# TRIANGULATION_MASK Remove Triangles from a Triangulation

TRIANGULATION_MASK is a MATLAB 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:

In the following discussion, the user masking routine is called "triangle_mask", but the actual name is arbitrary. The actual name is passed as the third argument into the program. It must be preceded by an "@" sign, so that MATLAB knows that this is the name of a function (technically, a MATLAB "function handle").

The user masking routine has the form:

with arguments:
• dim_num, the spatial dimension, always equal to 2.
• element_order, the number of nodes in the triangle, 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 command to invoke the program has the form:

where prefix is the common filename prefix:
• prefix_nodes.txt contains the node coordinates,
• prefix_elements.txt contains the element definitions.
and
reads the triangulation described by the node file and the triangle file, calls the user triangle mask routine for each triangle, and writes out the new node and triangle files.

### 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 MATLAB library which carries out various operations on order 3 ("linear") or order 6 ("quadratic") triangulations.

TRIANGULATION_BOUNDARY_EDGES, a MATLAB program which reads data defining a triangulation, determines which edges lie on the boundary, organizes them into connected components, and writes this information to a file.

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

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

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

TRIANGULATION_DISPLAY, a MATLAB program which displays the nodes and elements of a triangulation on the MATLAB graphics screen;

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

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

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

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_ORIENT, a MATLAB program which reads data defining a triangulation, makes sure that every triangle has positive orientation, and if not, writes a corrected triangle file.

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

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

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

TRIANGULATION_RCM, a MATLAB 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 MATLAB 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 MATLAB 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,
Volume 13, 1991, pages 325-331.
4. Albert Nijenhuis, Herbert Wilf,
Combinatorial Algorithms for Computers and Calculators,
Second Edition,
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.

### 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.

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.

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

Last revised on 04 October 2009.