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.

Routines are available to

• evaluate "quality measures" for the triangulation;
• create a "node neighbor array" for each node;
• create a "triangle neighbor array" for each triangle;
• estimate the integral of a function over a triangulation;
• plot the nodes and triangles of a triangulation;
• determine the parts of the triangulation that lie on the boundary;
• sample points at random from the total triangulated region;
• search a triangulation to determine which triangle 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.

Related Data and Programs:

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

DISTMESH is 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.

DUNAVANT is a MATLAB library of routines for defining Dunavant rules for quadrature on a triangle.

FEKETE is a MATLAB library of routines for defining a Fekete rule for quadrature or interpolation over a triangle.

FEM_SAMPLE is a MATLAB library of routines for evaluating a finite element function defined on an order 3 or order 6 triangulation.

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

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

NCC_TRIANGLE is a MATLAB library defining Newton-Cotes closed quadrature rules on a triangle.

NCO_TRIANGLE is a MATLAB library defining Newton-Cotes open quadrature rules on a triangle.

QUADRATURE_RULES_TRI is a collection of files defining quadrature rules to be applied to triangular regions.

STROUD is a MATLAB library of routines which includes some quadrature rules for triangles.

TABLE_DELAUNAY is an executable FORTRAN90 program for the triangulation of a set of nodes whose coordinates are stored in a file.

TEST_TRI_INT is a FORTRAN90 library of functions that can be used to test algorithms for quadrature over a triangle.

TEST_TRIANGULATION is a FORTRAN90 library of routines which can set up a number of triangulation test problems.

TOMS612 is a FORTRAN77 library of routines which can estimate the integral of a function over a triangle.

TOMS706 is a FORTRAN77 library which estimates the integral of a function over a triangulated region.

TRIANGLE is an executable C program which computes a triangulation of a geometric region.

TRIANGULATION is available in a C++ version and a FORTRAN90 version and a MATLAB version.

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

TRIANGULATION_DISPLAY_OPEN_GL is an executable C++ program which reads files defining a triangulation and displays an image using Open GL.

TRIANGULATION_L2Q is an executable MATLAB program that reads data defining a 3-node triangulation and generates midside nodes and writes out the corresponding 6-node triangulation.

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

TRIANGULATION_ORDER3 is a directory which contains a description and examples of order 3 triangulations.

TRIANGULATION_ORDER6 is a directory which contains a description and examples of order 6 triangulations.

TRIANGULATION_ORIENT is an executable MATLAB program that reads data defining a triangulation, makes sure that every triangle has positive orientation, and if not, writes a corrected triangle file.

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

TRIANGULATION_Q2L is an executable MATLAB program that reads data defining a 6-node triangulation, and subdivides each triangle into 4 3-node triangles, writing the resulting triangulation to a file.

TRIANGULATION_QUALITY is an executable MATLAB program that reads data defining a triangulation and computes a number of quality measures.

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

WANDZURA is a MATLAB library of routines for defining Wandzura rules for quadrature on a triangle.

Reference:

1. Franz Aurenhammer,
   Voronoi diagrams - a study of a fundamental geometric data structure,
   ACM Computing Surveys,
   Volume 23, Number 3, September 1991, pages 345-405.
2. Paul Bratley, Bennett Fox, Linus Schrage,
   A Guide to Simulation,
   Second Edition,
   Springer, 1987,
   ISBN: 0387964673.
3. Marc deBerg, Marc Krevald, Mark Overmars, Otfried Schwarzkopf,
   Computational Geometry,
   Springer, 2000,
   ISBN: 3-540-65620-0.
4. 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.
5. Albert Nijenhuis, Herbert Wilf,
   Combinatorial Algorithms for Computers and Calculators,
   Second Edition,
   Academic Press, 1978,
   ISBN: 0-12-519260-6,
   LC: QA164.N54.
6. Atsuyuki Okabe, Barry Boots, Kokichi Sugihara, Sung Nok Chiu,
   Spatial Tesselations: Concepts and Applications of Voronoi Diagrams,
   Second Edition,
   Wiley, 2000,
   ISBN: 0-471-98635-6,
   LC: QA278.2.O36.
7. Joseph ORourke,
   Computational Geometry,
   Second Edition,
   Cambridge, 1998,
   ISBN: 0521649765,
   LC: QA448.D38.
8. Per-Olof Persson, Gilbert Strang,
   A Simple Mesh Generator in MATLAB,
   SIAM Review,
   Volume 46, Number 2, June 2004, pages 329-345.

Tar File:

A GZIP'ed TAR file of the contents of this directory is available. This is only done as a convenience for users who want ALL the files, and don't want to download them individually. This is not a convenience for me, so don't be surprised if the tar file is somewhat out of date.

• triangulation_m_src.tar.csh the commands used to create the GZIP'ed TAR file.
• triangulation_m_src.tar.gz the GZIP'ed TAR file.

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;
• arc_cosine.m, computes the inverse cosine of C;
• area_measure.m, measures the triangulated point set area ratio quality measure.
• bandwidth.m, determines the bandwidth associated with the finite element mesh;
• 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.
• 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 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.
• r4_uniform_01.m, returns a random R4 in [0,1];
• 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;
• 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;
• timestring.m, returns the current YMDHMS date as a string;
• 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_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_neighhbor_triangles.m, determines triangle neighbors in an order 3 triangulation;
• 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_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_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_neighhbor_triangles.m, determines triangle neighbors 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_quad.m, approximates the integral of a function over a triangulated region.
• triangulation_search.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.out, output from a run of the test routines;
• triangulation_test01.m, tests ALPHA_MEASURE;
• triangulation_test02.m, tests AREA_MEASURE;
• 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_ORDER3_NEIGHBOR_TRIANGLES.
• triangulation_test20.m, tests TRIANGULATION_ORDER3_PLOT;
• triangulation_test21.m, tests TRIANGULATION_ORDER3_PRINT;
• triangulation_test215.m, tests TRIANGULATION_ORDER3_REFINE_COMPUTE and TRIANGULATION_ORDER3_REFINE_SIZE;
• 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;
• triangulation_test28.m, tests TRIANGULATION_QUAD;
• quad_fun.m, a sample integrand function for TRIANGULATION_QUAD;
• triangulation_test29.m, tests TRIANGULATION_SEARCH.
• triangulation_test30.m, tests TRIANGULATION_SEARCH and TRIANGULATION_SEARCH_NAIVE.
• 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.

You can go up one level to the MATLAB source codes.