TRIANGULATION
Triangulation of 2D data

TRIANGULATION is a MATLAB 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;
create a "node neighbor array" for each node;
create an "element neighbor array" for each element;
estimate the integral of a function over the region covered by the mesh;
plot the nodes and elements of a mesh;
determine the parts of the mesh that lie on the boundary;
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 FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CVT_TRIANGULATION, a FORTRAN90 program which uses routines from the TEST_TRIANGULATION library to create a CVT-based triangularization.

DISTMESH, a MATLAB program which takes the definition of a 2D region, and fills it up with a set of nodes, and triangulates those nodes to make a triangulation of the region. The region may be nonconvex and may include holes; the user may request a specific density for the nodes, and may require certain points to be in the set of nodes.

FEM2D_SAMPLE, a MATLAB library which evaluates a finite element function defined on an order 3 or order 6 triangulation.

HEX_GRID_TRIANGULATE, a FORTRAN90 program which uses this library of test regions and computes points on a hexagonal grid inside each region.

MESH_BANDWIDTH, a MATLAB program which returns the geometric bandwidth associated with a mesh of elements of any order and in a space of arbitrary dimension.

MESH2D, a MATLAB library which can automatically create a triangular mesh for a given polygonal region, by Darren Engwirda.

TABLE_DELAUNAY, a MATLAB program which triangulates a set of nodes whose coordinates are stored in a file.

TEST_TRIANGULATION, a MATLAB library which can set up a number of triangulation test problems.

TRIANGLE, a C program which computes a triangulation of a geometric region, by Jonathan Shewchuk.

TRIANGULATION_BOUNDARY_NODES, a MATLAB program which reads data defining a triangulation, determines which nodes lie on the boundary, and writes their coordinates to a file.

TRIANGULATION_CORNER, a MATLAB program which patches triangulations so that no triangle has two sides on the boundary.

TRIANGULATION_DELAUNAY_DISCREPANCY, a MATLAB program which measures the amount by which a triangulation fails the local Delaunay test;

TRIANGULATION_DISPLAY, a MATLAB program which displays the nodes and elements of a triangulation on the MATLAB graphics screen;

TRIANGULATION_DISPLAY_OPENGL, a C++ program which reads files defining a triangulation and displays an image using Open GL.

TRIANGULATION_HISTOGRAM, a MATLAB program which computes histograms of data over a triangulation.

TRIANGULATION_L2Q, a MATLAB program which reads data defining a 3-node triangulation and generates midside nodes and writes out the corresponding 6-node triangulation.

TRIANGULATION_MASK, a MATLAB program which takes an existing triangulation and deletes triangles and their corresponding nodes as requested by the user.

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 MATLAB 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 MATLAB program that reads data defining a triangulation and creates a PostScript image of the nodes and triangles.

TRIANGULATION_Q2L, a MATLAB 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 MATLAB program which estimates the integral of a function over a triangulated region.

TRIANGULATION_QUALITY, a MATLAB program which reads data defining a triangulation and computes a number of quality measures.

TRIANGULATION_RCM, a MATLAB 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 MATLAB 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 MATLAB program which reads data defining a triangulation, determines the neighboring triangles of each triangle, and writes that information to a file.

Reference:

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, Number 5, 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:

alpha_measure.m, measures the triangulated point set quality measure ALPHA.
angle_rad_2d.m, returns the angle swept out by two rays in 2D;
area_measure.m, measures the triangulated point set area ratio quality measure.
bandwidth.m, determines the bandwidth associated with the finite element mesh;
delaunay_swap_test.m, chooses a diagonal edge;
diaedg.m, chooses a diagonal edge;
get_seed.m, chooses an arbitrary seed for the random number generator;
i4_modp.m, returns the nonnegative remainder of I4 division;
i4_sign.m, returns the sign of an I4;
i4_swap.m, swaps two I4's;
i4_uniform.m, returns a random I4 in a given range;
i4_wrap.m, forces an I4 to lie in a given range;
i4col_compare.m, compares two columns of an I4COL;
i4col_sort_a.m, ascending sorts the columns of an I4COL;
i4col_sorted_unique_count.m, counts the unique columns in a sorted I4COL;
i4col_swap.m, swaps two columns of an I4COL;
i4i4_sort_a.m, ascending sorts a pair of I4's;
i4mat_transpose_print.m, prints an I4MAT, transposed;
i4mat_transpose_print_some.m, prints some of an I4MAT, transposed;
i4vec_heap_d.m, reorders an I4VEC into a descending heap;
i4vec_indicator.m, sets an I4VEC to the indicator vector;
i4vec_print.m, prints an I4VEC;
i4vec_sort_heap_a.m, ascending sorts an I4VEC using heap sort;
i4vec_sorted_unique.m, returns the unique elements in a sorted I4VEC;
i4vec2_compare.m, compares pairs of elements in an I4VEC2;
i4vec_sorta.m, ascending sorts an I4VEC2;
i4vec2_sorted_unique.m, returns the unique elements in a sorted I4VEC2;
lrline.m, determines if a point is left or right of a directed line;
lvec_print.m, prints an LVEC.
mesh_base_one.m detects and corrects 0-based indexing.
node_merge.m, detects nodes that should be merged.
ns_adj_col_set.m, sets up the column vector associated with the sparse compressed column storage of the variable adjacencies in a Taylor-Hood order 3/order 6 triangulation for Navier Stokes velocity and pressure.
ns_adj_count.m, counts the variable adjacencies in a Taylor-Hood order 3/order 6 triangulation for Navier Stokes velocity and pressure.
ns_adj_insert.m, inserts an adjacency into a compressed column adjacency matrix.
ns_adj_row_set.m, sets the Navier Stokes sparse compressed column row indices.
perm_check.m, checks that a permutation of 1:N is valid;
perm_check2.m, checks that a permutation of BASE:BASE+N-1 is valid;
perm_inverse.m, computes the inverse of a permutation;
points_delaunay_naive_2d.m, a "naive" algorithm for computing the Delaunay triangulation of a set of points in 2D;
points_hull_2d.m, computes the convex hull of a set of points in 2D;
points_point_near_naive_nd.m, finds the nearest of a set of points to a point in ND;
q_measure.m, measures the triangulated point set quality measure Q.
quad_convex_random.m, returns a random convex quadrilateral in the unit square.
r4_uniform_01.m, returns a random R4 in [0,1];
r8_acos.m, computes the inverse cosine;
r8_epsilon.m, returns the machine precision for R8's;
r8_huge.m, returns a huge R8;
r8_uniform_01.m, returns a random R8 in [0,1];
r82vec_permute.m, permutes an R82VEC in place;
r82vec_sort_heap_index_a.m, does an indexed heap ascending sort of an R82VEC;
r8mat_print.m, prints an R8MAT;
r8mat_print_some.m, prints some of an R8MAT;
r8mat_transpose_print.m, prints the transpose of an R8MAT;
r8mat_transpose_print_some.m, prints some of the transpose of an R8MAT;
r8mat_uniform_01.m, returns a unit pseudorandom R8MAT.
r8tris2.m, computes the Delaunay triangulation of a set of points in 2D;
r8vec_bracket.m, searches a sorted R8VEC for brackets of a value;
s_len_trim.m, returns the length of a string to the last nonblank;
sort_heap_external.m, externally sorts a list of values into ascending order;
swapec.m, swaps diagonal edges until all triangles are Delaunay;
timestamp.m, prints the current YMDHMS date as a timestamp;
triangle_angles_2d.m, returns the angles of a triangle in 2D;
triangle_area_2d.m, returns the area of a triangle in 2D;
triangle_circumcenter_2d.m, returns the circumcenter of a triangle in 2D;
triangle_reference_sample.m, randomly samples a point from the reference triangle;
triangle_sample.m, randomly samples a point from a triangle;
triangulation_area.m returns the area of a triangulation.
triangulation_areas.m returns the triangle areas in a triangulation.
triangulation_delaunay_discrepancy_compute.m reports if a triangulation is Delaunay.
triangulation_neighhbor_elements.m, determines element neighbors in a triangulation;
triangulation_node_order.m, determines the order of the nodes in a triangulation;
triangulation_order3_adj_count.m, counts adjacencies in an order 3 triangulation;
triangulation_order3_adj_set.m, records adjacencies in an order 3 triangulation;
triangulation_order3_adj_set2.m, records adjacencies in an order 3 triangulation;
triangulation_order3_boundary_edge_count.m, determines the number of boundary edges in an order 3 triangulation;
triangulation_order3_boundary_edge_count_euler.m, uses Euler's formula to determine the number of boundary edges in an order 3 triangulation;
triangulation_order3_boundary_node.m, determines the boundary nodes in an order 3 triangulation;
triangulation_order3_check.m, carries out some simple checks on an order 3 triangulation;
triangulation_order3_edge_check.m, checks the edges of an order 3 triangulation;
triangulation_order3_example1.m, returns an example order 3 triangulation;
triangulation_order3_example_size.m, returns the sizes of arrays in an example order 3 triangulation;
triangulation_order3_example.m, returns an example order 3 triangulation;
triangulation_order3_example_size.m, returns the sizes of arrays in an example order 3 triangulation;
triangulation_order3_neighhbor.m, determines a neighbor of a given triangle in an order 3 triangulation;
triangulation_order3_neighhbor_nodes.m, determines the neighbors of nodes in an order 3 triangulation;
triangulation_order3_neighhbor_nodes_print.m, prints information about an order 3 triangulation neighbor nodes;
triangulation_order3_plot.m, creates a EPS image of an order 3 triangulation;
triangulation_order3_print.m, prints information about an order 3 triangulation;
triangulation_order3_quad.m, approximates the integral of a function over an order 3 triangulation.
triangulation_order3_refine_compute.m, computes a refinement of an order3 mesh
triangulation_order3_refine_size.m, sizes a refinement of an order3 mesh;
triangulation_order3_sample.m, returns random points in an order 3 triangulation;
triangulation_order4_plot.m, creates a EPS image of an order 4 triangulation;
triangulation_order6_adj_count.m, counts adjacencies in an order 6 triangulation;
triangulation_order6_adj_set.m, records adjacencies in an order 6 triangulation;
triangulation_order6_boundary_edge_count.m, determines the number of boundary edges in an order 6 triangulation;
triangulation_order6_boundary_edge_count_euler.m, uses Euler's formula to determine the number of boundary edges in an order 6 triangulation;
triangulation_order6_boundary_node.m, determines the boundary nodes in an order 6 triangulation;
triangulation_order6_example1.m, returns an example order 6 triangulation;
triangulation_order6_example1_size.m, returns the sizes of arrays in an example order 6 triangulation;
triangulation_order6_example2.m, returns an example order 6 triangulation;
triangulation_order6_example2_size.m, returns the sizes of arrays in an example order 6 triangulation;
triangulation_order6_neighhbor.m, determines a neighbor of a given triangle in an order 6 triangulation;
triangulation_order6_plot.m, creates a EPS image of an order 6 triangulation;
triangulation_order6_print.m, prints information about an order 6 triangulation;
triangulation_order6_refine_compute.m, computes a refinement of an order6 mesh
triangulation_order6_refine_size.m, sizes a refinement of an order6 mesh;
triangulation_order6_to_order3.m, converts an order 6 triangulation to an order 3 triangulation;
triangulation_order6_vertex_count.m, counts the number of vertex and midside nodes in an order 6 triangulation;
triangulation_search_delaunay.m, finds the triangle containing a point, using a Delaunay search;
triangulation_search_naive.m, finds the triangle containing a point, using a naive search;
vbedg.m, determines which boundary edges are visible to a point;
voronoi_polygon_area.m, computes the area of a Voronoi polygon.
voronoi_polygon_centroid.m, computes the centroid of a Voronoi polygon.
voronoi_polygon_vertices.m, computes the vertices of a Voronoi polygon.

Examples and Tests:

triangulation_test.m, calls all the other test routines;
triangulation_test_output.txt, the output ffile;
triangulation_test01.m, tests ALPHA_MEASURE;
triangulation_test02.m, tests AREA_MEASURE;
triangulation_test025.m, tests DELAUNAY_SWAP_TEST;
triangulation_test026.m, tests DIAEDG;
triangulation_test03.m, tests NODE_MERGE;
triangulation_test04.m, tests NS_ADJ_COL_SET, NS_ADJ_COUNT and NS_ADJ_ROW_SET;
triangulation_test05.m, tests POINTS_DELAUNAY_NAIVE_2D;
triangulation_test06.m, tests POINTS_HULL_2D;
triangulation_test07.m, tests Q_MEASURE;
triangulation_test08.m, tests R8TRIS2;
triangulation_test09.m, tests TRIANGLE_ORDER3_PHYSICAL_TO_REFERENCE, TRIANGLE_ORDER3_REFERENCE_TO_PHYSICAL;
triangulation_test10.m, tests TRIANGLE_ORDER6_PHYSICAL_TO_REFERENCE, TRIANGLE_ORDER6_REFERENCE_TO_PHYSICAL;
triangulation_test11.m, tests TRIANGULATION_NODE_ORDER;
triangulation_test12.m, tests TRIANGULATION_ORDER3_ADJ_COUNT and TRIANGULATION_ORDER3_ADJ_SET;
triangulation_test125.m, tests TRIANGULATION_ORDER3_ADJ_COUNT and TRIANGULATION_ORDER3_ADJ_SET2;
triangulation_test13.m, tests TRIANGULATION_ORDER3_BOUNDARY_EDGE_COUNT and TRIANGULATION_ORDER3_PLOT.
triangulation_test14.m, tests TRIANGULATION_ORDER3_BOUNDARY_EDGE_COUNT_EULER;
triangulation_test15.m, tests TRIANGULATION_ORDER3_BOUNDARY_NODE;
triangulation_test16.m, tests TRIANGULATION_ORDER3_CHECK;
triangulation_test17.m, tests TRIANGULATION_ORDER3_EXAMPLE1 and TRIANGULATION_ORDER3_EXAMPLE1_SIZE;
triangulation_test18.m, tests TRIANGULATION_ORDER3_NEIGHBOR;
triangulation_test19.m, tests TRIANGULATION_NEIGHBOR_ELEMENTS.
triangulation_test20.m, tests TRIANGULATION_ORDER3_PLOT;
triangulation_test21.m, tests TRIANGULATION_ORDER3_PRINT;
triangulation_test213.m, tests TRIANGULATION_ORDER3_QUAD;
triangulation_test215.m, tests TRIANGULATION_ORDER3_REFINE_COMPUTE and TRIANGULATION_ORDER3_REFINE_SIZE;
triangulation_test217.m, tests TRIANGULATION_SEARCH_DELAUNAY.
triangulation_test219.m, tests TRIANGULATION_SEARCH_DELAUNAY and TRIANGULATION_SEARCH_NAIVE.
triangulation_test22.m, tests TRIANGULATION_ORDER6_ADJ_COUNT and TRIANGULATION_ORDER6_ADJ_SET;
triangulation_test23.m, tests TRIANGULATION_ORDER6_BOUNDARY_EDGE_COUNT..
triangulation_test24.m, tests TRIANGULATION_ORDER6_BOUNDARY_EDGE_COUNT_EULER;
triangulation_test25.m, tests TRIANGULATION_ORDER6_BOUNDARY_NODE and TRIANGULATION_ORDER6_PLOT.
triangulation_test26.m, tests TRIANGULATION_ORDER6_PRINT;
triangulation_test265.m, tests TRIANGULATION_ORDER3_REFINE_COMPUTE and TRIANGULATION_ORDER3_REFINE_SIZE;
triangulation_test27.m, tests TRIANGULATION_ORDER6_VERTEX_COUNT;
quad_fun.m, a sample integrand function for TRIANGULATION_QUAD;
triangulation_test31.m, tests VORONOI_POLYGON_AREA;
triangulation_test32.m, tests VORONOI_POLYGON_CENTROID;
triangulation_test33.m, tests VORONOI_POLYGON_VERTICES;
ns_triangulation.png, a PNG image of a "tiny" triangulation used for the Navier-Stokes examples.
triangulation_order3_plot.png, a PNG image of a triangulation.
triangulation_order3_plot2.png, a PNG image of a triangulation.
triangulation_order6_plot.png, a PNG image of a triangulation.

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Last revised 29 August 2011.