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<html>
<head>
<title>
TRIANGULATION - Triangulation of 2D data
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGULATION <br> Triangulation of 2D data
</h1>
<hr>
<p>
<b>TRIANGULATION</b>
is a C++ library which
computes a triangulation of
a set of points in 2D, and carries out various other related
operations on triangulations of order 3 or 6.
</p>
<p>
The mesh is the collection of triangles. Each triangle
is termed an "element". The points used to define the shape of the
triangle (the corners, and sometimes a few more points) are called
the "nodes".
</p>
<p>
Routines are available to:
<ul>
<li>
evaluate "quality measures" for the mesh;
</li>
<li>
create a "node neighbor array" for each node;
</li>
<li>
create an "element neighbor array" for each element;
</li>
<li>
estimate the integral of a function over the region covered by the mesh;
</li>
<li>
plot the nodes and elements of a mesh;
</li>
<li>
determine the parts of the mesh that lie on the boundary;
</li>
<li>
sample points at random from the region covered by the mesh;
</li>
<li>
search a mesh to determine which element contains a point.
</li>
</ul>
</p>
<p>
Since triangulations are often used to define a finite element
mesh, which in turn defines a sparse matrix, there are routines
available which can define the sparse compressed column arrays
needed for a sparse matrix associated with a mesh of order 3
or 6. The special case of the Taylor-Hood mixed element is also
handled, which is essentially an order 6 grid counted twice
and an order 3 grid that only uses the vertices of the order 6 grid.
</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</b> is available in
<a href = "../../c_src/triangulation/triangulation.html">a C version</a> and
<a href = "../../cpp_src/triangulation/triangulation.html">a C++ version</a> and
<a href = "../../f77_src/triangulation/triangulation.html">a FORTRAN77 version</a> and
<a href = "../../f_src/triangulation/triangulation.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangulation/triangulation.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Programs:
</h3>
<p>
<a href = "../../cpp_src/mesh_to_xml/mesh_to_xml.html">
MESH_TO_XML</a>,
a C++ 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 = "../../cpp_src/table_delaunay/table_delaunay.html">
TABLE_DELAUNAY</a>,
a C++ program which
reads a file of 2d point coordinates and computes the Delaunay triangulation.
</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 = "../../cpp_src/triangulation_boundary_nodes/triangulation_boundary_nodes.html">
TRIANGULATION_BOUNDARY_NODES</a>,
a C++ 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 = "../../cpp_src/triangulation_corner/triangulation_corner.html">
TRIANGULATION_CORNER</a>,
a C++ program which
patches triangulations so that no triangle has two sides on the boundary.
</p>
<p>
<a href = "../../cpp_src/triangulation_delaunay_discrepancy/triangulation_delaunay_discrepancy.html">
TRIANGULATION_DELAUNAY_DISCREPANCY</a>,
a C++ 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 = "../../cpp_src/triangulation_histogram/triangulation_histogram.html">
TRIANGULATION_HISTOGRAM</a>,
a C++ program which
computes histograms of data over a triangulation.
</p>
<p>
<a href = "../../cpp_src/triangulation_l2q/triangulation_l2q.html">
TRIANGULATION_L2Q</a>,
a C++ 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 = "../../cpp_src/triangulation_mask/triangulation_mask.html">
TRIANGULATION_MASK</a>,
a C++ program, which
takes an existing triangulation and deletes triangles and
their corresponding nodes as requested by the user.
</p>
<p>
<a href = "../../cpp_src/triangulation_node_to_element/triangulation_node_to_element.html">
TRIANGULATION_NODE_TO_ELEMENT</a>,
a C++ program which
reads files describing a set of nodes, their triangulation, and the
value of one or more quantities at each node, and outputs a file
that averages the quantities for each element. This operation
in effect creates an "order1" finite element model of the data.
</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 = "../../cpp_src/triangulation_orient/triangulation_orient.html">
TRIANGULATION_ORIENT</a>,
a C++ 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 = "../../cpp_src/triangulation_plot/triangulation_plot.html">
TRIANGULATION_PLOT</a>,
a C++ program which
reads data defining a triangulation and creates a
PostScript image of the nodes and triangles.
</p>
<p>
<a href = "../../cpp_src/triangulation_q2l/triangulation_q2l.html">
TRIANGULATION_Q2L</a>,
a C++ 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 = "../../cpp_src/triangulation_quad/triangulation_quad.html">
TRIANGULATION_QUAD</a>,
a C++ program which
estimates the integral of a function over a triangulated region.
</p>
<p>
<a href = "../../cpp_src/triangulation_quality/triangulation_quality.html">
TRIANGULATION_QUALITY</a>,
a C++ program which
reads data defining a triangulation and computes a number
of quality measures.
</p>
<p>
<a href = "../../cpp_src/triangulation_rcm/triangulation_rcm.html">
TRIANGULATION_RCM</a>,
a C++ 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 = "../../cpp_src/triangulation_refine/triangulation_refine.html">
TRIANGULATION_REFINE</a>,
a C++ 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 = "../../cpp_src/triangulation_triangle_neighbors/triangulation_triangle_neighbors.html">
TRIANGULATION_TRIANGLE_NEIGHBORS</a>,
a C++ 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">
References:
</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>
Paul Bratley, Bennett Fox, Linus Schrage,<br>
A Guide to Simulation,<br>
Second Edition,<br>
Springer, 1987,<br>
ISBN: 0387964673.
</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, 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 Tessellations:
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.cpp">triangulation.cpp</a>, the source code.
</li>
<li>
<a href = "triangulation.hpp">triangulation.hpp</a>, the include file.
</li>
<li>
<a href = "triangulation.sh">triangulation.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "triangulation_prb.cpp">triangulation_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "triangulation_prb.sh">triangulation_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "triangulation_prb_output.txt">triangulation_prb_output.txt</a>,
the output file.
</li>
<li>
<a href = "ns_triangulation.png">
ns_triangulation.png</a>,
a PNG image of
a tiny triangulation used for the Navier Stokes calculations.
</li>
<li>
<a href = "triangulation_order3_plot.png">
triangulation_order3_plot.png</a>,
a PNG image of
an order 3 triangulation.
</li>
<li>
<a href = "triangulation_order3_plot2.png">
triangulation_order3_plot2.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
a second order 3 triangulation.
</li>
<li>
<a href = "triangulation_order6_plot.png">
triangulation_order6_plot.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
an order 6 triangulation.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>ALPHA_MEASURE</b> determines the triangulated pointset quality measure ALPHA.
</li>
<li>
<b>ANGLE_RAD_2D</b> returns the angle in radians swept out between two rays in 2D.
</li>
<li>
<b>ARC_COSINE</b> computes the arc cosine function, with argument truncation.
</li>
<li>
<b>AREA_MEASURE</b> determines the area ratio quality measure.
</li>
<li>
<b>BANDWIDTH</b> determines the bandwidth associated with the finite element mesh.
</li>
<li>
<b>DELAUNAY_SWAP_TEST</b> performs the Delaunay swap test.
</li>
<li>
<b>DIAEDG</b> chooses a diagonal edge.
</li>
<li>
<b>GET_SEED</b> returns a random seed for the random number generator.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two I4's.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of I4 division.
</li>
<li>
<b>I4_POWER</b> returns the value of I^J.
</li>
<li>
<b>I4_SIGN</b> returns the sign of an I4.
</li>
<li>
<b>I4_SWAP</b> switches two I4's.
</li>
<li>
<b>I4_UNIFORM</b> returns a scaled pseudorandom I4.
</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_SORTED_UNIQUE_COUNT</b> counts unique elements in an I4COL.
</li>
<li>
<b>I4COL_SWAP</b> swaps two columns of an I4COL.
</li>
<li>
<b>I4I4_SORT_A</b> ascending sorts a pair of I4's.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT</b> prints an I4MAT, transposed.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of an I4MAT, transposed.
</li>
<li>
<b>I4VEC_HEAP_D</b> reorders an I4VEC into a descending heap.
</li>
<li>
<b>I4VEC_INDICATOR_NEW</b> sets an I4VEC to the indicator vector.
</li>
<li>
<b>I4VEC_MIN</b> returns the value of the minimum element in an I4VEC.
</li>
<li>
<b>I4VEC_PRINT</b> prints an I4VEC.
</li>
<li>
<b>I4VEC_REVERSE</b> reverses the elements of an I4VEC.
</li>
<li>
<b>I4VEC_SORT_HEAP_A</b> ascending sorts an I4VEC using heap sort.
</li>
<li>
<b>I4VEC_SORTED_UNIQUE</b> finds unique elements in a sorted I4VEC.
</li>
<li>
<b>I4VEC2_COMPARE</b> compares pairs of integers stored in two vectors.
</li>
<li>
<b>I4VEC2_SORT_A</b> ascending sorts a vector of pairs of integers.
</li>
<li>
<b>I4VEC2_SORTED_UNIQUE</b> keeps the unique elements in a array of pairs of integers.
</li>
<li>
<b>LRLINE</b> determines where a point lies in relation to a directed line.
</li>
<li>
<b>LVEC_PRINT</b> prints a logical vector.
</li>
<li>
<b>NODE_MERGE</b> detects nodes that should be merged.
</li>
<li>
<b>NS_ADJ_COL_SET</b> sets the COL array in a Navier Stokes triangulation.
</li>
<li>
<b>NS_ADJ_COUNT</b> counts adjacencies in a Navier Stokes triangulation.
</li>
<li>
<b>NS_ADJ_INSERT</b> inserts an adjacency into a compressed column adjacency matrix.
</li>
<li>
<b>NS_ADJ_ROW_SET</b> sets the Navier Stokes sparse compressed column row indices.
</li>
<li>
<b>PERM_CHECK</b> checks that a vector represents a permutation.
</li>
<li>
<b>PERM_INVERSE</b> inverts a permutation "in place".
</li>
<li>
<b>POINTS_DELAUNAY_NAIVE_2D</b> computes the Delaunay triangulation in 2D.
</li>
<li>
<b>POINTS_HULL_2D</b> computes the convex hull of a set of nodes in 2D.
</li>
<li>
<b>POINTS_POINT_NEAR_NAIVE_ND</b> finds the nearest point to a given point in ND.
</li>
<li>
<b>Q_MEASURE</b> determines the triangulated pointset quality measure Q.
</li>
<li>
<b>QUAD_CONVEX_RANDOM</b> returns a random convex quadrilateral.
</li>
<li>
<b>R4_ABS</b> returns the absolute value of an R4.
</li>
<li>
<b>R4_NINT</b> returns the nearest integer to an R4.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_EPSILON</b> returns the roundoff unit for R8 arithmetic.
</li>
<li>
<b>R8_HUGE</b> returns the largest legal R8.
</li>
<li>
<b>R8_MAX</b> returns the maximum of two R8's.
</li>
<li>
<b>R8_MIN</b> returns the minimum of two R8's.
</li>
<li>
<b>R8_NINT</b> returns the nearest integer to an R8.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</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_PRINT</b> prints an R8MAT, with an optional title.
</li>
<li>
<b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT</b> prints an R8MAT, transposed.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8MAT_UNIFORM_01</b> returns a unit pseudorandom R8MAT.
</li>
<li>
<b>R8TRIS2</b> constructs a Delaunay triangulation of 2D vertices.
</li>
<li>
<b>R8VEC_BRACKET</b> searches a sorted array for successive brackets of a value.
</li>
<li>
<b>R8VEC_MAX</b> returns the value of the maximum element in an R8VEC.
</li>
<li>
<b>R8VEC_MIN</b> returns the value of the minimum element in an R8VEC.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>SORT_HEAP_EXTERNAL</b> externally sorts a list of items into ascending order.
</li>
<li>
<b>SWAPEC</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>TRIANGLE_ANGLES_2D</b> computes the angles of a triangle in 2D.
</li>
<li>
<b>TRIANGLE_AREA_2D</b> computes the area of a triangle in 2D.
</li>
<li>
<b>TRIANGLE_CIRCUMCENTER_2D</b> computes the circumcenter of a triangle in 2D.
</li>
<li>
<b>TRIANGLE_ORDER3_PHYSICAL_TO_REFERENCE</b> maps physical points to reference points.
</li>
<li>
<b>TRIANGLE_ORDER3_REFERENCE_TO_PHYSICAL</b> maps reference points to physical points.
</li>
<li>
<b>TRIANGLE_ORDER6_PHYSICAL_TO_REFERENCE</b> maps a physical point to a reference point.
</li>
<li>
<b>TRIANGLE_ORDER6_REFERENCE_TO_PHYSICAL</b> maps reference points to physical points.
</li>
<li>
<b>TRIANGLE_REFERENCE_SAMPLE</b> returns random points in the reference triangle.
</li>
<li>
<b>TRIANGLE_SAMPLE</b> returns random points in a triangle.
</li>
<li>
<b>TRIANGULATION_DELAUNAY_DISCREPANCY</b> reports if a triangulation is Delaunay.
</li>
<li>
<b>TRIANGULATION_ORDER3_NEIGHBOR_TRIANGLES</b> determines triangle neighbors.
</li>
<li>
<b>TRIANGULATION_NODE_ORDER</b> determines the order of nodes in a triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_ADJ_COUNT</b> counts adjacencies in a triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_ADJ_SET</b> sets adjacencies in a triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_ADJ_SET2</b> sets adjacencies in a triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_BOUNDARY_EDGE_COUNT</b> counts the boundary edges.
</li>
<li>
<b>TRIANGULATION_ORDER3_BOUNDARY_EDGE_COUNT_EULER</b> counts boundary edges.
</li>
<li>
<b>TRIANGULATION_ORDER3_BOUNDARY_NODE</b> indicates nodes on the boundary.
</li>
<li>
<b>TRIANGULATION_ORDER3_CHECK</b> makes some simple checks on a triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_EDGE_CHECK</b> checks the edges of a triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_EXAMPLE1</b> sets up a sample triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_EXAMPLE1_SIZE</b> sets sizes for a sample triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_EXAMPLE2</b> sets up a sample triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_EXAMPLE2_SIZE</b> sets sizes for a sample triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_NEIGHBOR</b> determines a neighbor of a given triangle.
</li>
<li>
<b>TRIANGULATION_ORDER3_NEIGHBOR_NODES</b> determines node neighbors.
</li>
<li>
<b>TRIANGULATION_ORDER3_NEIGHBOR_NODES_PRINT</b> prints a node neighbor array.
</li>
<li>
<b>TRIANGULATION_ORDER3_PLOT</b> plots a triangulation of a set of nodes.
</li>
<li>
<b>TRIANGULATION_ORDER3_PRINT</b> prints information defining a triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_QUAD</b> approximates an integral over a triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_REFINE_COMPUTE</b> computes a refined order 3 triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_REFINE_SIZE</b> sizes a refined order 3 triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_SAMPLE</b> returns random points in a triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER4_PLOT</b> plots a 4-node triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER6_ADJ_COUNT</b> counts adjacencies in a triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER6_ADJ_SET</b> sets adjacencies in a triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER6_BOUNDARY_EDGE_COUNT</b> counts the boundary edges.
</li>
<li>
<b>TRIANGULATION_ORDER6_BOUNDARY_EDGE_COUNT_EULER</b> counts boundary edges.
</li>
<li>
<b>TRIANGULATION_ORDER6_BOUNDARY_NODE</b> indicates nodes on the boundary.
</li>
<li>
<b>TRIANGULATION_ORDER6_EXAMPLE1</b> sets up a sample triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER6_EXAMPLE1_SIZE</b> sets sizes for a sample triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER6_EXAMPLE2</b> sets up a sample triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER6_EXAMPLE2_SIZE</b> sets sizes for a sample triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER6_NEIGHBOR</b> determines a neighbor of a given triangle.
</li>
<li>
<b>TRIANGULATION_ORDER6_PLOT</b> plots a 6-node triangulation of a set of nodes.
</li>
<li>
<b>TRIANGULATION_ORDER6_PRINT</b> prints information defining a triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER6_REFINE_COMPUTE</b> computes a refined order 6 triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER6_REFINE_SIZE</b> sizes a refined order 6 triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER6_TO_ORDER3</b> linearizes a quadratic triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER6_VERTEX_COUNT</b> counts vertex nodes in a triangulation.
</li>
<li>
<b>TRIANGULATION_SEARCH_DELAUNAY</b> searches a triangulation for a point.
</li>
<li>
<b>TRIANGULATION_SEARCH_NAIVE</b> naively searches a triangulation for a point.
</li>
<li>
<b>VBEDG</b> determines which boundary edges are visible to a point.
</li>
<li>
<b>VORONOI_POLYGON_AREA</b> computes the area of a Voronoi polygon.
</li>
<li>
<b>VORONOI_POLYGON_CENTROID_2D</b> computes the centroid of a Voronoi polygon.
</li>
<li>
<b>VORONOI_POLYGON_VERTICES_2D</b> computes the vertices of a Voronoi polygon.
</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 11 September 2009.
</i>
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