TRIANGULATION_MASK
Remove Triangles from a Triangulation

TRIANGULATION_MASK is a FORTRAN90 program 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 prefix_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 prefix_elements.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.

Usage:

The user masking subroutine has the form:

subroutine triangle_mask ( dim_num, triangle_order, nodes, coord, mask )

• dim_num, the spatial dimension, always equal to 2.
• element_order, the number of nodes in the element, usually 3 or 6;
• nodes, an integer array of dimension element_order, containing the indices of each node of the triangle;
• coord, a real array of dimension dim_num by element_order, containing the x and y coordinates of each node of the triangle;
• mask, a logical value, which is true if the triangle should be deleted or "masked", and false if the triangle should be preserved;

The user masking routine must be compiled and linked with the software, perhaps with a command like:

F90 triangulation_mask.o triangle_mask.f90

The program is the executed with a command like:

triangulation_mask prefix

• prefix_nodes.txt contains the node coordinates,
• prefix_elements.txt contains the element definitions.
• prefix_mask_nodes.txt is an output file which will contain the node coordinates retained by the mask;
• prefix_mask_elements.txt is an output file which will contain the defintions of elements retained by the mask.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

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

Related Data and Programs:

TABLE_DELAUNAY, a FORTRAN90 program which can compute the Delaunay triangulation of a set of points.

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

TRIANGULATION, a FORTRAN90 library which carries out various operations on order 3 ("linear") or order 6 ("quadratic") triangulations.

TRIANGULATION_BOUNDARY_NODES, a FORTRAN90 program which reads data defining a triangulation, determines which nodes lie on the boundary, and writes their coordinates to a file.

TRIANGULATION_CORNER, a FORTRAN90 program which patches triangulations so that no triangle has two sides on the boundary.

TRIANGULATION_DELAUNAY_DISCREPANCY, a FORTRAN90 program which measures the amount by which a triangulation fails the local Delaunay test;

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

TRIANGULATION_HISTOGRAM, a FORTRAN90 program which computes histograms of data over a triangulation.

TRIANGULATION_L2Q, a FORTRAN90 program which reads data defining a 3-node triangulation and generates midside nodes and writes out the corresponding 6-node triangulation.

TRIANGULATION_ORIENT, a FORTRAN90 program which reads data defining a triangulation and reorients those triangles that have negative orientation.

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_PLOT, a FORTRAN90 program which reads data defining a triangulation and creates a PostScript image of the nodes and triangles.

TRIANGULATION_Q2L, a FORTRAN90 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 FORTRAN90 program which estimates the integral of a function over a triangulated region.

TRIANGULATION_QUALITY, a FORTRAN90 program which reads data defining a triangulation and computes a number of quality measures.

TRIANGULATION_RCM, a FORTRAN90 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 FORTRAN90 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 FORTRAN90 program which reads data defining a triangulation, determines the neighboring triangles of each triangle, and writes that information to a file.

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. Marc deBerg, Marc Krevald, Mark Overmars, Otfried Schwarzkopf,
   Computational Geometry,
   Springer, 2000,
   ISBN: 3-540-65620-0.
3. Barry Joe,
   GEOMPACK - a software package for the generation of meshes using geometric algorithms,
   Advances in Engineering Software,
   Volume 13, 1991, pages 325-331.
4. Albert Nijenhuis, Herbert Wilf,
   Combinatorial Algorithms for Computers and Calculators,
   Second Edition,
   Academic Press, 1978,
   ISBN: 0-12-519260-6,
   LC: QA164.N54.
5. 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.
6. Joseph ORourke,
   Computational Geometry,
   Second Edition,
   Cambridge, 1998,
   ISBN: 0521649765,
   LC: QA448.D38.

Source Code:

• triangulation_mask.f90, the source code.

Examples and Tests:

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_elements.txt, a set of order 3 triangles.
• p15_elements.png, a PNG image of the original data.
• p15_mask.f90, a routine which masks the triangles by dropping those whose centroids lie outside the region.
• p15_mask_output.txt, the output from a run of the program.
• p15_mask_nodes.txt, the nodes after masking.
• p15_mask_elements.txt, the elements after masking.
• p15_mask_elements.png, PNG image of the elements after masking.

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, an image of the original nodes.
• small_elements.txt, a set of order 3 triangles.
• small_elements.png, a PNG image of the original elements.
• small_mask.f90, a routine which masks the triangles by dropping those whose centroids lie outside the region.
• small_mask_output.txt, the output from a run of the program.
• small_mask_nodes.txt, the nodes after masking.
• small_mask_nodes.png, an image of the nodes after masking.
• small_mask_elements.txt, the elements after masking.
• small_mask_elements.png, a PNG image of the elements after masking.

List of Routines:

• MAIN is the main program for TRIANGULATION_MASK.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• 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.
• GET_UNIT returns a free FORTRAN unit number.
• I4MAT_DATA_READ reads data from an I4MAT file.
• I4MAT_HEADER_READ reads the header from an I4MAT.
• I4MAT_TRANSPOSE_PRINT_SOME prints some of the transpose of an I4MAT.
• I4MAT_WRITE writes an I4MAT file.
• LVEC_PRINT prints an LVEC.
• LVEC_PRINT_SOME prints "some" of an LVEC.
• MESH_BASE_ONE ensures that the element definition is one-based.
• R8MAT_DATA_READ reads data from an R8MAT file.
• R8MAT_HEADER_READ reads the header from an R8MAT file.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT transposed.
• R8MAT_WRITE writes an R8MAT file.
• S_INDEX_LAST_C finds the LAST occurrence of a given character.
• 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.

You can go up one level to the FORTRAN90 source codes.