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<html>
<head>
<title>
TRIANGULATION_REFINE - Refine a Triangulation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGULATION_REFINE <br> Refine a Triangulation
</h1>
<hr>
<p>
<b>TRIANGULATION_REFINE</b>
is a C++ program which
reads information
describing a triangulation of a set of points and creates
a refined triangulation.
</p>
<p>
The refined triangulation is created by dividing each triangle
into four similar subtriangles. The mesh size parameter <b>h</b>
will be halved by such a procedure. If the input triangulation
is Delaunay, then so will be the output triangulation.
</p>
<p>
The program can handle triangulations that use 3 or 6 nodes
per triangle.
</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 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>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>triangulation_refine</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>_ref_nodes.txt is the refined node file;
</li>
<li>
<i>prefix</i>_ref_elements.txt is the refined element fle.
</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_REFINE</b> is available in
<a href = "../../cpp_src/triangulation_rcm/triangulation_rcm.html">a C++ version</a> and
<a href = "../../f_src/triangulation_rcm/triangulation_rcm.html">a FORTRAN90 version</a> and
. <a href = "../../m_src/triangulation_rcm/triangulation_rcm.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 read a file of point coordinates and construct the
Delaunay triangulation for that 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
that 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_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_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,<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_refine.cpp">triangulation_refine.cpp</a>,
the source code.
</li>
<li>
<a href = "triangulation_refine.sh">triangulation_refine.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>BOX3</b> is an example of the refinement of a
3-node triangulation of a rectangular box.
<ul>
<li>
<a href = "box3_nodes.txt">box3_nodes.txt</a>,
the nodes for a 3-node triangulation.
</li>
<li>
<a href = "box3_elements.txt">box3_elements.txt</a>,
the triangles for an order 3 triangulation.
</li>
<li>
<a href = "box3.png">box3.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the triangulation, created by
<a href = "../triangulation_plot/triangulation_plot.html">
TRIANGULATION_PLOT</a>.
</li>
<li>
<a href = "box3_ref_nodes.txt">box3_ref_nodes.txt</a>,
the nodes for the refined triangulation.
</li>
<li>
<a href = "box3_ref_elements.txt">
box3_ref_elements.txt</a>,
the triangles for the refined triangulation.
</li>
<li>
<a href = "box3_ref.png">
box3_ref.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the refined triangulation, created by
<a href = "../triangulation_plot/triangulation_plot.html">
TRIANGULATION_PLOT</a>.
</li>
<li>
<a href = "box3_output.txt">
box3_output.txt</a>,
output printed by the program.
</li>
</ul>
</p>
<p>
<b>IRREG6</b> is an example of the refinement of a
6-node triangulation of an irregular region with a hole:
<ul>
<li>
<a href = "irreg6_nodes.txt">irreg6_nodes.txt</a>,
the nodes for a 6-node triangulation.
</li>
<li>
<a href = "irreg6_elements.txt">irreg6_elements.txt</a>,
the triangles for an order 6 triangulation.
</li>
<li>
<a href = "irreg6.png">irreg6.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the triangulation, created by
<a href = "../triangulation_plot/triangulation_plot.html">
TRIANGULATION_PLOT</a>.
</li>
<li>
<a href = "irreg6_ref_nodes.txt">irreg6_ref_nodes.txt</a>,
the nodes for the refined triangulation.
</li>
<li>
<a href = "irreg6_ref_elements.txt">
irreg6_ref_elements.txt</a>,
the triangles for the refined triangulation.
</li>
<li>
<a href = "irreg6_ref.png">
irreg6_ref.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the refined triangulation, created by
<a href = "../triangulation_plot/triangulation_plot.html">
TRIANGULATION_PLOT</a>.
</li>
<li>
<a href = "irreg6_output.txt">
irreg6_output.txt</a>,
output printed by the program.
</li>
</ul>
</p>
<p>
<b>HOUSE6</b> is an example of the refinement of a
simple 6-node triangulation of a "house" diagram that involved
3 triangles.
<ul>
<li>
<a href = "house6_nodes.txt">house6_nodes.txt</a>,
the nodes for a 6-node triangulation.
</li>
<li>
<a href = "house6_elements.txt">house6_elements.txt</a>,
the triangles for an order 6 triangulation.
</li>
<li>
<a href = "house6.png">house6.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the triangulation, created by
<a href = "../triangulation_plot/triangulation_plot.html">
TRIANGULATION_PLOT</a>.
</li>
<li>
<a href = "house6_ref_nodes.txt">house6_ref_nodes.txt</a>,
the nodes for the refined triangulation.
</li>
<li>
<a href = "house6_ref_elements.txt">
house6_ref_elements.txt</a>,
the triangles for the refined triangulation.
</li>
<li>
<a href = "house6_ref.png">
house6_ref.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the refined triangulation, created by
<a href = "../triangulation_plot/triangulation_plot.html">
TRIANGULATION_PLOT</a>.
</li>
<li>
<a href = "house6_output.txt">
house6_output.txt</a>,
output printed by the program.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for TRIANGULATION_REFINE.
</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>I4COL_COMPARE</b> compares columns I and J of an I4COL.
</li>
<li>
<b>I4COL_SORT_A</b> ascending sorts the columns of an I4COL.
</li>
<li>
<b>I4COL_SORTED_UNIQUE_COUNT</b> counts unique elements in an I4COL.
</li>
<li>
<b>I4COL_SWAP</b> swaps two columns of an I4COL.
</li>
<li>
<b>I4I4_SORT_A</b> ascending sorts a pair of 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>I4MAT_WRITE</b> writes an I4MAT file with no header.
</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>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>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_ORDER3_REFINE_COMPUTE</b> computes a refined order 3 triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_REFINE_SIZE</b> sizes a refined order 3 triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER6_REFINE_COMPUTE</b> computes a refined order 6 triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER6_REFINE_SIZE</b> sizes a refined order 6 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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