TRIANGULATION_QUALITY
Interactive Program for
Triangulation Quality

TRIANGULATION_QUALITY is an executable C++ 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 C++ 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 C++ library of routines for carrying out various operations on order 3 ("linear") or order 6 ("quadratic") triangulations.

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

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

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

TRIANGULATION_ORIENT is an executable C++ 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 C++ program that reads data defining a triangulation and creates a PostScript image of the nodes and triangles.

TRIANGULATION_Q2L is an executable C++ 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 FORTRAN90 version and a MATLAB version.

TRIANGULATION_RCM is an executable C++ 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 C++ 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 C++ 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.

Source Code:

• triangulation_quality.C, the source code.
• triangulation_quality.csh, commands to compile, link and load the source code.

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.

List of Routines:

• MAIN is the main program for TRIANGULATION_QUALITY.
• ALPHA_MEASURE determines the triangulated pointset quality measure ALPHA.
• ARC_COSINE computes the arc cosine function, with argument truncation.
• AREA_MEASURE determines the area ratio quality measure.
• BANDWIDTH_MESH determines the bandwidth of the coefficient matrix.
• CH_CAP capitalizes a single character.
• CH_EQI is true if two characters are equal, disregarding case.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• DTABLE_DATA_READ reads the data from a real TABLE file.
• DTABLE_HEADER_READ reads the header from a real TABLE file.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• I4_MAX returns the maximum of two integers.
• I4_MIN returns the smaller of two integers.
• I4MAT_TRANSPOSE_PRINT prints an I4MAT, transposed.
• I4MAT_TRANSPOSE_PRINT_SOME prints some of an I4MAT, transposed.
• ITABLE_DATA_READ reads data from an ITABLE file.
• ITABLE_HEADER_READ reads the header from an ITABLE file.
• Q_MEASURE determines the triangulated pointset quality measure Q.
• R8_HUGE returns a "huge" double precision value.
• R8_MAX returns the maximum of two double precision values.
• R8_MIN returns the minimum of two double precision values.
• R8MAT_TRANSPOSE_PRINT prints a real matrix, transposed.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of a real matrix, transposed.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• S_TO_I4 reads an I4 from a string.
• S_TO_I4VEC reads an I4VEC from a string.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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