TRIANGULATION_MASK
Remove Triangles from a Triangulation

TRIANGULATION_MASK is a C++ program, using double precision arithmetic, which reads the nodes and triangles that define a triangulation, calls a user routine which determines whether each triangle is to be preserved or discarded ("masked") from the triangulation, and writes out new node and triangle files that define the masked triangulation.

The input file nodes.txt contains the node information for the triangulation. Each data line contains the X and Y coordinates of a single node.

The input file triangles.txt contains the triangle information for the triangulation. Each line contains the indices of 3 or 6 nodes that form a triangle.

One motivation for creating this program is as follows. Suppose we have a set of points that lie on the boundary or inside of a non-convex region. If we naively call an unconstrained Delaunay triangulation routine, such as TABLE_DELAUNAY, then because the region is not convex, it is possible to create triangles which lie outside the region.

An easy way to correct this problem is to call a user routine and pass it the indices and coordinates of each triangle. The user can then decide to drop any triangle whose centroid, say, lies outside the region.

Other masking criteria might drop triangles that are too small, or that have too small an angle, or that lie inside some interior hole. These choices are entirely up to the user.

The user masking subroutine has the form:

bool triangle_mask ( int dim_num, int triangle_order, int nodes[], double coord[] )

with arguments:

dim_num: input, the spatial dimension, always equal to 2.
triangle_order: input, the number of nodes in the triangle, usually 3 or 6;
nodes: input, an integer array of dimension triangle_order, containing the indices of each node of the triangle;
coord: input, a double array of dimension dim_num by triangle_order, containing the x and y coordinates of each node of the triangle;
triangle_mask: output, a boolean value, which is true if the triangle should be deleted or "masked", and false if the triangle should be preserved;

Usage:

g++ triangulation_mask.o triangle_mask.C: compiles the user routine triangle_mask.C and links it with the partially compiled triangulation_mask.o to create an executable a.out. We will assume that a.out is renamed to triangulation_mask.
triangulation_mask node_file triangle_file: reads the triangulation described by the node file and the triangle file, calls the user mask routine for each triangle, and writes out the new node and triangle files.

Related Data and Programs:

TABLE is the format used for the input and output files.

TABLE_DELAUNAY is an executable C++ program which can compute the Delaunay triangulation of a set of points.

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 also available in a FORTRAN90 version and a MATLAB version.

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 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 an executable C++ program that reads data defining a triangulation and computes a number of quality measures.

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.

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,
../../pdf/aurenhammer.pdf
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 Tesselations: 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.

Source Code:

triangulation_mask.C, the source code.
triangulation_mask.csh, commands to compile the source code.

Examples and Tests:

SMALL is a triangulation of the 25 lattice points on the [0,4]x[0,4] square. Our masking operation should cut out a lower left triangular corner and a section from the upper right.

small_nodes.txt, a set of nodes.
small_nodes.png, a PNG image of the nodes.
small_triangles.txt, a set of order 3 triangles.
small_triangles.png, a PNG image of the original data.
small_mask.C, a routine which masks the triangles by dropping those whose centroids lie outside the region.
small_mask.csh, commands which compile the user routine with TRIANGULATION_MASK, and analyze the triangulation.
small_mask.out, the output from a run of the program.
small_nodes.mask.txt, the set of nodes after TRIANGULATION_MASK has masked the triangles.
small_nodes.mask.png, a PNG image of the remaining nodes.
small_triangles.mask.txt, the set of triangles after TRIANGULATION_MASK has masked the triangles.
small_triangles.mask.png, a PNG image of the remaining triangles.

P15 is a triangulation created by calling DISTMESH, then removing duplicate points by calling TABLE_MERGE, then creating a Delaunay triangulation by calling TABLE_DELAUNAY, Unfortunately, this results in many triangles that lie outside the region of interest.

p15_nodes.txt, a set of nodes.
p15_nodes.png, a PNG image of the nodes.
p15_triangles.txt, a set of order 3 triangles.
p15_triangles.png, a PNG image of the original data.
p15_mask.C, a routine which masks the triangles by dropping those whose centroids lie outside the region.
p15_mask.csh, commands which compile the user routine with TRIANGULATION_MASK, and analyze the triangulation.
p15_mask.out, the output from a run of the program.
p15_nodes.mask.txt, the set of nodes after TRIANGULATION_MASK has masked the triangles.
p15_triangles.mask.txt, the set of triangles after TRIANGULATION_MASK has masked the triangles.
p15_triangles.mask.png, a PNG image of the remaining triangles.

List of Routines:

MAIN is the main program for TRIANGULATION_MASK.
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_CLOSE_WRITE closes a file to which a DTABLE was to be written.
DTABLE_DATA_READ reads the data from a real TABLE file.
DTABLE_DATA_WRITE writes data to a DTABLE file.
DTABLE_HEADER_READ reads the header from a real TABLE file.
DTABLE_HEADER_WRITE writes the header of a DTABLE file.
DTABLE_WRITE writes information to a DTABLE file.
FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
FILE_NAME_EXT_GET determines the "extension" of a file name.
FILE_NAME_EXT_SWAP replaces the current "extension" of a file name.
FILE_ROW_COUNT counts the number of row records in a file.
I4_MAX returns the maximum of two I4's.
I4_MIN returns the minimum of two I4's.
I4MAT_TRANSPOSE_PRINT_SOME prints some of an I4MAT, transposed.
ITABLE_DATA_READ reads data from an ITABLE file.
ITABLE_DATA_WRITE writes data to an ITABLE file.
ITABLE_HEADER_READ reads the header from an ITABLE file.
ITABLE_HEADER_WRITE writes the header of an ITABLE file.
ITABLE_WRITE writes information to an ITABLE file.
R8_EPSILON returns the R8 roundoff unit.
R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
S_BLANK_DELETE removes blanks from a string, left justifying the remainder.
S_INDEX_LAST_C returns a pointer to the last occurrence of a given character.
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 vector 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.
TIMESTRING returns the current YMDHMS date as a string.

You can go up one level to the C++ source codes.

Last revised on 14 January 2007.

TRIANGULATION_MASK Remove Triangles from a Triangulation