TRIANGULATION_L2Q
6-Node Triangulation from 3-Node Triangulation

TRIANGULATION_L2Q is an executable FORTRAN90 program, using double precision arithmetic, which reads information describing a triangulation of a set of points using 3-node ("linear") triangles, and creates a 6-node ("quadratic") triangulation.

The same number of triangles are used, but each triangle is given three extra midside nodes. The coordinates of these nodes are determined by averaging the coordinates of pairs of vertices of the triangles.

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 nodes.txt contains the node information for the 3-node 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 3-node triangulation. Each line contains the indices of three nodes that form a triangle, in counterclockwise order.

The output file nodes.l2q.txt contains the node information for the 6-node triangulation. It begins with the node information from nodes.txt, followed by the coordinates of the new nodes.

The output file triangles.l2q.txt contains the triangle information for the 6-node triangulation. There are exactly as many triangles as before, but now each triangle uses six nodes. Each line of the file contains the indices of 6 nodes that form the triangle, listed in a particular 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. It should be the case that the first three columns of triangles.l2q.txt are the same as the three columns of triangles.txt.

Related Data and Programs:

TABLE_DELAUNAY is an interactive FORTRAN90 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 FORTRAN90 library of routines for carrying out various operations on order 3 ("linear") or order 6 ("quadratic") triangulations.

TRIANGULATION_BOUNDARY_NODES is an executable FORTRAN90 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 also available in a C++ version and a MATLAB version.

TRIANGULATION_MASK is an executable FORTRAN90 program, which takes an existing triangulation and deletes triangles and their corresponding nodes as requested by the user.

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

TRIANGULATION_Q2L is an executable FORTRAN90 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 FORTRAN90 program that reads data defining a triangulation and computes a number of quality measures.

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

Usage:

triangulation_l2q nodes.txt triangles.txt

creates a new node file called nodes.l2q.txt and a new triangle file called triangles.l2q.txt.

Reference:

1. Marc deBerg, Marc Krevald, Mark Overmars, Otfried Schwarzkopf,
   Computational Geometry,
   Springer, 2000,
   ISBN: 3-540-65620-0.
2. Joseph ORourke,
   Computational Geometry,
   Second Edition,
   Cambridge, 1998,
   ISBN: 0521649765,
   LC: QA448.D38.

Source Code:

• triangulation_l2q.f90, the source code.
• triangulation_l2q.csh, commands to compile the source code.

Examples and Tests:

You may copy data defining an initial 3-node triangulation:

• nodes_order3.txt, the nodes for the 3-node triangulation.
• triangles_order3.txt, the triangles for the 3-node triangulation.
• triangulation_order3.png, a PNG image of the 3-node triangulation, created by TRIANGULATION_PLOT.

You may copy data defining the resultant 6-node triangulation:

• nodes_order6.txt, the nodes for the 6-node triangulation.
• triangles_order6.txt, the triangles for the 6-node triangulation.
• triangulation_order6.png, a PNG image of the 6-node triangulation, created by TRIANGULATION_PLOT.

List of Routines:

• MAIN is the main program for TRIANGULATION_L2Q.
• 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.
• DTABLE_CLOSE_WRITE closes a file used to write a DTABLE.
• DTABLE_DATA_READ reads data from a double precision table file.
• DTABLE_DATA_WRITE writes data to a double precision table file.
• DTABLE_HEADER_READ reads the header from a double precision table file.
• DTABLE_HEADER_WRITE writes the header to a double precision table file.
• DTABLE_OPEN_WRITE opens a file to write a DTABLE.
• DTABLE_WRITE writes a DTABLE to a 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.
• GET_UNIT returns a free FORTRAN unit number.
• I4_MODP returns the nonnegative remainder of integer division.
• I4_WRAP forces an integer to lie between given limits by wrapping.
• I4COL_COMPARE compares columns I and J of an I4COL.
• I4COL_SORT_A ascending sorts an I4COL.
• I4COL_SORTED_UNIQUE_COUNT counts unique elements in an I4COL.
• I4COL_SWAP swaps columns I and J of an I4COL.
• I4MAT_TRANSPOSE_PRINT prints an I4MAT, transposed.
• I4MAT_TRANSPOSE_PRINT_SOME prints some of the transpose of an I4MAT.
• I4ROW_COMPARE compares two rows of a integer array.
• I4ROW_SORT_A ascending sorts the rows of an integer array.
• I4ROW_SWAP swaps two rows of an integer array.
• ITABLE_CLOSE_WRITE closes a file used to write an ITABLE.
• ITABLE_DATA_READ reads data from an integer table file.
• ITABLE_DATA_WRITE writes data to an integer table file.
• ITABLE_HEADER_READ reads the header from an integer table file.
• ITABLE_HEADER_WRITE writes the header to an integer table file.
• ITABLE_WRITE writes an ITABLE file.
• ITABLE_OPEN_WRITE opens a file to write ITABLE data.
• R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
• 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 finds the LAST occurrence of a given substring.
• 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.
• SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING writes the current YMDHMS date into a string.
• TRIANGULATION_ORDER3_BOUNDARY_COUNT counts the number of boundary edges.
• TRIANGULATION_ORDER3_NEIGHBOR_TRIANGLES determines triangle neighbors.

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