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<html>
<head>
<title>
TRIANGULATION_BOUNDARY_EDGES - Boundary Edges of a Triangulation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGULATION_BOUNDARY_EDGES <br> Boundary Edges of a Triangulation
</h1>
<hr>
<p>
<b>TRIANGULATION_BOUNDARY_EDGES</b>
is a MATLAB program which
analyzes the triangulation of a region, identifies the edges that
form the boundaries, organizes the boundary edges into one or more
connected components, and writes a file describing the components
and the sequence of boundary edges.
</p>
<p>
Either a 3-node or 6-node triangulation may be used.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>triangulation_boundary_edges</b> ( <i>'prefix'</i> )
</blockquote>
where <i>prefix</i> is the common filename prefix:
<ul>
<li>
<i>prefix</i><b>_nodes.txt</b> contains the node coordinates,
</li>
<li>
<i>prefix</i><b>_elements.txt</b> contains the element definitions.
</li>
<li>
<i>prefix</i><b>_boundary_edges.txt</b>
is a file created by the program. Each line lists a component
index, and a pair of node indices.
</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_BOUNDARY_EDGES</b> is available in
<a href = "../../m_src/triangulation_boundary_edges/triangulation_boundary_edges.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Programs:
</h3>
<p>
<a href = "../../m_src/mesh_to_xml/mesh_to_xml.html">
MESH_TO_XML</a>,
a MATLAB 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 = "../../m_src/triangulation/triangulation.html">
TRIANGULATION</a>,
a MATLAB library which
carries out various operations on order 3 ("linear") or order 6
("quadratic") triangulations.
</p>
<p>
<a href = "../../m_src/triangulation_boundary_nodes/triangulation_boundary_nodes.html">
TRIANGULATION_BOUNDARY_NODES</a>,
a MATLAB 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 = "../../m_src/triangulation_corner/triangulation_corner.html">
TRIANGULATION_CORNER</a>,
a MATLAB program which
patches triangulations so that no triangle has two sides on the boundary.
</p>
<p>
<a href = "../../m_src/triangulation_delaunay_discrepancy/triangulation_delaunay_discrepancy.html">
TRIANGULATION_DELAUNAY_DISCREPANCY</a>,
a MATLAB program which
measures the amount by which a triangulation fails the local Delaunay test;
</p>
<p>
<a href = "../../m_src/triangulation_display/triangulation_display.html">
TRIANGULATION_DISPLAY</a>,
a MATLAB program which
displays the nodes and elements of a triangulation on the MATLAB graphics screen;
</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 = "../../m_src/triangulation_histogram/triangulation_histogram.html">
TRIANGULATION_HISTOGRAM</a>,
a MATLAB program which
computes histograms of data over a triangulation.
</p>
<p>
<a href = "../../m_src/triangulation_l2q/triangulation_l2q.html">
TRIANGULATION_L2Q</a>,
a MATLAB 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 = "../../m_src/triangulation_mask/triangulation_mask.html">
TRIANGULATION_MASK</a>,
a MATLAB 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 data 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 data directory which
contains a description and examples of order 6 triangulations.
</p>
<p>
<a href = "../../m_src/triangulation_orient/triangulation_orient.html">
TRIANGULATION_ORIENT</a>,
a MATLAB 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 = "../../m_src/triangulation_plot/triangulation_plot.html">
TRIANGULATION_PLOT</a>,
a MATLAB program which
reads data defining a triangulation and creates a
PostScript image of the nodes and triangles.
</p>
<p>
<a href = "../../m_src/triangulation_q2l/triangulation_q2l.html">
TRIANGULATION_Q2L</a>,
a MATLAB 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 = "../../m_src/triangulation_quad/triangulation_quad.html">
TRIANGULATION_QUAD</a>,
a MATLAB program which
estimates the integral of a function over a triangulated region.
</p>
<p>
<a href = "../../m_src/triangulation_quality/triangulation_quality.html">
TRIANGULATION_QUALITY</a>,
a MATLAB program which
reads data defining a triangulation and computes a number
of quality measures.
</p>
<p>
<a href = "../../m_src/triangulation_rcm/triangulation_rcm.html">
TRIANGULATION_RCM</a>,
a MATLAB 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 = "../../m_src/triangulation_refine/triangulation_refine.html">
TRIANGULATION_REFINE</a>,
a MATLAB 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 = "../../m_src/triangulation_triangle_neighbors/triangulation_triangle_neighbors.html">
TRIANGULATION_TRIANGLE_NEIGHBORS</a>,
a MATLAB 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_boundary_edges.m">triangulation_boundary_edges.m</a>
is the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>ELBOW</b> is a curved region, looking like an elbow, that lies inside
the unit square. It is triangulated using order 3 triangles. There are
320 nodes and 568 triangles. There are 66 nodes on the boundary.
<ul>
<li>
<a href = "elbow_nodes.txt">elbow_nodes.txt</a>,
the node file.
</li>
<li>
<a href = "elbow_elements.txt">elbow_elements.txt</a>,
the triangles.
</li>
<li>
<a href = "elbow_output.txt">elbow_output.txt</a>,
the printed output from a run of the sample program.
</li>
<li>
<a href = "elbow_boundary_edges.txt">
elbow_boundary_edges.txt</a>,
the boundary edge file.
</li>
</ul>
</p>
<p>
<b>ELL</b> is an L-shaped region, using 96 triangles and 65 nodes.
It is triangulated using order 3 triangles.
<ul>
<li>
<a href = "ell_nodes.txt">ell_nodes.txt</a>,
the node file.
</li>
<li>
<a href = "ell_elements.txt">ell_elements.txt</a>,
the triangles.
</li>
<li>
<a href = "ell_output.txt">ell_output.txt</a>,
the printed output from a run of the sample program.
</li>
<li>
<a href = "ell_boundary_edges.txt">
ell_boundary_edges.txt</a>,
the boundary edge file.
</li>
</ul>
</p>
<p>
<b>LAKE</b> is an irregular lake-shaped region with an "island",
which means the boundary is made of two components.
It is triangulated using order 3 triangles.
<ul>
<li>
<a href = "lake_nodes.txt">lake_nodes.txt</a>,
the node file.
</li>
<li>
<a href = "lake_elements.txt">lake_elements.txt</a>,
the triangles.
</li>
<li>
<a href = "lake_output.txt">lake_output.txt</a>,
the printed output from a run of the sample program.
</li>
<li>
<a href = "lake_boundary_edges.txt">
lake_boundary_edges.txt</a>,
the boundary edge file.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 02 December 2010.
</i>
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