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<html>
<head>
<title>
TRIANGULATION_DELAUNAY_DISCREPANCY - Local Delaunay Discrepancy of a Triangulation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGULATION_DELAUNAY_DISCREPANCY <br> Local Delaunay Discrepancy of a Triangulation
</h1>
<hr>
<p>
<b>TRIANGULATION_DELAUNAY_DISCREPANCY</b>
is a FORTRAN90 program which
computes the local Delaunay discrepancy of a triangulation.
</p>
<p>
A (maximal) triangulation is Delaunay if and only if it is locally
Delaunay.
</p>
<p>
A triangulation is Delaunay if the minimum angle over all triangles
in the triangulation is maximized. That is, there is no other
triangulation of the points which has a larger minimum angle.
</p>
<p>
A triangulation is locally Delaunay if, for every pair of triangles
that share an edge, the minimum angle in the two triangles is larger
than the minimum angle in the two triangles formed by removing the
common edge and joining the two opposing vertices.
</p>
<p>
This function examines the question of whether a given triangulation
is locally Delaunay. It does this by looking at every pair of
neighboring triangles and comparing the minimum angle attained
for the current triangle pair and the alternative triangle pair.
</p>
<p>
Let A(i,j) be the minimum angle formed by triangles T(i) and T(j),
which are two triangles in the triangulation which share a common edge.
Let B(I,J) be the minimum angle formed by triangles S(i) and S(j),
where S(i) and S(j) are formed by removing the common edge of T(i)
and T(j), and joining the opposing vertices.
</p>
<p>
Then the triangulation is Delaunay if B(i,j) <= A(i,j) for every
pair of neighbors T(i) and T(j).
</p>
<p>
If A(i,j) < B(i,j) for at least one pair of neighbors, the triangulation
is not a Delaunay triangulation.
</p>
<p>
This program returns VALUE = min ( A(i,j) - B(i,j) ) over all
triangle neighbors. VALUE is scaled to be in degrees, for
comprehensibility. If VALUE is negative, then at least one pair
of triangles violates the Delaunay condition, and so the entire
triangulation is not a Delaunay triangulation. If VALUE is nonnegative,
then the triangulation is a Delaunay triangulation.
</p>
<p>
It is useful to return VALUE, rather than a simple True/False value,
because there can be cases where the Delaunay condition is only
"slightly" violated. A simple example is a triangulation formed
by starting with a mesh of squares and dividing each square into
two triangles by choosing one of the diagonals of the square.
The Delaunay discrepancy for this mesh, if computed exactly, is 0,
but roundoff could easily result in discrepancies that were very
slightly negative.
</p>
<p>
If VALUE is positive, and not very small in magnitude, then every
pair of triangles in the triangulation satisfies the local Delaunay
condition, and so the triangulation is a Delaunay triangulation.
</p>
<p>
If VALUE is negative, and not very small in magnitude, then at least
one pair of triangles violates the Delaunay condition, and to a
significant degree. The triangulation is not a Delaunay triangulation.
</p>
<p>
If the magnitude of VALUE is very close to zero, then the triangulation
is numerically ambiguous. At least one pair of triangles violates
or almost violates the condition, but no triangle violates the
condition to a great extent. The user must judge whether the
violation is significant or not.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>triangulation_delaunay_discrepancy</b> <i>prefix</i>
</blockquote>
where <i>prefix</i> is the common prefix for the node and element files
<ul>
<li>
<i>prefix</i><b>_nodes.txt</b>, the node coordinates;
</li>
<li>
<i>prefix</i><b>_elements.txt</b>, the 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_DELAUNAY_DISCREPANCY</b> is available in
<a href = "../../cpp_src/triangulation_delaunay_discrepancy/triangulation_delaunay_discrepancy.html">a C++ version</a> and
<a href = "../../f_src/triangulation_delaunay_discrepancy/triangulation_delaunay_discrepancy.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangulation_delaunay_discrepancy/triangulation_delaunay_discrepancy.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 a set of 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 = "../../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_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_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,<br>
LC: QA448.D38.C65.
</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>
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_delaunay_discrepancy.f90">triangulation_delaunay_discrepancy.f90</a>, the source code.
</li>
<li>
<a href = "triangulation_delaunay_discrepancy.sh">triangulation_delaunay_discrepancy.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>TEN3</b> is a triangulation of 10 nodes. It is a Delaunay triangulation.
<ul>
<li>
<a href = "ten3_nodes.txt">ten3_nodes.txt</a>,
the nodes.
</li>
<li>
<a href = "ten3_elements.txt">ten3_elements.txt</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "ten3.png">ten3.png</a>,
an image of the triangulation.
</li>
<li>
<a href = "ten3_output.txt">ten3_output.txt</a>,
output from the program.
</li>
</ul>
</p>
<p>
<b>TED3</b> is a triangulation of 10 nodes. It is NOT a Delaunay triangulation.
<ul>
<li>
<a href = "ted3_nodes.txt">ted3_nodes.txt</a>,
the nodes.
</li>
<li>
<a href = "ted3_elements.txt">ted3_elements.txt</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "ted3.png">ted3.png</a>,
an image of the triangulation.
</li>
<li>
<a href = "ted3_output.txt">ted3_output.txt</a>,
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_DELAUNAY_DISCREPANCY.
</li>
<li>
<b>ARC_COSINE</b> computes the arc cosine function, with argument truncation.
</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>FILE_COLUMN_COUNT</b> counts the number of 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 rows 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 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 an I4COL.
</li>
<li>
<b>I4COL_SWAP</b> swaps columns J1 and J2 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>R8_HUGE</b> returns a very large R8.
</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 an 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>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>TRIANGLE_ANGLES_2D</b> computes the angles of a triangle in 2D.
</li>
<li>
<b>TRIANGULATION_DELAUNAY_DISCREPANCY</b> reports if a triangulation is Delaunay.
</li>
<li>
<b>TRIANGULATION_NEIGHBOR_TRIANGLES</b> determines triangle neighbors.
</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 06 September 2009.
</i>
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