than 20 degrees occur. An alternative minimum angle may be
+ specified after the `q'. If the minimum angle is 20.7 degrees or
+ smaller, the triangulation algorithm is theoretically guaranteed to
+ terminate (assuming infinite precision arithmetic - Triangle may
+ fail to terminate if you run out of precision). In practice, the
+ algorithm often succeeds for minimum angles up to 33.8 degrees.
+ For highly refined meshes, however, it may be necessary to reduce
+ the minimum angle to well below 20 to avoid problems associated
+ with insufficient floating-point precision. The specified angle
+ may include a decimal point.
+ -a Imposes a maximum triangle area. If a number follows the `a', no
+ triangle will be 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 number 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
+ enforcing triangle densities in a first triangulation. 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.
+ -A Assigns an additional 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 point set, this switch causes a .poly file to
+ be written, containing all edges in the convex hull. (By default,
+ a .poly file is written only if a .poly file is read.) If you are
+ triangulating a PSLG, this switch specifies that the interior of
+ the convex hull of the PSLG should be triangulated. If you do not
+ use this switch when triangulating a PSLG, it is assumed 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 or failure if Triangle runs out of precision. If
+ you are refining a mesh, the -c switch works differently; it
+ generates the set of boundary edges of the mesh, rather than the
+ convex hull.
+ -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.
+ -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 impose segment constraints 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 will be written.) Cannot be used
+ with the -r switch, because that would overwrite your input .ele
+ file. Shouldn't be used with the -s, -q, or -a switch if you are
+ using a .node file for input, because no .node file will be
+ written, so there will be no record of any added 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 will cause a small
+ improvement in speed and create the possibility (albeit small) 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 overrided by the value used to number the
+ first point 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 points 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 some of the
+ quality of the mesh; Triangle will try, but the resulting mesh may
+ contain triangles of poor aspect ratio. Works well if all the
+ boundary points are closely spaced. Specify this switch twice
+ (`-YY') to prevent all segment splitting, including internal
+ boundaries.
+ -S Specifies the maximum number of Steiner points (points that are not
+ in the input, but are added to meet the constraints of 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 points at segment
+ intersections, even if it needs to use more points than the limit
+ you set. When Triangle inserts segments by splitting (-s), it
+ always adds enough points to ensure that all the segments appear in
+ the triangulation, again ignoring the limit. Be forewarned that
+ the -S switch may result in a conforming triangulation that is not
+ truly Delaunay, because Triangle may be forced to stop adding
+ points when the mesh is in a state where a segment is non-Delaunay
+ and needs to be split. If so, Triangle will print a warning.
+ -i Uses an incremental rather than divide-and-conquer algorithm to
+ form a Delaunay triangulation. Try it if the divide-and-conquer
+ algorithm fails.
+ -F Uses Steven Fortune's sweepline algorithm to form 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 uses alternating vertical and horizontal cuts,
+ which usually improve the speed except with point 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 near external small features.
+ -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' gives
+ information on algorithmic progress and more detailed statistics.
+ `-VV' gives point-by-point details, and will print so much that
+ Triangle will run much more slowly. `-VVV' gives information only
+ a debugger could love.
+ -h Help: Displays these instructions.
+
+Definitions:
+
+ A Delaunay triangulation of a point set is a triangulation whose vertices
+ are the point set, having the property that no point in the point set
+ falls in the interior of the circumcircle (circle that passes through all
+ three vertices) of any triangle in the triangulation.
+
+ A Voronoi diagram of a point set is a subdivision of the plane into
+ polygonal regions (some of which may be infinite), where each region is
+ the set of points in the plane that are closer to some input point than
+ to any other input point. (The Voronoi diagram is the geometric dual of
+ the Delaunay triangulation.)
+
+ A Planar Straight Line Graph (PSLG) is a collection of points and
+ segments. Segments are simply edges, whose endpoints are points in the
+ PSLG. The file format for PSLGs (.poly files) is described below.
+
+ A constrained Delaunay triangulation of a PSLG is similar to a Delaunay
+ triangulation, but each PSLG segment is present as a single edge in the
+ triangulation. (A constrained Delaunay triangulation is not truly a
+ Delaunay triangulation.)
+
+ A conforming Delaunay triangulation of a PSLG is a true Delaunay
+ triangulation in which each PSLG segment may have been subdivided into
+ several edges by the insertion of additional points. These inserted
+ points are necessary to allow the segments to exist in the mesh while
+ maintaining the Delaunay property.
+
+File Formats:
+
+ All files may contain comments prefixed by the character '#'. Points,
+ 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 nodes 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 points> <dimension (must be 2)> <# of attributes>
+ <# of boundary markers (0 or 1)>
+ Remaining lines: <point #> <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 -s,
+ -q, or -a is selected, each new Steiner point added to the mesh will
+ have 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 points and points resting on PSLG
+ segments; a complete description appears in a section below. The .node
+ file produced by Triangle will contain boundary markers in the last
+ column unless they are suppressed by the -B switch.
+
+ .ele files:
+ First line: <# of triangles> <points per triangle> <# of attributes>
+ Remaining lines: <triangle #> <point> <point> <point> ... [attributes]
+
+ Points are indices into the corresponding .node file. The first three
+ points are the corners, and are listed in counterclockwise order around
+ each triangle. (The remaining points, 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, an attempt is made to interpolate attributes, which may
+ result in a good deal of diffusion of attributes among nearby triangles
+ as the triangulation is refined. Diffusion does not occur across
+ segments, so attributes used to identify segment-bounded regions remain
+ intact. In output .ele files, all triangles have three points each
+ unless the -o2 switch is used, in which case they have six, and the
+ fourth, fifth, and sixth points lie on the midpoints of the edges
+ opposite the first, second, and third corners.
+
+ .poly files:
+ First line: <# of points> <dimension (must be 2)> <# of attributes>
+ <# of boundary markers (0 or 1)>
+ Following lines: <point #> <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: <constraint #> <x> <y> <attrib> <max area>
+
+ A .poly file represents a PSLG, as well as some additional information.
+ The first section lists all the points, and is identical to the format
+ of .node files. <# of points> may be set to zero to indicate that the
+ points are listed in a separate .node file; .poly files produced by
+ Triangle always have this format. This has the advantage that a point
+ set may easily be triangulated with or without segments. (The same
+ effect can be achieved, albeit using more disk space, by making a copy
+ of the .poly file with the extension .node; all sections of the file
+ but the first are ignored.)
+
+ The second section lists the segments. Segments are edges whose
+ presence in the triangulation is enforced. Each segment is specified
+ by listing the indices of its two endpoints. This means that you must
+ include its endpoints in the point list. If -s, -q, and -a are not
+ selected, Triangle will produce a constrained Delaunay triangulation,
+ in which each segment appears as a single edge in the triangulation.
+ If -q or -a is selected, Triangle will produce a conforming Delaunay
+ triangulation, in which segments may be subdivided into smaller edges.
+ Each segment, like each point, may have a boundary marker.
+
+ 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 PSLG segments; you must be careful to enclose
+ each hole in segments, or your whole triangulation may 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 will choose 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 will read 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 will
+ affect the whole region (bounded by segments) containing the point. If
+ two values are written on a line after the x and y coordinate, the
+ former is assumed to be a regional attribute (but will only be applied
+ if the -A switch is selected), and the latter is assumed to be a
+ regional area constraint (but will only be applied if the -a switch is
+ selected). You may also 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 encloses the convex hull of the input point
+ set. If you do not use the -c switch, Triangle will eat all triangles
+ on the outer boundary that are not protected 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 appropriate
+ placement of holes just inside the convex hull.
+
+ An ideal PSLG has no intersecting segments, nor any points 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 will calculate the intersection points for you and add 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 point at the middle of a segment, and hope that Triangle will
+ break up the segment at that point, you might get lucky. On the other
+ hand, Triangle might decide that the point doesn't lie precisely on the
+ line, 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 edges that are part of input segments. If the -c switch
+ is used, the output .poly file will also include all of the edges on
+ the convex hull. Hence, the output .poly file is useful for finding
+ edges associated with input segments and setting boundary conditions in
+ finite element simulations. More importantly, you will need it 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 they 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 a mesh boundary, and therefore no neighbor. Triangle can
+ produce .neigh files (use the -n switch), but cannot read them.
+
+ The first neighbor of triangle i is opposite the first corner of
+ triangle i, and so on.
+
+Boundary Markers:
+
+ Boundary markers are tags used mainly to identify which output points and
+ edges are associated with which PSLG segment, and to identify which
+ points 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
+ or 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 occurs on a boundary of the triangulation
+ (including boundaries of holes), then the edge is assigned the marker
+ one (1).
+ - Otherwise, the edge is assigned the marker zero (0).
+ The boundary marker associated with each point in an output .node file is
+ chosen as follows:
+ - If a point 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 point lies on a PSLG segment (including the
+ segment's endpoints) with a nonzero boundary marker, then the point
+ is assigned the same marker. If the point lies on several such
+ segments, one of the markers is chosen arbitrarily.
+ - Otherwise, if the point occurs on a boundary of the triangulation,
+ then the point is assigned the marker one (1).
+ - Otherwise, the point is assigned the marker zero (0).
+
+ If you want Triangle to determine for you which points and edges are on
+ the boundary, assign them the boundary marker zero (or use no markers at
+ all) in your input files. Alternatively, you can mark some of them and
+ leave others marked zero, allowing Triangle to label them.
+
+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 will
+ also disable 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 will be written.)
+
+Examples of How to Use Triangle:
+
+ `triangle dots' will read points from dots.node, and write their Delaunay
+ triangulation to dots.1.node and dots.1.ele. (dots.1.node will be
+ 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' will read a PSLG from object.1.poly (and possibly
+ object.1.node, if the points are omitted from object.1.poly) and write
+ their constrained Delaunay triangulation to object.2.node and
+ object.2.ele. The segments will be copied to object.2.poly, and all
+ edges will be written to object.2.edge.
+
+ `triangle -pq31.5a.1 object' will read a PSLG from object.poly (and
+ possibly object.node), generate a mesh whose angles are all greater than
+ 31.5 degrees and whose triangles all have area smaller than 0.1, and
+ write the mesh to object.1.node and object.1.ele. Each segment may have
+ been broken up into multiple edges; the resulting constrained edges are
+ written to object.1.poly.
+
+ Here is a sample file `box.poly' describing a square with a square hole:
+
+ # A box with eight points 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 point.
+ # 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.
+ 2 5 7 0 # Segments 2 through 5 enclose the hole.
+ 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 one must
+ use the `-c' switch. After `triangle -pqc box.poly', here is the output
+ file `box.1.node', with twelve points. The last four points were added
+ to meet the angle constraint. Points 1, 2, and 9 have markers from
+ segment 1. Points 6 and 8 have markers from segment 4. All the other
+ points 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 split by newly
+ added points. Note also that <# of points> is set to zero to indicate
+ that the points 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 divided into smaller edges) by the refinement process.
+
+ When you refine a mesh, you generally want to impose tighter quality
+ 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.
+
+ If you are refining a mesh composed of linear (three-node) elements, the
+ output mesh will contain all the nodes present in the input mesh, in the
+ same order, with new nodes added at the end of the .node file. However,
+ there is no guarantee that each output element is contained in a single
+ input element. Often, output elements will overlap two input elements,
+ and input edges are not present in the output mesh. Hence, a sequence of
+ refined meshes will form a hierarchy of nodes, but not a hierarchy of
+ elements. If you a refining a mesh of higher-order elements, the
+ hierarchical property applies only to the nodes at the corners of an
+ element; other nodes may not be present in the refined mesh.
+
+ It is important to understand that maximum area constraints in .poly
+ files are handled 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' will read object.1.node and object.1.ele, then
+ refine the triangulation to enforce a 25 degree minimum angle, and then
+ write the refined triangulation to object.2.node and object.2.ele.
+
+ `triangle -rpaa6.2 z.3' will read z.3.node, z.3.ele, z.3.poly, and
+ z.3.area. After reconstructing the mesh and its segments, Triangle will
+ refine 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. 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 point set (rather than 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 if you use the -c switch.
+
+Voronoi Diagrams:
+
+ The -v switch produces a Voronoi diagram, in files suffixed .v.node and
+ .v.edge. For example, `triangle -v points' will read points.node,
+ produce its Delaunay triangulation in points.1.node and points.1.ele,
+ and produce 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 points, crossing
+ edges, or infinite rays whose direction vector is zero. Also, if you
+ generate a constrained (as opposed to conforming) Delaunay triangulation,
+ or if the triangulation has holes, the corresponding Voronoi diagram is
+ likely to have crossing edges and unlikely to make sense.
+
+Mesh Topology:
+
+ You may wish to know which triangles are adjacent to a certain Delaunay
+ edge in an .edge file, which Voronoi regions are adjacent to a certain
+ Voronoi edge in a .v.edge file, or which Voronoi regions 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 diagrams 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; and Voronoi region k is the
+ dual of point 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; their dual triangles are on
+ the left and right of the Delaunay edge, respectively. To find the
+ Voronoi regions adjacent to a Voronoi edge, look at the endpoints of the
+ corresponding Delaunay edge; their dual regions are on the right and left
+ of the Voronoi edge, respectively. To find which Voronoi regions are
+ adjacent to each other, just read the list of Delaunay edges.
+
+Statistics:
+
+ After generating a mesh, Triangle prints a count of the number of points,
+ triangles, edges, boundary edges, and segments in the output mesh. If
+ you've forgotten the statistics for an existing mesh, the -rNEP switches
+ (or -rpNEP if you've got a .poly file for the existing mesh) will
+ regenerate these statistics without writing any output.
+
+ The -V switch produces extended statistics, including a rough estimate
+ of memory use and a histogram of triangle aspect ratios and angles 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
+ registers, then your output is guaranteed to be an absolutely true
+ Delaunay or conforming 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. The exact tests can be disabled with the -X switch. On
+ most inputs, this switch will reduce the computation time by about eight
+ percent - it's not worth the risk. There are rare difficult inputs
+ (having many collinear and cocircular points), however, for which the
+ difference could be a factor of two. These are precisely the inputs most
+ likely to cause errors if you use the -X switch.
+
+ Unfortunately, these routines don't solve every numerical problem. Exact
+ arithmetic is not used to compute the positions of points, because the
+ bit complexity of point 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 compute your own intersection points in your .poly files.) Similarly,
+ exact arithmetic is not used to compute the vertices of the Voronoi
+ diagram.
+
+ Underflow and overflow can also cause difficulties; the exact arithmetic
+ routines do not ameliorate out-of-bounds exponents, which can arise
+ during the orientation and incircle tests. As a rule of thumb, you
+ should ensure that your input values are within a range such that their
+ third powers can be taken without underflow or overflow. Underflow can
+ silently prevent the tests from being performed exactly, while overflow
+ will typically cause 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. To test these possibilities, try again with
+ the -c and -O switches. Alternatively, all your input points 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 point set, or even inside the densest part of your
+ mesh. On the other hand, 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 a very small angle, or if such an
+ angle is introduced by the -c switch, which may occur if a point lies
+ ever-so-slightly inside the convex hull, and is connected by a PSLG
+ segment to a point on the convex hull. If you don't realize that a
+ small 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 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.
+
+ `The numbering of the output points doesn't match the input points.'
+
+ You may have eaten some of your input points with a hole, or by placing
+ them outside the area enclosed by segments.
+
+ `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's divide-and-conquer Delaunay
+ triangulation algorithm 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.
+
+ Strange things can happen if you've taken liberties with your PSLG. Do
+ you have a point 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 points? Have you simply defined
+ one long segment connecting the leftmost point to the rightmost point,
+ and a bunch of points 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 points 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
+ points 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
+ points are their corners in counterclockwise order?
+
+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. Show Me requires that you have the X
+ Windows system. If you didn't receive Show Me with Triangle, complain to
+ whomever you obtained Triangle from, then send me mail.
+
+Triangle on the Web:
+
+ To see an illustrated, updated 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.cmu.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 and on the three-dimensional
+ successor to Triangle, 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.
+
+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, Boris Delaunay, Rex A. Dwyer, David
+ Eppstein, Steven Fortune, Leonidas J. Guibas, Donald E. Knuth, C. L.
+ Lawson, Der-Tsai Lee, Ernst P. Mucke, Douglas M. Priest, Jim Ruppert,
+ Isaac Saias, Bruce J. Schachter, Micha Sharir, Jorge Stolfi, Christopher
+ J. Van Wyk, David F. Watson, and Binhai Zhu. See the comments at the
+ beginning of the source code for references.
+