TRIANGULATION_Q2L
6-Node Triangulation to 3-Node Triangulation

TRIANGULATION_Q2L is a MATLAB program which reads information describing a triangulation of a set of points using 6-node ("quadratic") triangles, and creates a 3-node ("linear") triangulation.

The same nodes are used, but each 6-node triangle is broken up into four smaller 3-node triangles.

Thus, the program might be given the following 4 triangles:

        11-12-13-14-15
         |\    |\    |
         | \   | \   |
         6  7  8  9 10
         |   \ |   \ |
         |    \|    \|
         1--2--3--4--5
      

It would make a new triangulation involving 16 triangles:

        11-12-13-14-15
         |\ |\ |\ |\ |
         | \| \| \| \|
         6--7--8--9-10
         |\ |\ |\ |\ |
         | \| \| \| \|
         1--2--3--4--5
      

The input and output files use the simple TABLE format; comment lines begin with a "#" character. Otherwise, each line of the file contains one set of information, either the coordinates of a node (for a node file), or the indices of nodes that make up a triangle, (for a triangle file).

The input file prefix_elements.txt contains the triangle information for the 6-node triangulation. Each line contains the indices of six nodes that form a triangle, in counterclockwise order. The first three indices are the vertices, in counterclockwise order. The fourth index is the midside node between vertices 1 and 2, the fifth the midside between vertices 2 and 3, and the sixth the midside between vertices 3 and 1.

The output file prefix_q2l_elements.txt contains the triangle information for the 3-node triangulation. The vertices for each triangle are listed in counterclockwise order. There are 4 times as many triangles in this triangulation.

Usage:

triangulation_q2l ( 'prefix' )

• prefix_nodes.txt contains the node coordinates (not needed by this program),
• prefix_elements.txt contains the quadratic element definitions.
• prefix_q2l_elements.txt will contain the linear element definitions, based on the same set of nodes.

Licensing:

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

Languages:

TRIANGULATION_Q2L is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

TABLE_DELAUNAY, a FORTRAN90 program which triangulates a set of nodes whose coordinates are stored in a file.

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_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, 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_MASK, a MATLAB program which takes an existing triangulation and deletes triangles and their corresponding nodes as requested by the user.

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_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. Marc deBerg, Marc Krevald, Mark Overmars, Otfried Schwarzkopf,
   Computational Geometry,
   Springer, 2000.
2. Joseph ORourke,
   Computational Geometry,
   Cambridge University Press,
   Second Edition, 1998.

Source Code:

• triangulation_q2l.m, is the source code.

Examples and Tests:

EXAMPLE is a set of demonstration data.

• example_nodes.txt, the nodes.
• example_elements.txt, the triangles for the 6-node triangulation.
• example.png, a PNG image of the 6-node triangulation, created by TRIANGULATION_PLOT.
• example_q2l_elements.txt, the triangles for the 3-node triangulation.
• example_q2l.png, a PNG image of the 3-node triangulation, created by TRIANGULATION_PLOT.

Here is the output printed by the program:

• triangulation_q2l_output.txt, the output file.

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