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<html>
<head>
<title>
TRIANGULATION_MASK - Remove Triangles from a Triangulation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGULATION_MASK <br> Remove Triangles from a Triangulation
</h1>
<hr>
<p>
<b>TRIANGULATION_MASK</b>
is a C++ program which
reads the nodes and triangles that define a triangulation,
calls a user routine which determines whether each triangle is to
be preserved or discarded ("masked") from the triangulation,
and writes out new node and triangle files that define the
masked triangulation.
</p>
<p>
The input file <i>prefix</i>_nodes.txt contains the node information
for the triangulation. Each data line contains the X and
Y coordinates of a single node.
</p>
<p>
The input file <i>prefix</i>_elements.txt contains the triangle information
for the triangulation. Each line contains the indices of 3 or 6
nodes that form a triangle.
</p>
<p>
One motivation for creating this program is as follows. Suppose
we have a set of points that lie on the boundary or inside of
a non-convex region. If we naively call an unconstrained Delaunay
triangulation routine, such as <b>TABLE_DELAUNAY</b>, then because
the region is not convex, it is possible to create triangles which
lie outside the region.
</p>
<p>
An easy way to correct this problem is to call a user routine
and pass it the indices and coordinates of each triangle. The
user can then decide to drop any triangle whose centroid, say,
lies outside the region.
</p>
<p>
Other masking criteria might drop triangles that are too small,
or that have too small an angle, or that lie inside some interior
hole. These choices are entirely up to the user.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
The user masking routine has the form:
<blockquote>
bool <b>triangle_mask</b> ( int <i>dim_num</i>, int <i>triangle_order</i>,
int <i>nodes</i>[], double <i>coord</i>[] )
</blockquote>
with arguments:
<ul>
<li>
<i>dim_num</i>, the spatial dimension, always equal to 2.
</li>
<li>
<i>element_order</i>, the number of nodes in the triangle, usually 3
or 6;
</li>
<li>
<i>nodes</i>, an integer array of dimension <i>element_order</i>,
containing the indices of each node of the triangle;
</li>
<li>
<i>coord</i>, a double array of dimension <i>dim_num</i>
by <i>element_order</i>, containing the <i>x</i> and <i>y</i>
coordinates of each node of the triangle;
</li>
<li>
<i>triangle_mask</i>, a boolean value, which is <b>true</b> if the
triangle should be deleted or "masked", and <b>false</b> if
the triangle should be preserved;
</li>
</ul>
</p>
<p>
The user masking routine must be compiled and linked with the
software, perhaps with a command like:
<blockquote>
g++ <b>triangulation_mask.o</b> <i>triangle_mask.C</i>
</blockquote>
We will assume that the executable is renamed to <b>triangulation_mask</b>.
</p>
<p>
<blockquote>
<b>triangulation_mask</b> <i>prefix</i>
</blockquote>
where <i>prefix</i> is the common filename prefix:
<ul>
<li>
<i>prefix</i>_nodes.txt contains the node coordinates,
</li>
<li>
<i>prefix</i>_elements.txt contains the element definitions.
</li>
<li>
<i>prefix</i>_mask_nodes.txt is an output file which
will contain the node coordinates retained by the mask;
</li>
<li>
<i>prefix</i>_mask_elements.txt is an output file which
will contain the defintions of elements retained by the mask.
</li>
</ul>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>TRIANGULATION_MASK</b> is available in
<a href = "../../cpp_src/triangulation_mask/triangulation_mask.html">a C++ version</a> and
<a href = "../../f_src/triangulation_mask/triangulation_mask.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangulation_mask/triangulation_mask.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../data/table/table.html">
TABLE</a>,
a data format which
is used for the input and output files.
</p>
<p>
<a href = "../../cpp_src/table_delaunay/table_delaunay.html">
TABLE_DELAUNAY</a>,
a C++ program which
can compute the
Delaunay triangulation of a set of points.
</p>
<p>
<a href = "../../c_src/triangle/triangle.html">
TRIANGLE</a>,
a C program which
computes a triangulation of a geometric region.
</p>
<p>
<a href = "../../cpp_src/triangulation/triangulation.html">
TRIANGULATION</a>,
a C++ library which
carries out various operations on order 3 ("linear") or order 6
("quadratic") triangulations.
</p>
<p>
<a href = "../../cpp_src/triangulation_boundary_nodes/triangulation_boundary_nodes.html">
TRIANGULATION_BOUNDARY_NODES</a>,
a C++ program which
reads data defining a triangulation, determines which nodes
lie on the boundary, and writes their coordinates to a file.
</p>
<p>
<a href = "../../cpp_src/triangulation_corner/triangulation_corner.html">
TRIANGULATION_CORNER</a>,
a C++ program which
patches triangulations so that no triangle has two sides on the boundary.
</p>
<p>
<a href = "../../cpp_src/triangulation_delaunay_discrepancy/triangulation_delaunay_discrepancy.html">
TRIANGULATION_DELAUNAY_DISCREPANCY</a>,
a C++ program which
measures the amount by which a triangulation fails the local Delaunay test;
</p>
<p>
<a href = "../../cpp_src/triangulation_display_opengl/triangulation_display_opengl.html">
TRIANGULATION_DISPLAY_OPENGL</a>,
a C++ program which
reads files defining a triangulation and displays an image using Open GL.
</p>
<p>
<a href = "../../cpp_src/triangulation_histogram/triangulation_histogram.html">
TRIANGULATION_HISTOGRAM</a>,
a C++ program which
computes histograms of data over a triangulation.
</p>
<p>
<a href = "../../cpp_src/triangulation_l2q/triangulation_l2q.html">
TRIANGULATION_L2Q</a>,
a C++ program which
reads data defining a 3-node triangulation and generates
midside nodes and writes out the corresponding 6-node triangulation.
</p>
<p>
<a href = "../../cpp_src/triangulation_node_to_element/triangulation_node_to_element.html">
TRIANGULATION_NODE_TO_ELEMENT</a>,
a C++ program which
reads files describing a set of nodes, their triangulation, and the
value of one or more quantities at each node, and outputs a file
that averages the quantities for each element. This operation
in effect creates an "order1" finite element model of the data.
</p>
<p>
<a href = "../../data/triangulation_order3/triangulation_order3.html">
TRIANGULATION_ORDER3</a>,
a directory which
contains a description and
examples of order 3 triangulations.
</p>
<p>
<a href = "../../data/triangulation_order6/triangulation_order6.html">
TRIANGULATION_ORDER6</a>,
a directory which
contains a description and
examples of order 6 triangulations.
</p>
<p>
<a href = "../../cpp_src/triangulation_orient/triangulation_orient.html">
TRIANGULATION_ORIENT</a>,
a C++ program which
reads data defining a triangulation, makes sure that
every triangle has positive orientation, and if not, writes a
corrected triangle file.
</p>
<p>
<a href = "../../cpp_src/triangulation_plot/triangulation_plot.html">
TRIANGULATION_PLOT</a>,
a C++ program which
reads data defining a triangulation and creates a
PostScript image of the nodes and triangles.
</p>
<p>
<a href = "../../cpp_src/triangulation_q2l/triangulation_q2l.html">
TRIANGULATION_Q2L</a>,
a C++ program which
reads data defining a 6-node triangulation, and subdivides
each triangle into 4 3-node triangles, writing the resulting
triangulation to a file.
</p>
<p>
<a href = "../../cpp_src/triangulation_quad/triangulation_quad.html">
TRIANGULATION_QUAD</a>,
a C++ program which
estimates the integral of a function over a triangulated region.
</p>
<p>
<a href = "../../cpp_src/triangulation_quality/triangulation_quality.html">
TRIANGULATION_QUALITY</a>,
a C++ program which
reads data defining a triangulation and computes a number
of quality measures.
</p>
<p>
<a href = "../../cpp_src/triangulation_rcm/triangulation_rcm.html">
TRIANGULATION_RCM</a>,
a C++ program which
reads data defining a triangulation, determines an ordering
of the nodes that will reduce the bandwidth of the adjacency
matrix, and writes the new triangulation information to a file.
</p>
<p>
<a href = "../../cpp_src/triangulation_refine/triangulation_refine.html">
TRIANGULATION_REFINE</a>,
a C++ program which
reads data defining a triangulation, replaces each triangle
by four congruent smaller ones, and writes the new triangulation
information to a file.
</p>
<p>
<a href = "../../cpp_src/triangulation_triangle_neighbors/triangulation_triangle_neighbors.html">
TRIANGULATION_TRIANGLE_NEIGHBORS</a>,
a C++ program which
reads data defining a triangulation, determines the neighboring
triangles of each triangle, and writes that information to a file.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Franz Aurenhammer,<br>
Voronoi diagrams -
a study of a fundamental geometric data structure,<br>
ACM Computing Surveys,<br>
Volume 23, Number 3, September 1991, pages 345-405.
</li>
<li>
Marc deBerg, Marc Krevald, Mark Overmars,
Otfried Schwarzkopf,<br>
Computational Geometry,<br>
Springer, 2000,<br>
ISBN: 3-540-65620-0.
</li>
<li>
Barry Joe, <br>
GEOMPACK - a software package for the generation of meshes
using geometric algorithms, <br>
Advances in Engineering Software,<br>
Volume 13, 1991, pages 325-331.
</li>
<li>
Albert Nijenhuis, Herbert Wilf,<br>
Combinatorial Algorithms for Computers and Calculators,<br>
Second Edition,<br>
Academic Press, 1978,<br>
ISBN: 0-12-519260-6,<br>
LC: QA164.N54.
</li>
<li>
Atsuyuki Okabe, Barry Boots, Kokichi Sugihara, Sung Nok Chiu,<br>
Spatial Tesselations:
Concepts and Applications of Voronoi Diagrams,<br>
Second Edition,<br>
Wiley, 2000,<br>
ISBN: 0-471-98635-6,<br>
LC: QA278.2.O36.
</li>
<li>
Joseph ORourke,<br>
Computational Geometry,<br>
Second Edition,<br>
Cambridge, 1998,<br>
ISBN: 0521649765,<br>
LC: QA448.D38.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "triangulation_mask.cpp">triangulation_mask.cpp</a>,
the source code.
</li>
<li>
<a href = "triangulation_mask.sh">triangulation_mask.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>P15</b> is a triangulation created by calling
<a href = "../../m_src/distmesh/distmesh.html">DISTMESH</a>,
then removing duplicate points by calling
<a href = "../../f_src/table_merge/table_merge.html">TABLE_MERGE</a>,
then creating a Delaunay triangulation by calling
<a href = "../../f_src/table_delaunay/table_delaunay.html">
TABLE_DELAUNAY</a>,
Unfortunately, this results in many triangles that lie outside
the region of interest.
<ul>
<li>
<a href = "p15_nodes.txt">p15_nodes.txt</a>,
a set of nodes.
</li>
<li>
<a href = "p15_nodes.png">p15_nodes.png</a>,
a PNG image of the nodes.
</li>
<li>
<a href = "p15_elements.txt">p15_elements.txt</a>,
a set of order 3 triangles.
</li>
<li>
<a href = "p15_elements.png">p15_elements.png</a>,
a PNG image of the
original data.
</li>
<li>
<a href = "p15_mask.cpp">p15_mask.cpp</a>,
a routine which masks the triangles by dropping those whose
centroids lie outside the region.
</li>
<li>
<a href = "p15_mask.sh">p15_mask.sh</a>,
commands which compile the user routine with
<b>TRIANGULATION_MASK</b>, and analyze the triangulation.
</li>
<li>
<a href = "p15_mask_output.txt">p15_mask_output.txt</a>,
the output from a run of the program.
</li>
<li>
<a href = "p15_mask_nodes.txt">p15_mask_nodes.txt</a>,
the nodes after masking.
</li>
<li>
<a href = "p15_mask_elements.txt">p15_mask_elements.txt</a>,
the elements after masking.
</li>
<li>
<a href = "p15_mask_elements.png">p15_mask_elements.png</a>,
PNG image of the elements after masking.
</li>
</ul>
</p>
<p>
<b>SMALL</b> is a triangulation of the 25 lattice points on
the [0,4]x[0,4] square. Our masking operation should cut out
a lower left triangular corner and a section from the upper right.
<ul>
<li>
<a href = "small_nodes.txt">small_nodes.txt</a>,
a set of nodes.
</li>
<li>
<a href = "small_nodes.png">small_nodes.png</a>,
an image of the original nodes.
</li>
<li>
<a href = "small_elements.txt">small_elements.txt</a>,
a set of order 3 triangles.
</li>
<li>
<a href = "small_elements.png">small_elements.png</a>,
a PNG image of the original elements.
</li>
<li>
<a href = "small_mask.cpp">small_mask.cpp</a>,
a routine which masks the triangles by dropping those whose
centroids lie outside the region.
</li>
<li>
<a href = "small_mask.sh">small_mask.sh</a>,
commands which compile the user routine with
<b>TRIANGULATION_MASK</b>, and analyze the triangulation.
</li>
<li>
<a href = "small_mask_output.txt">small_mask_output.txt</a>,
the output from a run of the program.
</li>
<li>
<a href = "small_mask_nodes.txt">small_mask_nodes.txt</a>,
the nodes after masking.
</li>
<li>
<a href = "small_mask_nodes.png">small_mask_nodes.png</a>,
an image of the nodes after masking.
</li>
<li>
<a href = "small_mask_elements.txt">small_mask_elements.txt</a>,
the elements after masking.
</li>
<li>
<a href = "small_mask_elements.png">small_mask_elements.png</a>,
a PNG image of the elements after masking.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for TRIANGULATION_MASK.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is true if two characters are equal, disregarding case.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the columns in the first line of a file.
</li>
<li>
<b>FILE_EXIST</b> reports whether a file exists.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the minimum of two I4's.
</li>
<li>
<b>I4MAT_DATA_READ</b> reads data from an I4MAT file.
</li>
<li>
<b>I4MAT_HEADER_READ</b> reads the header from an I4MAT file.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of an I4MAT, transposed.
</li>
<li>
<b>I4MAT_WRITE</b> writes an I4MAT file with no header.
</li>
<li>
<b>LVEC_PRINT</b> prints a logical vector.
</li>
<li>
<b>R8MAT_DATA_READ</b> reads the data from an R8MAT file.
</li>
<li>
<b>R8MAT_HEADER_READ</b> reads the header from an R8MAT file.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file with no header.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_I4VEC</b> reads an I4VEC from a string.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>S_WORD_COUNT</b> counts the number of "words" in a string.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 30 September 2010.
</i>
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