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<html>
<head>
<title>
TRIANGULATION_Q2L - 6-Node Triangulation to 3-Node Triangulation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGULATION_Q2L <br> 6-Node Triangulation to 3-Node Triangulation
</h1>
<hr>
<p>
<b>TRIANGULATION_Q2L</b>
is a C++ program which
reads information
describing a triangulation of a set of points using 6-node
("quadratic") triangles, and creates a 3-node ("linear")
triangulation.
</p>
<p>
The same nodes are used, but each 6-node triangle is broken
up into four smaller 3-node triangles.
</p>
<p>
Thus, the program might be given the following 4 triangles:
<pre>
11-12-13-14-15
|\ |\ |
| \ | \ |
6 7 8 9 10
| \ | \ |
| \| \|
1--2--3--4--5
</pre>
</p>
<p>
It would make a new triangulation involving 16 triangles:
<pre>
11-12-13-14-15
|\ |\ |\ |\ |
| \| \| \| \|
6--7--8--9-10
|\ |\ |\ |\ |
| \| \| \| \|
1--2--3--4--5
</pre>
</p>
<p>
The input and output files use the simple TABLE format;
comment lines begin with a "#" character. Otherwise, each line
of the file contains one set of information, either the coordinates
of a node (for a node file), or the indices of nodes that make up
a triangle, (for a triangle file).
</p>
<p>
The input file <i>prefix</i>_elements.txt</i> contains the triangle information
for the 6-node triangulation. Each line contains the indices of six
nodes that form a triangle, in counterclockwise order. The first three
indices are the vertices, in counterclockwise order. The fourth
index is the midside node between vertices 1 and 2, the fifth
the midside between vertices 2 and 3, and the sixth the midside
between vertices 3 and 1.
</p>
<p>
The output file <i>prefix</i>_q2l_elements.txt contains the triangle
information for the 3-node triangulation. The vertices for
each triangle are listed in counterclockwise order. There are
4 times as many triangles in this triangulation.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>triangulation_q2l</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 (not needed by this program),
</li>
<li>
<i>prefix</i>_elements.txt contains the quadratic element definitions.
</li>
<li>
<i>prefix</i>_q2l_elements.txt will contain the linear element
definitions, based on the same set of nodes.
</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_Q2L</b> is available in
<a href = "../../cpp_src/triangulation_q2l/triangulation_q2l.html">a C++ version</a> and
<a href = "../../f_src/triangulation_q2l/triangulation_q2l.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangulation_q2l/triangulation_q2l.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/mesh_to_xml/mesh_to_xml.html">
MESH_TO_XML</a>,
a C++ program which
reads information defining a 1D, 2D or 3D mesh, namely
a file of node coordinates and a file of elements defined by
node indices, and creates a corresponding XML file for input
to DOLFIN or FENICS.
</p>
<p>
<a href = "../../cpp_src/table_delaunay/table_delaunay.html">
TABLE_DELAUNAY</a>,
a C++ program which
triangulates a set of nodes whose coordinates are stored in a file.
</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_mask/triangulation_mask.html">
TRIANGULATION_MASK</a>,
a C++ program which
takes an existing triangulation and deletes triangles and
their corresponding nodes as requested by the user.
</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_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>
Marc deBerg, Marc Krevald, Mark Overmars,
Otfried Schwarzkopf,<br>
Computational Geometry,<br>
Springer, 2000.
</li>
<li>
Joseph ORourke,<br>
Computational Geometry,<br>
Cambridge University Press,<br>
Second Edition, 1998.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "triangulation_q2l.cpp">triangulation_q2l.cpp</a>,
the source code.
</li>
<li>
<a href = "triangulation_q2l.sh">triangulation_q2l.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>EXAMPLE</b> is a set of demonstration data.
<ul>
<li>
<a href = "example_nodes.txt">example_nodes.txt</a>,
the nodes.
</li>
<li>
<a href = "example_elements.txt">example_elements.txt</a>,
the triangles for the 6-node triangulation.
</li>
<li>
<a href = "example.png">example.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the 6-node triangulation, created by
<a href = "../triangulation_plot/triangulation_plot.html">
TRIANGULATION_PLOT</a>.
</li>
<li>
<a href = "example_q2l_elements.txt">example_q2l_elements.txt</a>,
the triangles for the 3-node triangulation.
</li>
<li>
<a href = "example_q2l.png">example_q2l.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the 3-node triangulation, created by
<a href = "../triangulation_plot/triangulation_plot.html">
TRIANGULATION_PLOT</a>.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for TRIANGULATION_Q2L.
</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>I4_MODP</b> returns the nonnegative remainder of I4 division.
</li>
<li>
<b>I4_WRAP</b> forces an I4 to lie between given limits by wrapping.
</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>I4MAT_WRITE</b> writes an I4MAT 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_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>
<li>
<b>TRIANGULATION_ORDER6_TO_ORDER3</b> linearizes a quadratic triangulation.
</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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