TRIANGULATION
Triangulation of 2D data
TRIANGULATION
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.
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".
Routines are available to:
-
evaluate "quality measures" for the mesh;
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create a "node neighbor array" for each node;
-
create an "element neighbor array" for each element;
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estimate the integral of a function over the region covered by the mesh;
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plot the nodes and elements of a mesh;
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determine the parts of the mesh that lie on the boundary;
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sample points at random from the region covered by the mesh;
-
search a mesh to determine which element contains a point.
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.
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
Languages:
TRIANGULATION is available in
a C version and
a C++ version and
a FORTRAN90 version and
a MATLAB version.
Related Programs:
MESH_TO_XML,
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.
TABLE_DELAUNAY,
a C++ program which
reads a file of 2d point coordinates and computes the Delaunay triangulation.
TRIANGLE,
a C program which
computes a triangulation of a geometric region.
TRIANGULATION_BOUNDARY_NODES,
a C++ program which
reads data defining a triangulation, determines which nodes
lie on the boundary, and writes their coordinates to a file.
TRIANGULATION_CORNER,
a C++ program which
patches triangulations so that no triangle has two sides on the boundary.
TRIANGULATION_DELAUNAY_DISCREPANCY,
a C++ program which
measures the amount by which a triangulation fails the local Delaunay test;
TRIANGULATION_DISPLAY_OPENGL,
a C++ program which
reads files defining a triangulation and displays an image using Open GL.
TRIANGULATION_HISTOGRAM,
a C++ program which
computes histograms of data over a triangulation.
TRIANGULATION_L2Q,
a C++ program which
reads data defining a 3-node triangulation and generates
midside nodes and writes out the corresponding 6-node triangulation.
TRIANGULATION_MASK,
a C++ program, which
takes an existing triangulation and deletes triangles and
their corresponding nodes as requested by the user.
TRIANGULATION_NODE_TO_ELEMENT,
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.
TRIANGULATION_ORDER3,
a directory which
contains a description and
examples of order 3 triangulations.
TRIANGULATION_ORDER6,
a directory which
contains a description and
examples of order 6 triangulations.
TRIANGULATION_ORIENT,
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.
TRIANGULATION_PLOT,
a C++ program which
reads data defining a triangulation and creates a
PostScript image of the nodes and triangles.
TRIANGULATION_Q2L,
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.
TRIANGULATION_QUAD,
a C++ program which
estimates the integral of a function over a triangulated region.
TRIANGULATION_QUALITY,
a C++ program which
reads data defining a triangulation and computes a number
of quality measures.
TRIANGULATION_RCM,
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.
TRIANGULATION_REFINE,
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.
TRIANGULATION_TRIANGLE_NEIGHBORS,
a C++ program which
reads data defining a triangulation, determines the neighboring
triangles of each triangle, and writes that information to a file.
References:
-
Franz Aurenhammer,
Voronoi diagrams -
a study of a fundamental geometric data structure,
ACM Computing Surveys,
Volume 23, Number 3, September 1991, pages 345-405.
-
Paul Bratley, Bennett Fox, Linus Schrage,
A Guide to Simulation,
Second Edition,
Springer, 1987,
ISBN: 0387964673.
-
Marc deBerg, Marc Krevald, Mark Overmars,
Otfried Schwarzkopf,
Computational Geometry,
Springer, 2000,
ISBN: 3-540-65620-0.
-
Barry Joe,
GEOMPACK - a software package for the generation of meshes
using geometric algorithms,
Advances in Engineering Software,
Volume 13, 1991, pages 325-331.
-
Albert Nijenhuis, Herbert Wilf,
Combinatorial Algorithms for Computers and Calculators,
Second Edition,
Academic Press, 1978,
ISBN: 0-12-519260-6,
LC: QA164.N54.
-
Atsuyuki Okabe, Barry Boots, Kokichi Sugihara, Sung Nok Chiu,
Spatial Tessellations:
Concepts and Applications of Voronoi Diagrams,
Second Edition,
Wiley, 2000,
ISBN: 0-471-98635-6,
LC: QA278.2.O36.
-
Joseph ORourke,
Computational Geometry,
Second Edition,
Cambridge, 1998,
ISBN: 0521649765,
LC: QA448.D38.
-
Per-Olof Persson, Gilbert Strang,
A Simple Mesh Generator in MATLAB,
SIAM Review,
Volume 46, Number 2, June 2004, pages 329-345.
Source Code:
Examples and Tests:
List of Routines:
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ALPHA_MEASURE determines the triangulated pointset quality measure ALPHA.
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ANGLE_RAD_2D returns the angle in radians swept out between two rays in 2D.
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ARC_COSINE computes the arc cosine function, with argument truncation.
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AREA_MEASURE determines the area ratio quality measure.
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BANDWIDTH determines the bandwidth associated with the finite element mesh.
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DELAUNAY_SWAP_TEST performs the Delaunay swap test.
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DIAEDG chooses a diagonal edge.
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GET_SEED returns a random seed for the random number generator.
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I4_MAX returns the maximum of two I4's.
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I4_MIN returns the smaller of two I4's.
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I4_MODP returns the nonnegative remainder of I4 division.
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I4_POWER returns the value of I^J.
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I4_SIGN returns the sign of an I4.
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I4_SWAP switches two I4's.
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I4_UNIFORM returns a scaled pseudorandom I4.
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I4_WRAP forces an I4 to lie between given limits by wrapping.
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I4COL_COMPARE compares columns I and J of an I4COL.
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I4COL_SORT_A ascending sorts an I4COL.
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I4COL_SORTED_UNIQUE_COUNT counts unique elements in an I4COL.
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I4COL_SWAP swaps two columns of an I4COL.
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I4I4_SORT_A ascending sorts a pair of I4's.
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I4MAT_TRANSPOSE_PRINT prints an I4MAT, transposed.
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I4MAT_TRANSPOSE_PRINT_SOME prints some of an I4MAT, transposed.
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I4VEC_HEAP_D reorders an I4VEC into a descending heap.
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I4VEC_INDICATOR_NEW sets an I4VEC to the indicator vector.
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I4VEC_MIN returns the value of the minimum element in an I4VEC.
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I4VEC_PRINT prints an I4VEC.
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I4VEC_REVERSE reverses the elements of an I4VEC.
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I4VEC_SORT_HEAP_A ascending sorts an I4VEC using heap sort.
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I4VEC_SORTED_UNIQUE finds unique elements in a sorted I4VEC.
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I4VEC2_COMPARE compares pairs of integers stored in two vectors.
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I4VEC2_SORT_A ascending sorts a vector of pairs of integers.
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I4VEC2_SORTED_UNIQUE keeps the unique elements in a array of pairs of integers.
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LRLINE determines where a point lies in relation to a directed line.
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LVEC_PRINT prints a logical vector.
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NODE_MERGE detects nodes that should be merged.
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NS_ADJ_COL_SET sets the COL array in a Navier Stokes triangulation.
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NS_ADJ_COUNT counts adjacencies in a Navier Stokes triangulation.
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NS_ADJ_INSERT inserts an adjacency into a compressed column adjacency matrix.
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NS_ADJ_ROW_SET sets the Navier Stokes sparse compressed column row indices.
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PERM_CHECK checks that a vector represents a permutation.
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PERM_INVERSE inverts a permutation "in place".
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POINTS_DELAUNAY_NAIVE_2D computes the Delaunay triangulation in 2D.
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POINTS_HULL_2D computes the convex hull of a set of nodes in 2D.
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POINTS_POINT_NEAR_NAIVE_ND finds the nearest point to a given point in ND.
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Q_MEASURE determines the triangulated pointset quality measure Q.
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QUAD_CONVEX_RANDOM returns a random convex quadrilateral.
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R4_ABS returns the absolute value of an R4.
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R4_NINT returns the nearest integer to an R4.
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R8_ABS returns the absolute value of an R8.
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R8_EPSILON returns the roundoff unit for R8 arithmetic.
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R8_HUGE returns the largest legal R8.
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R8_MAX returns the maximum of two R8's.
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R8_MIN returns the minimum of two R8's.
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R8_NINT returns the nearest integer to an R8.
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R8_UNIFORM_01 returns a unit pseudorandom R8.
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R82VEC_PERMUTE permutes an R82VEC in place.
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R82VEC_SORT_HEAP_INDEX_A does an indexed heap ascending sort of an R82VEC.
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R8MAT_PRINT prints an R8MAT, with an optional title.
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R8MAT_PRINT_SOME prints some of an R8MAT.
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R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
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R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
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R8MAT_UNIFORM_01 returns a unit pseudorandom R8MAT.
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R8TRIS2 constructs a Delaunay triangulation of 2D vertices.
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R8VEC_BRACKET searches a sorted array for successive brackets of a value.
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R8VEC_MAX returns the value of the maximum element in an R8VEC.
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R8VEC_MIN returns the value of the minimum element in an R8VEC.
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S_LEN_TRIM returns the length of a string to the last nonblank.
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SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
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SWAPEC swaps diagonal edges until all triangles are Delaunay.
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TIMESTAMP prints the current YMDHMS date as a time stamp.
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TRIANGLE_ANGLES_2D computes the angles of a triangle in 2D.
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TRIANGLE_AREA_2D computes the area of a triangle in 2D.
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TRIANGLE_CIRCUMCENTER_2D computes the circumcenter of a triangle in 2D.
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TRIANGLE_ORDER3_PHYSICAL_TO_REFERENCE maps physical points to reference points.
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TRIANGLE_ORDER3_REFERENCE_TO_PHYSICAL maps reference points to physical points.
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TRIANGLE_ORDER6_PHYSICAL_TO_REFERENCE maps a physical point to a reference point.
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TRIANGLE_ORDER6_REFERENCE_TO_PHYSICAL maps reference points to physical points.
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TRIANGLE_REFERENCE_SAMPLE returns random points in the reference triangle.
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TRIANGLE_SAMPLE returns random points in a triangle.
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TRIANGULATION_DELAUNAY_DISCREPANCY reports if a triangulation is Delaunay.
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TRIANGULATION_ORDER3_NEIGHBOR_TRIANGLES determines triangle neighbors.
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TRIANGULATION_NODE_ORDER determines the order of nodes in a triangulation.
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TRIANGULATION_ORDER3_ADJ_COUNT counts adjacencies in a triangulation.
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TRIANGULATION_ORDER3_ADJ_SET sets adjacencies in a triangulation.
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TRIANGULATION_ORDER3_ADJ_SET2 sets adjacencies in a triangulation.
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TRIANGULATION_ORDER3_BOUNDARY_EDGE_COUNT counts the boundary edges.
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TRIANGULATION_ORDER3_BOUNDARY_EDGE_COUNT_EULER counts boundary edges.
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TRIANGULATION_ORDER3_BOUNDARY_NODE indicates nodes on the boundary.
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TRIANGULATION_ORDER3_CHECK makes some simple checks on a triangulation.
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TRIANGULATION_ORDER3_EDGE_CHECK checks the edges of a triangulation.
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TRIANGULATION_ORDER3_EXAMPLE1 sets up a sample triangulation.
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TRIANGULATION_ORDER3_EXAMPLE1_SIZE sets sizes for a sample triangulation.
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TRIANGULATION_ORDER3_EXAMPLE2 sets up a sample triangulation.
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TRIANGULATION_ORDER3_EXAMPLE2_SIZE sets sizes for a sample triangulation.
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TRIANGULATION_ORDER3_NEIGHBOR determines a neighbor of a given triangle.
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TRIANGULATION_ORDER3_NEIGHBOR_NODES determines node neighbors.
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TRIANGULATION_ORDER3_NEIGHBOR_NODES_PRINT prints a node neighbor array.
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TRIANGULATION_ORDER3_PLOT plots a triangulation of a set of nodes.
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TRIANGULATION_ORDER3_PRINT prints information defining a triangulation.
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TRIANGULATION_ORDER3_QUAD approximates an integral over a triangulation.
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TRIANGULATION_ORDER3_REFINE_COMPUTE computes a refined order 3 triangulation.
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TRIANGULATION_ORDER3_REFINE_SIZE sizes a refined order 3 triangulation.
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TRIANGULATION_ORDER3_SAMPLE returns random points in a triangulation.
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TRIANGULATION_ORDER4_PLOT plots a 4-node triangulation.
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TRIANGULATION_ORDER6_ADJ_COUNT counts adjacencies in a triangulation.
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TRIANGULATION_ORDER6_ADJ_SET sets adjacencies in a triangulation.
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TRIANGULATION_ORDER6_BOUNDARY_EDGE_COUNT counts the boundary edges.
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TRIANGULATION_ORDER6_BOUNDARY_EDGE_COUNT_EULER counts boundary edges.
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TRIANGULATION_ORDER6_BOUNDARY_NODE indicates nodes on the boundary.
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TRIANGULATION_ORDER6_EXAMPLE1 sets up a sample triangulation.
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TRIANGULATION_ORDER6_EXAMPLE1_SIZE sets sizes for a sample triangulation.
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TRIANGULATION_ORDER6_EXAMPLE2 sets up a sample triangulation.
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TRIANGULATION_ORDER6_EXAMPLE2_SIZE sets sizes for a sample triangulation.
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TRIANGULATION_ORDER6_NEIGHBOR determines a neighbor of a given triangle.
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TRIANGULATION_ORDER6_PLOT plots a 6-node triangulation of a set of nodes.
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TRIANGULATION_ORDER6_PRINT prints information defining a triangulation.
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TRIANGULATION_ORDER6_REFINE_COMPUTE computes a refined order 6 triangulation.
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TRIANGULATION_ORDER6_REFINE_SIZE sizes a refined order 6 triangulation.
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TRIANGULATION_ORDER6_TO_ORDER3 linearizes a quadratic triangulation.
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TRIANGULATION_ORDER6_VERTEX_COUNT counts vertex nodes in a triangulation.
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TRIANGULATION_SEARCH_DELAUNAY searches a triangulation for a point.
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TRIANGULATION_SEARCH_NAIVE naively searches a triangulation for a point.
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VBEDG determines which boundary edges are visible to a point.
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VORONOI_POLYGON_AREA computes the area of a Voronoi polygon.
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VORONOI_POLYGON_CENTROID_2D computes the centroid of a Voronoi polygon.
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VORONOI_POLYGON_VERTICES_2D computes the vertices of a Voronoi polygon.
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the C++ source codes.
Last revised 11 September 2009.