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<html>
<head>
<title>
TRIANGULATION_L2Q - 6-Node Triangulation from 3-Node Triangulation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGULATION_L2Q <br> 6-Node Triangulation from 3-Node Triangulation
</h1>
<hr>
<p>
<b>TRIANGULATION_L2Q</b>
is a FORTRAN90 program which
reads information describing a triangulation of a set of points using 3-node
("linear") triangles, and creates a 6-node ("quadratic")
triangulation.
</p>
<p>
The same number of triangles are used, but each triangle
is given three extra midside nodes. The coordinates of
these nodes are determined by averaging the coordinates of
pairs of vertices of the triangles.
</p>
<p>
The input and output files use the simple
<a href = "../../data/table/table.html">TABLE format</a>;
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>_nodes.txt contains the node information
for the 3-node 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 3-node triangulation. Each line contains the indices of three
nodes that form a triangle, in counterclockwise order.
</p>
<p>
The output file <i>prefix</i>_l2q_nodes.txt contains the node information
for the 6-node triangulation. It begins with the node information
from <i>nodes.txt</i>, followed by the coordinates of the new nodes.
</p>
<p>
The output file <i>prefix</i>_l2q_elements.txt contains the triangle
information for the 6-node triangulation. There are exactly
as many triangles as before, but now each triangle uses six nodes.
Each line of the file contains the indices of 6 nodes that form the
triangle, listed in a particular 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. It should be the case that the first
three columns of <i>prefix</i>_l2q_elements.txt are the same as the
three columns of <i>prefix</i>elements.txt.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>triangulation_l2q</b> <i>prefix</i>
</blockquote>
where <i>prefix</i> is the common filename prefix:
<ul>
<li>
<i>prefix</i>_nodes.txt contains the linear node coordinates,
</li>
<li>
<i>prefix</i>_elements.txt contains the linear element definitions.
</li>
<li>
<i>prefix</i>_l2q_nodes.txt will contain the quadratic node coordinates,
</li>
<li>
<i>prefix</i>_l2q_elements.txt will contain the quadratic element definitions.
</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_L2Q</b> is available in
<a href = "../../cpp_src/triangulation_l2q/triangulation_l2q.html">a C++ version</a> and
<a href = "../../f_src/triangulation_l2q/triangulation_l2q.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangulation_l2q/triangulation_l2q.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/table_delaunay/table_delaunay.html">
TABLE_DELAUNAY</a>,
a FORTRAN90 program which
triangulates 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 = "../../f_src/triangulation/triangulation.html">
TRIANGULATION</a>,
a FORTRAN90 library which
carries out various operations on order 3 ("linear") or order 6
("quadratic") triangulations.
</p>
<p>
<a href = "../../f_src/triangulation_boundary_nodes/triangulation_boundary_nodes.html">
TRIANGULATION_BOUNDARY_NODES</a>,
a FORTRAN90 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 = "../../f_src/triangulation_corner/triangulation_corner.html">
TRIANGULATION_CORNER</a>,
a FORTRAN90 program which
patches triangulations so that no triangle has two sides on the boundary.
</p>
<p>
<a href = "../../f_src/triangulation_delaunay_discrepancy/triangulation_delaunay_discrepancy.html">
TRIANGULATION_DELAUNAY_DISCREPANCY</a>,
a FORTRAN90 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 = "../../f_src/triangulation_histogram/triangulation_histogram.html">
TRIANGULATION_HISTOGRAM</a>,
a FORTRAN90 program which
computes histograms of data over a triangulation.
</p>
<p>
<a href = "../../f_src/triangulation_mask/triangulation_mask.html">
TRIANGULATION_MASK</a>,
a FORTRAN90 program which
takes an existing triangulation and deletes triangles and
their corresponding nodes as requested by the user.
</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 = "../../f_src/triangulation_orient/triangulation_orient.html">
TRIANGULATION_ORIENT</a>,
a FORTRAN90 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 = "../../f_src/triangulation_plot/triangulation_plot.html">
TRIANGULATION_PLOT</a>,
a FORTRAN90 program which
reads data defining a triangulation and creates a
PostScript image of the nodes and triangles.
</p>
<p>
<a href = "../../f_src/triangulation_q2l/triangulation_q2l.html">
TRIANGULATION_Q2L</a>,
a FORTRAN90 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 = "../../f_src/triangulation_quad/triangulation_quad.html">
TRIANGULATION_QUAD</a>,
a FORTRAN90 program which
estimates the integral of a function over a triangulated region.
</p>
<p>
<a href = "../../f_src/triangulation_quality/triangulation_quality.html">
TRIANGULATION_QUALITY</a>,
a FORTRAN90 program which
reads data defining a triangulation and computes a number
of quality measures.
</p>
<p>
<a href = "../../f_src/triangulation_rcm/triangulation_rcm.html">
TRIANGULATION_RCM</a>,
a FORTRAN90 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 = "../../f_src/triangulation_refine/triangulation_refine.html">
TRIANGULATION_REFINE</a>,
a FORTRAN90 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 = "../../f_src/triangulation_t3_to_t4/triangulation_t3_to_t4.html">
TRIANGULATION_T3_TO_T4</a>,
a FORTRAN90 program which
reads information about a 3-node triangulation and creates data
defining a corresponding 4-node triangulation (vertices + centroid);
</p>
<p>
<a href = "../../f_src/triangulation_triangle_neighbors/triangulation_triangle_neighbors.html">
TRIANGULATION_TRIANGLE_NEIGHBORS</a>,
a FORTRAN90 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,<br>
ISBN: 3-540-65620-0.
</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_l2q.f90">triangulation_l2q.f90</a>,
the source code.
</li>
<li>
<a href = "triangulation_l2q.sh">triangulation_l2q.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>EXAMPLE</b> is an example data set.
<ul>
<li>
<a href = "example_nodes.txt">example_nodes.txt</a>,
the nodes for the 3-node triangulation.
</li>
<li>
<a href = "example_elements.txt">example_elements.txt</a>,
the elements for the 3-node triangulation.
</li>
<li>
<a href = "example.png">example.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>
<li>
<a href = "example_output.txt">example_output.txt</a>,
the output file.
</li>
<li>
<a href = "example_l2q_nodes.txt">example_l2q_nodes.txt</a>,
the nodes for the 6-node triangulation.
</li>
<li>
<a href = "example_l2q_elements.txt">example_l2q_elements.txt</a>,
the elements for the 6-node triangulation.
</li>
<li>
<a href = "example_l2q.png">example_l2q.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>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for TRIANGULATION_L2Q.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is a case insensitive comparison of two characters for equality.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer ( kind = 4 ) value of a base 10 digit.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of integer ( kind = 4 ) division.
</li>
<li>
<b>I4_WRAP</b> forces an I4 to lie between given limits by wrapping.
</li>
<li>
<b>I4COL_COMPARE</b> compares columns I and J of an I4COL.
</li>
<li>
<b>I4COL_SORT_A</b> ascending sorts an I4COL.
</li>
<li>
<b>I4COL_SORTED_UNIQUE_COUNT</b> counts unique elements in an I4COL.
</li>
<li>
<b>I4COL_SWAP</b> swaps columns I and J of an I4COL.
</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.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT</b> prints an I4MAT, transposed.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of the transpose of an I4MAT.
</li>
<li>
<b>I4MAT_WRITE</b> writes an I4MAT file.
</li>
<li>
<b>MESH_BASE_ONE</b> ensures that the element definition is one-based.
</li>
<li>
<b>R8MAT_DATA_READ</b> reads 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>R8MAT_WRITE</b> writes an R8MAT file.
</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>SORT_HEAP_EXTERNAL</b> externally sorts a list of items into ascending order.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TRIANGULATION_NEIGHBOR_TRIANGLES</b> determines triangle neighbors.
</li>
<li>
<b>TRIANGULATION_ORDER3_BOUNDARY_EDGE_COUNT</b> counts the boundary edges.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 11 November 2011.
</i>
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