TRIANGULATION_QUALITY
Interactive Program for
Triangulation Quality

TRIANGULATION_QUALITY is a MATLAB program, using double precision arithmetic, which computes and prints a variety of quality measures for a given triangulation of a set of points in 2D.

The triangulation is defined by a node file containing the coordinates of nodes, and a triangle file containing sets of 3 or 6 node indices.

The quality measures computed include:

• Alpha, the minimum angle divided by the maximum possible minimum angle.
• Area, the ratio of the minimum area to the maximum area;
• Q, the triangle shape uniformity measure, twice the radius of the inscribed circle divided by the radius of the circumscribed circle;

Each quality measure is defined as the minimum of its value over all the triangles; the maximum and best possible value is 1, and the minimum and worst possible value is 0.

Related Data and Programs:

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

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

TRIANGULATION is a MATLAB library of routines for carrying out various operations on order 3 ("linear") or order 6 ("quadratic") triangulations.

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 contains a description and examples of order 3 triangulations.

TRIANGULATION_ORDER6 is a directory 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 also available in a C++ version and a FORTRAN90 version..

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.

Usage:

triangulation_quality ( 'node_file', 'triangle_file' )

reads the node and triangle information, computes the quality measures, and prints them.

Reference:

1. Marc deBerg, Marc Krevald, Mark Overmars, Otfried Schwarzkopf,
   Computational Geometry,
   Springer, 2000,
   ISBN: 3-540-65620-0.
2. David Field,
   Qualitative Measures for Initial Meshes,
   International Journal of Numerical Methods in Engineering,
   Volume 47, 2000, pages 887-906.
3. Max Gunzburger,
   Uniformity Measures for Point Samples in Hypercubes,
   PDF version.
4. Joseph ORourke,
   Computational Geometry,
   Second Edition,
   Cambridge, 1998,
   ISBN: 0521649765,
   LC: QA448.D38.
5. Per-Olof Persson, Gilbert Strang,
   A Simple Mesh Generator in MATLAB,
   SIAM Review,
   Volume 46, Number 2, pages 329-345, June 2004.

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.

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

Source Code:

• triangulation_quality.m, is the main program.
• alpha_measure.m, determines the triangulated pointset quality measure ALPHA.
• arc_cosine.m, computes the arc cosine function, with argument truncation.
• area_measure.m, determines the area ratio quality measure.
• bandwidth_mesh.m, determines the bandwidth associated with the finite element mesh.
• dtable_data_read.m, reads data from a double precision table file.
• dtable_header_read.m, reads the header from a double precision table file.
• file_column_count.m, counts the number of columns in the first line of a file.
• file_row_count.m, counts the number of row records in a file.
• i4_transpose_print.m, prints an I4MAT, transposed.
• i4mat_transpose_print_some.m, prints some of the transpose of an I4MAT.
• itable_data_read.m, reads data from an integer table file.
• itable_header_read.m, reads the header from an integer table file.
• q_measure.m, determines the triangulated pointset quality measure Q.
• r8_huge.m, returns a huge R8 value.
• r8mat_transpose_print.m, prints an R8MAT, transposed.
• r8mat_transpose_print_some.m, prints some of an R8MAT, transposed.
• s_len_trim.m, returns the length of a string to the last nonblank.
• s_word_count.m, counts the number of "words" in a string.
• timestamp.m, TIMESTAMP prints the current YMDHMS date as a time stamp.

Examples and Tests:

ELL is a set of nodes in an L-shaped region, which have been arranged into an order 3 triangulation.

• ell_nodes.txt, the node file.
• ell_tri3.txt, the triangle file.
• ell_quality.txt, the report from the program.

DOUBLE_HEX is a set of nodes in an square region, with two hexagonal holes, which have been arranged into an order 6 triangulation.

• double_hex_nodes.txt, the node file.
• double_hex_tri6.txt, the triangle file.
• double_hex_quality.txt, the report from the program.

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