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<html>
<head>
<title>
TRIANGULATION_ORIENT - Positively Orient Triangles
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGULATION_ORIENT <br> Positively Orient Triangles
</h1>
<hr>
<p>
<b>TRIANGULATION_ORIENT</b>
is a FORTRAN90 program which
reads a triangulation, and reorients each triangle that
has a negative area. If at least one such triangle is encountered,
the program writes out a new copy of the triangle file in which
all the triangles have been correctly oriented.
</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>
<p>
For many applications, including computer graphics and finite
element computations, it is assumed that the triangles are
described with a positive orientation. That is, the nodes
are listed in clockwise order.
</p>
<p>
<b>TRIANGULATION_ORIENT</b> can check whether every triangle
in a triangulation has positive orientation, and can "repair"
the file if it finds one or more triangles with a negative orientation.
</p>
<p>
A misoriented order 3 triangle:
<pre>
2
/|
/ |
/ |
/ |
/ |
1-----3
</pre>
The corrected order 3 triangle:
<pre>
3
/|
/ |
/ |
/ |
/ |
1-----2
</pre>
</p>
<p>
A misoriented order 6 triangle:
<pre>
2
/|
/ |
4 5
/ |
/ |
1--6--3
</pre>
The corrected order 6 triangle:
<pre>
3
/|
/ |
6 5
/ |
/ |
1--4--2
</pre>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>triangulation_orient</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>_orient_elements.txt will contains the oriented 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_ORIENT</b> is available in
<a href = "../../cpp_src/triangulation_orient/triangulation_orient.html">a C++ version</a> and
<a href = "../../f_src/triangulation_orient/triangulation_orient.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangulation_orient/triangulation_orient.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
computes the Delaunay triangulation of a 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 = "../../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_l2q/triangulation_l2q.html">
TRIANGULATION_L2Q</a>,
a FORTRAN90 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 = "../../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_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_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>
Franz Aurenhammer,<br>
Voronoi diagrams -
a study of a fundamental geometric data structure,<br>
ACM Computing Surveys,<br>
Volume 23, Number 3, September 1991, pages 345-405.
</li>
<li>
Marc deBerg, Marc Krevald, Mark Overmars,
Otfried Schwarzkopf,<br>
Computational Geometry,<br>
Springer, 2000,<br>
ISBN: 3-540-65620-0.
</li>
<li>
Barry Joe, <br>
GEOMPACK - a software package for the generation of meshes
using geometric algorithms, <br>
Advances in Engineering Software,<br>
Volume 13, Number 5, 1991, pages 325-331.
</li>
<li>
Albert Nijenhuis, Herbert Wilf,<br>
Combinatorial Algorithms for Computers and Calculators,<br>
Second Edition,<br>
Academic Press, 1978,<br>
ISBN: 0-12-519260-6,<br>
LC: QA164.N54.
</li>
<li>
Atsuyuki Okabe, Barry Boots, Kokichi Sugihara, Sung Nok Chiu,<br>
Spatial Tesselations:
Concepts and Applications of Voronoi Diagrams,<br>
Second Edition,<br>
Wiley, 2000,<br>
ISBN: 0-471-98635-6,<br>
LC: QA278.2.O36.
</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, June 2004, pages 329-345.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "triangulation_orient.f90">triangulation_orient.f90</a>, the source code.
</li>
<li>
<a href = "triangulation_orient.sh">triangulation_orient.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>P15</b> is a triangulation created by
<a href = "../../m_src/distmesh/distmesh.html">DISTMESH</a>.
Unfortunately, 512 of the triangles have a negative orientation.
In this example, <b>TRIANGULATION_ORIENT</b> was used to reorient
those triangles.
<ul>
<li>
<a href = "p15_nodes.txt">p15_nodes.txt</a>,
a set of nodes.
</li>
<li>
<a href = "p15_elements.txt">p15_elements.txt</a>,
a set of order 3 triangles.
</li>
<li>
<a href = "p15_orient_elements.txt">p15_orient_elements.txt</a>,
the set of order 3 triangles after <b>TRIANGULATION_ORIENT</b>
has reoriented them.
</li>
<li>
<a href = "p15_output.txt">p15_output.txt</a>,
the printed output 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_ORIENT.
</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>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_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_SOME</b> prints some of a R8MAT, transposed.
</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>
<li>
<b>TRIANGLE_AREA_2D</b> computes the area of a triangle in 2D.
</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 26 June 2010.
</i>
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