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<html>
<head>
<title>
POLYGON_TRIANGULATE - Triangulate a Polygon
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
POLYGON_TRIANGULATE <br> Triangulate a Polygon
</h1>
<hr>
<p>
<b>POLYGON_TRIANGULATE</b>
is a C++ library which
triangulates a possibly nonconvex polygon in 2D,
and which can use gnuplot to display the external edges and
internal diagonals of the triangulation.
</p>
<p>
The polygon is defined by an input file which gives the coordinates
of the vertices of the polygon, in counterclockwise order.
</p>
<p>
No consecutive pair of vertices should be equal; when describing a
polygon, sometimes the first and last vertices are equal. For this
program, that is not the case. To describe a square, your input
file should contain four pairs of coordinates, for instance.
</p>
<p>
The vertices should be listed in counterclockwise order. If you list
them in clockwise order, then the function will refuse to operate
on the data.
</p>
<p>
It is possible to create a polygon that has zero area. The function
will refuse to process such an object.
</p>
<p>
The polygon does not need to be convex. However, you should be careful
not to specify a polygon which crosses itself, since this means the
interior of the polygon is not well defined, and hence a triangulation
is not well defined. As a simple example of such a problem, consider
the four vertices of a square in counterclockwise order: V1, V2, V3, V4,
and list them instead as V1, V3, V2, V4. This shape cannot be
triangulated in the usual sense. However, the function may not
realize this, in which case it will return what it thinks is a
reasonable triangulation of the (unreasonable) data.
</p>
<p>
The output of the program is a list of the N-3 triples of nodes that
form the triangles of the triangulation.
</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>POLYGON_TRIANGULATE</b> is available in
<a href = "../../c_src/polygon_triangulate/polygon_triangulate.html">a C version</a> and
<a href = "../../cpp_src/polygon_triangulate/polygon_triangulate.html">a C++ version</a> and
<a href = "../../f77_src/polygon_triangulate/polygon_triangulate.html">a FORTRAN77 version</a> and
<a href = "../../f_src/polygon_triangulate/polygon_triangulate.html">a FORTRAN90 version</a> and
<a href = "../../m_src/polygon_triangulate/polygon_triangulate.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/polygon_integrals/polygon_integrals.html">
POLYGON_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of a polygon in 2D.
</p>
<p>
<a href = "../../cpp_src/polygon_monte_carlo/polygon_monte_carlo.html">
POLYGON_MONTE_CARLO</a>,
a C++ library which
applies a Monte Carlo method to estimate the integral of a function
over the interior of a polygon in 2D.
</p>
<p>
<a href = "../../cpp_src/polygon_properties/polygon_properties.html">
POLYGON_PROPERTIES</a>,
a C++ library which
computes properties of an arbitrary polygon in the plane, defined
by a sequence of vertices, including interior angles, area, centroid,
containment of a point, convexity, diameter, distance to a point,
inradius, lattice area, nearest point in set, outradius, uniform sampling.
</p>
<h3 align = "center">
Author:
</h3>
<p>
Based on a C function by Joseph ORourke;
This C++ version by John Burkardt.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ul>
<li>
Joseph ORourke,<br>
Computational Geometry in C,<br>
Second Edition,<br>
Cambridge, 1998,<br>
ISBN: 0521649765,<br>
LC: QA448.D38.
</li>
</ul>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "polygon_triangulate.cpp">polygon_triangulate.cpp</a>, the source code.
</li>
<li>
<a href = "polygon_triangulate.hpp">polygon_triangulate.hpp</a>, the include file.
</li>
<li>
<a href = "polygon_triangulate.sh">polygon_triangulate.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "polygon_triangulate_prb.cpp">polygon_triangulate_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "polygon_triangulate_prb.sh">polygon_triangulate_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "polygon_triangulate_prb_output.txt">polygon_triangulate_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>COMB</b> is an example of a "comb" polygon of 10 vertices
<ul>
<li>
<a href = "comb_nodes.txt">comb_nodes.txt</a>,
the vertex coordinates.
</li>
<li>
<a href = "comb_elements.txt">comb_elements.txt</a>,
triples of vertex indices that make up the triangulation.
</li>
<li>
<a href = "comb_edges.txt">comb_edges.txt</a>,
pairs of vertex coordinates that form external edges.
</li>
<li>
<a href = "comb_diagonals.txt">comb_diagonals.txt</a>,
pairs of vertex coordinates that form internal diagonals.
</li>
<li>
<a href = "comb.png">comb.png</a>,
a PNG image of the triangulated polygon.
</li>
</ul>
</p>
<p>
<b>HAND</b> outlines a hand using 59 vertices.
<ul>
<li>
<a href = "hand_nodes.txt">hand_nodes.txt</a>,
the vertex coordinates.
</li>
<li>
<a href = "hand_elements.txt">hand_elements.txt</a>,
triples of vertex indices that make up the triangulation.
</li>
<li>
<a href = "hand_edges.txt">hand_edges.txt</a>,
pairs of vertex coordinates that form external edges.
</li>
<li>
<a href = "hand_diagonals.txt">hand_diagonals.txt</a>,
pairs of vertex coordinates that form internal diagonals.
</li>
<li>
<a href = "hand.png">hand.png</a>,
a PNG image of the triangulated polygon.
</li>
</ul>
</p>
<p>
<b>I18</b> is an example of a complicated nonconvex polygon, using 18 vertices.
<ul>
<li>
<a href = "i18_nodes.txt">i18_nodes.txt</a>,
the vertex coordinates.
</li>
<li>
<a href = "i18_elements.txt">i18_elements.txt</a>,
triples of vertex indices that make up the triangulation.
</li>
<li>
<a href = "i18_edges.txt">i18_edges.txt</a>,
pairs of vertex coordinates that form external edges.
</li>
<li>
<a href = "i18_diagonals.txt">i18_diagonals.txt</a>,
pairs of vertex coordinates that form internal diagonals.
</li>
<li>
<a href = "i18.png">i18.png</a>,
a PNG image of the triangulated polygon.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>BETWEEN</b> is TRUE if vertex C is between vertices A and B.
</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 value of a base 10 digit.
</li>
<li>
<b>COLLINEAR</b> returns a measure of collinearity for three points.
</li>
<li>
<b>DIAGONAL:</b> VERTEX(IM1) to VERTEX(IP1) is a proper internal diagonal.
</li>
<li>
<b>DIAGONALIE</b> is true if VERTEX(IM1):VERTEX(IP1) is a proper diagonal.
</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>I4MAT_PRINT</b> prints an I4MAT.
</li>
<li>
<b>I4MAT_PRINT_SOME</b> prints some of an I4MAT.
</li>
<li>
<b>I4MAT_WRITE</b> writes an I4MAT file.
</li>
<li>
<b>IN_CONE</b> is TRUE if the diagonal VERTEX(IM1):VERTEX(IP1) is strictly internal.
</li>
<li>
<b>INTERSECT</b> is true if lines VA:VB and VC:VD intersect.
</li>
<li>
<b>INTERSECT_PROP</b> is TRUE if lines VA:VB and VC:VD have a proper intersection.
</li>
<li>
<b>L4_XOR</b> returns the exclusive OR of two L4's.
</li>
<li>
<b>POLYGON_TRIANGULATE</b> determines a triangulation of a polygon.
</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>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</b> computes the signed area of a triangle.
</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 06 May 2014.
</i>
<!-- John Burkardt -->
</body>
</html>