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<html>
<head>
<title>
TRIANGULATION_QUALITY - Triangulation Quality Measures
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGULATION_QUALITY <br>
Triangulation Quality Measures
</h1>
<hr>
<p>
<b>TRIANGULATION_QUALITY</b>
is a C++ program which
computes and prints quality measures for a given triangulation of a set of points in 2D.
</p>
<p>
The triangulation is defined by a <i>node file</i> containing
the coordinates of nodes, and a <i>triangle file</i> containing
sets of 3 or 6 node indices.
</p>
<p>
The quality measures computed include:
<ul>
<li>
<b>Alpha</b>, the minimum angle divided by the maximum possible
minimum angle.
</li>
<li>
<b>Area</b>, the ratio of the minimum area to the maximum area;
</li>
<li>
<b>Q</b>, the triangle shape uniformity measure, twice the
radius of the inscribed circle divided by the radius of the
circumscribed circle;
</li>
</ul>
</p>
<p>
Each quality measure is defined as the minimum of its value
over all the triangles; the maximum and best possible value is
1, and the minimum and worst possible value is 0.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>triangulation_quality</b> <i>prefix</i>
</blockquote>
where <i>prefix</i> is the common filename prefix:
<ul>
<li>
<i>prefix</i><b>_nodes.txt</b>, the node coordinates;
</li>
<li>
<i>prefix</i><b>_elements.txt</b>, the triangulation.
</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_QUALITY</b> is available in
<a href = "../../cpp_src/triangulation_quality/triangulation_quality.html">a C++ version</a> and
<a href = "../../f_src/triangulation_quality/triangulation_quality.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangulation_quality/triangulation_quality.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/distmesh/distmesh.html">
DISTMESH</a>,
a MATLAB library which
carries out triangular or tetrahedral mesh generation,
by Per-Olof Persson and Gilbert Strang.
</p>
<p>
<a href = "../../m_src/mesh2d/mesh2d.html">
MESH2D</a>,
a MATLAB library which
can automatically create a triangular mesh for a given polygonal region,
by Darren Engwirda.
</p>
<p>
<a href = "../../cpp_src/tet_mesh_quality/tet_mesh_quality.html">
TET_MESH_QUALITY</a>,
a C++ program which
computes quality measures of a tetrahedral mesh.
</p>
<p>
<a href = "../../c_src/triangle/triangle.html">
TRIANGLE</a>
is a C program which
computes a triangulation of a geometric region.
</p>
<p>
<a href = "../../cpp_src/triangulation/triangulation.html">
TRIANGULATION</a>
is a C++ library which
carries out various operations on order 3 ("linear") or order 6
("quadratic") triangulations.
</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>
is 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>
is a directory which
contains a description and
examples of order 6 triangulations.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Marc deBerg, Marc Krevald, Mark Overmars,
Otfried Schwarzkopf,<br>
Computational Geometry,<br>
Springer, 2000,<br>
ISBN: 3-540-65620-0.
</li>
<li>
David Field,<br>
Qualitative Measures for Initial Meshes,<br>
International Journal of Numerical Methods in Engineering,<br>
Volume 47, 2000, pages 887-906.
</li>
<li>
Joseph ORourke,<br>
Computational Geometry,<br>
Second Edition,<br>
Cambridge, 1998,<br>
ISBN: 0521649765,<br>
LC: QA448.D38.
</li>
<li>
Per-Olof Persson, Gilbert Strang,<br>
A Simple Mesh Generator in MATLAB,<br>
SIAM Review,<br>
Volume 46, Number 2, pages 329-345, June 2004.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "triangulation_quality.cpp">triangulation_quality.cpp</a>,
the source code.
</li>
<li>
<a href = "triangulation_quality.sh">triangulation_quality.sh</a>,
commands to compile, link and load the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>ELL3</b> is a set of nodes in an L-shaped region, which have
been arranged into an order 3 triangulation.
<ul>
<li>
<a href = "../../data/triangulation_order3/ell3_nodes.txt">ell3_nodes.txt</a>,
the node file.
</li>
<li>
<a href = "../../data/triangulation_order3/ell3_elements.txt">ell3_elements.txt</a>,
the element file.
</li>
<li>
<a href = "ell3_quality.txt">ell3_quality.txt</a>,
the report from the program.
</li>
</ul>
</p>
<p>
<b>HEX_HOLES6</b> is a set of nodes in an square region, with
two hexagonal holes, which have
been arranged into an order 6 triangulation.
<ul>
<li>
<a href = "../../data/triangulation_order6/hex_holes6_nodes.txt">hex_holes6_nodes.txt</a>,
the node file.
</li>
<li>
<a href = "../../data/triangulation_order6/hex_holes6_elements.txt">hex_holes6_elements.txt</a>,
the element file.
</li>
<li>
<a href = "hex_holes6_quality.txt">hex_holes6_quality.txt</a>,
the report from the program.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for TRIANGULATION_QUALITY.
</li>
<li>
<b>ALPHA_MEASURE</b> determines the triangulated pointset quality measure ALPHA.
</li>
<li>
<b>ARC_COSINE</b> computes the arc cosine function, with argument truncation.
</li>
<li>
<b>AREA_MEASURE</b> determines the area ratio quality measure.
</li>
<li>
<b>BANDWIDTH_MESH</b> determines the bandwidth of the coefficient matrix.
</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</b> prints an I4MAT, transposed.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of an I4MAT, transposed.
</li>
<li>
<b>Q_MEASURE</b> determines the triangulated pointset quality measure Q.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8_MAX</b> returns the maximum of two R8's.
</li>
<li>
<b>R8_MIN</b> returns the minimum of two R8's.
</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</b> prints an R8MAT, transposed.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</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 22 August 2009.
</i>
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