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<html>
<head>
<title>
DELAUNAY_LMAP_2D - Delaunay Triangulations Under Linear Maps
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
DELAUNAY_LMAP_2D <br> Delaunay Triangulations Under Linear Maps
</h1>
<hr>
<p>
<b>DELAUNAY_LMAP_2D</b>
is a FORTRAN90 program which
computes the Delaunay triangulation of a set of
points in the plane that have been transformed by a linear map <b>A</b>.
</p>
<p>
Specifically, <b>DELAUNAY_LMAP_2D</b> reads two input files:
<ul>
<li>
a <i>node file</i> containing (many) point coordinates <b>(X,Y)</b>;
</li>
<li>
a <i>map file</i> containing the entries of the 2x2 matrix
<b>A</b> that describes the linear distortion of the geometry;
</li>
</ul>
The program then implicitly multiplies each point <b>(X,Y)</b>
by <b>A</b> and computes the Delaunay triangulation of the
transformed points <b>(AX,AY)</b>.
It then writes out
<ul>
<li>
an order 3 <i>triangulation file</i>, listing the indices of
the 3 nodes that form each Delaunay triangle,
</li>
<li>
an <a href = "../../data/eps/eps.html">
Encapsulated PostScript file</a> containing an image of
the triangulation.
</li>
</ul>
</p>
<p>
In a sense, there's no reason to do all this fancy stuff.
You can simply multiply your data by <b>A</b> yourself, get the Delaunay
triangulation of the transformed data, and then recover your
original data.
</p>
<p>
The main reason for setting up this code is
to prepare for cases that are a generalization of this one,
in which the matrix <b>A</b> actually varies spatially.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>delaunay_lmap_2d</b> <i>node_file</i> <i>matrix_file</i>
</blockquote>
where
<ul>
<li>
<i>node_file</i> contains the coordinates of the nodes;
</li>
<li>
<i>matrix_file</i> contains the 2x2 matrix;
</li>
</ul>
creating the files
<ul>
<li>
<i>node_file</i><b>.delaunay.txt</b>, with the triangulation information;
</li>
<li>
<i>node_file</i><b>.delaunay.eps</b>, with an image.
</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">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/geompack/geompack.html">
GEOMPACK</a>,
a FORTRAN90 library which
computes Delaunay triangulations, written by Barry Joe.
</p>
<p>
<a href = "../../f_src/stripack/stripack.html">
STRIPACK</a>,
a FORTRAN90 library which
computes the Delaunay triangulation or Voronoi diagram
of points on a sphere.
</p>
<p>
<a href = "../../f_src/table_delaunay/table_delaunay.html">
TABLE_DELAUNAY</a>,
a FORTRAN90 program which
reads a file of point coordinates in the TABLE format and writes out
the Delaunay triangulation.
</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 = "../../data/triangulation_order3/triangulation_order3.html">
TRIANGULATION_ORDER3</a>,
an data directory which
contains a description and examples of the
information needed to describe an order 3 triangulation of a set of nodes..
</p>
<p>
<a href = "../../f_src/triangulation_plot/triangulation_plot.html">
TRIANGULATION_PLOT</a>,
a FORTRAN90 program which
may be used to make a PostScript image of
the triangulation of the points.
</p>
<p>
<a href = "../../f_src/tripack/tripack.html">
TRIPACK</a>,
a FORTRAN90 library which
computes the Delaunay triangulation of points in the plane.
</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>
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 = "delaunay_lmap_2d.f90">delaunay_lmap_2d.f90</a>, the source code.
</li>
<li>
<a href = "delaunay_lmap_2d.sh">delaunay_lmap_2d.sh</a>,
commands to compile, link and load the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
Test matrices you can copy include:
<ul>
<li>
<a href = "matrix_identity.txt">matrix_identity.txt</a>,
the identity matrix.
</li>
<li>
<a href = "matrix_d21.txt">matrix_d21.txt</a>,
the (2,1) dilation.
</li>
<li>
<a href = "matrix_d41.txt">matrix_d41.txt</a>,
the (4,1) dilation.
</li>
<li>
<a href = "matrix_d81.txt">matrix_d81.txt</a>,
the (8,1) dilation.
</li>
<li>
<a href = "matrix_d15.txt">matrix_d15.txt</a>,
the (1,5) dilation.
</li>
<li>
<a href = "matrix_r30.txt">matrix_r30.txt</a>,
the 30 degree rotation (should have no effect on Delaunay).
</li>
<li>
<a href = "matrix_r30d21.txt">matrix_r30d21.txt</a>,
the (2,1) dilation and 30 degree rotation.
</li>
<li>
<a href = "matrix_r30d41.txt">matrix_r30d41.txt</a>,
the (4,1) dilation and 30 degree rotation.
</li>
<li>
<a href = "matrix_r30d81.txt">matrix_r30d81.txt</a>,
the (8,1) dilation and 30 degree rotation.
</li>
</ul>
</p>
<p>
<b>POINTS</b> is a set of 10 "random" points in the unit square. Test
files you may copy include:
<ul>
<li>
<a href = "points.txt">points.txt</a>,
a (real) TABLE file describing the coordinates of the points.
</li>
<li>
<a href = "points.identity.delaunay.txt">
points.identity.delaunay.txt</a>,
the (integer) TABLE file, describing
the Delaunay triangulation of the points under the identity map.
</li>
<li>
<a href = "points.identity.delaunay.png">
points.identity.delaunay.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the triangulated data.
</li>
<li>
<a href = "points.d21.delaunay.png">
points.d21.delaunay.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the triangulated data under the D21 matrix.
</li>
<li>
<a href = "points.d41.delaunay.png">
points.d41.delaunay.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the triangulated data under the D41 matrix.
</li>
<li>
<a href = "points.d81.delaunay.png">
points.d81.delaunay.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the triangulated data under the D81 matrix.
</li>
<li>
<a href = "points.r30.delaunay.png">
points.rot30.delaunay.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the triangulated data under the R30 matrix.
</li>
<li>
<a href = "points.r30d21.delaunay.png">
points.r30d21.delaunay.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the triangulated data under the R30D21 matrix.
</li>
<li>
<a href = "points.r30d41.delaunay.png">
points.r30d41.delaunay.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the triangulated data under the R30D41 matrix.
</li>
<li>
<a href = "points.r30d81.delaunay.png">
points.r30d81.delaunay.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the triangulated data under the R30D81 matrix.
</li>
</ul>
</p>
<p>
<b>DOUBLE_HEX2_CVT</b> is a set of 139 points in the unit square
with two hexagonal holes. Test files you may copy include:
<ul>
<li>
<a href = "double_hex2_cvt.txt">double_hex2_cvt.txt</a>,
a (real) TABLE file containing the point coordinates.
</li>
<li>
<a href = "double_hex2_cvt.identity.delaunay.txt">
double_hex2_cvt.identity.delaunay.txt</a>,
the Delaunay triangulation under the identity matrix.
</li>
<li>
<a href = "double_hex2_cvt.identity.delaunay.png">
double_hex2_cvt.identity.delaunay.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the Delaunay triangulation under the identity matrix.
</li>
<li>
<a href = "double_hex2_cvt.d21.delaunay.png">
double_hex2_cvt.d21.delaunay.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the Delaunay triangulation under the D21 matrix.
</li>
<li>
<a href = "double_hex2_cvt.d41.delaunay.png">
double_hex2_cvt.d41.delaunay.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the Delaunay triangulation under the D41 matrix.
</li>
<li>
<a href = "double_hex2_cvt.d81.delaunay.png">
double_hex2_cvt.d81.delaunay.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the Delaunay triangulation under the D81 matrix.
</li>
</ul>
</p>
<p>
<b>GRID105</b> is a set of 105 points in the unit square, arranged
into 5 columns of 21 points each. The points were originally
evenly spaced in the X direction, and evenly spaced in the Y
direction, but a uniform random perturbation of scale 0.03
was applied to both coordinates of every point. Because the
data has been compressed by a factor of 5 in the Y direction,
we get some "ugly" triangles if we do a Delaunay triangulation
directly on the data. However, if we use a (1,5) dilation
matrix to uncompress the Y direction, the triangles look much
nicer (except along the boundary, where we are powerless to help).
Test files you may copy include:
<ul>
<li>
<a href = "grid105.txt">grid105.txt</a>,
a (real) TABLE file containing the point coordinates.
</li>
<li>
<a href = "grid105.png">grid105.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the points.
</li>
<li>
<a href = "grid105.identity.delaunay.png">
grid105.identity.delaunay.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the Delaunay triangulation under the identity matrix.
</li>
<li>
<a href = "grid105.d15.delaunay.png">
grid105.d15.delaunay.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the Delaunay triangulation under the D15 matrix.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for DELAUNAY_LMAP_2D.
</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>DIAEDG_LMAP</b> chooses a diagonal edge under a linear map.
</li>
<li>
<b>DTRIS2_LMAP</b> constructs a Delaunay triangulation of 2D linear mapped vertices.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_NAME_EXT_GET</b> determines the "extension" of a file name.
</li>
<li>
<b>FILE_NAME_EXT_SWAP</b> replaces the current "extension" of a file name.
</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>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>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>I4VEC_INDICATOR</b> sets an I4VEC to the indicator vector.
</li>
<li>
<b>I4VEC_SORT_HEAP_INDEX_A</b> does an indexed heap ascending sort of an I4VEC.
</li>
<li>
<b>LRLINE</b> determines if a point is left of, right or, or on a directed line.
</li>
<li>
<b>PERM_INV</b> inverts a permutation "in place".
</li>
<li>
<b>R82VEC_PERMUTE</b> permutes an R82VEC in place.
</li>
<li>
<b>R82VEC_SORT_HEAP_INDEX_A</b> does an indexed heap ascending sort of an R82VEC.
</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_PRINT</b> prints an R8MAT.
</li>
<li>
<b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT transposed.
</li>
<li>
<b>S_BLANK_DELETE</b> removes blanks from a string, left justifying the remainder.
</li>
<li>
<b>S_INDEX_LAST</b> finds the LAST occurrence of a given substring.
</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>SWAPEC_LMAP</b> swaps diagonal edges until all triangles are Delaunay.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TRANSFORM_LMAP</b> applies a transformation matrix to a vector.
</li>
<li>
<b>TRIANGULATION_PLOT_EPS</b> creates an EPS file image of a triangulation.
</li>
<li>
<b>VBEDG</b> determines which boundary edges are visible to a point.
</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 08 June 2010.
</i>
<!-- John Burkardt -->
</body>
</html>