# TRIANGULATION_CORNER Handle triangulation corners.

TRIANGULATION_CORNER, a C++ code which tries to correct situations in which a triangulation includes corner triangles, that is, triangles which have two sides on boundary.

Especially in finite element settings, such a triangle is considered undesirable. Especially in the case when a linear (order 3) finite element is involved, the function approximation in the interior of the triangle will be entirely determined by boundary data. If, for instance, the solution is constrained to be zero on the boundary, then the finite element function will be zero throughout the interior of the corner triangle.

Presumably, the triangle has a neighbor triangle on the one non-boundary side. It is generally possible and reasonable to replace these two triangles but another two which fill the same quadrilateral, but which are separated by the other diagonal of the quadrilateral.

The initial situation at the corner is suggest by the following diagram:

```          |/  |/
A---C--
|\  |\
| \ |
|  \|
B---D--
```

By rearranging the corner triangle and its neighbor, we have the more satisfactory triangulation:

```          |/  |/
A---C--
|  /|\
| / |
|/  |
B---D--
```

### Usage:

triangulation_corner prefix
where prefix is the common filename prefix:
• prefix_nodes.txt contains the node coordinates,
• prefix_elements.txt contains the element definitions.
• prefix_corner_nodes.txt will contain the revised node coordinates,
• prefix_corner_elements.txt will contain the revised element definitions.

### Languages:

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

### Related Data and Programs:

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

TRIANGULATION, a C++ library which performs various operations on order 3 ("linear") or order 6 ("quadratic") triangulations.

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

TRIANGULATION_DELAUNAY_DISCREPANCY, a C++ 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 C++ program which computes histograms of data over a triangulation.

TRIANGULATION_L2Q, a C++ program which reads information about a 3-node (linear) triangulation and creates data defining a corresponding 6-node (quadratic) triangulation;

TRIANGULATION_MASK, a C++ program which reads a triangulation and calls a user-supplied routine to consider each triangle for deletion;

TRIANGULATION_NODE_TO_ELEMENT, a C++ program which reads files describing a set of nodes, their triangulation, and the value of one or more quantities at each node, and outputs a file that averages the quantities for each element. This operation in effect creates an "order1" finite element model of the data.

TRIANGULATION_ORDER3, a data directory which contains examples of TRIANGULATION_ORDER3 files, description of a linear triangulation of a set of 2D points, using a pair of files to list the node coordinates and the 3 nodes that make up each triangle;

TRIANGULATION_ORDER6, a data directory which contains examples of TRIANGULATION_ORDER6 files, a description of a quadratic triangulation of a set of 2D points, using a pair of files to list the node coordinates and the 6 nodes that make up each triangle.

TRIANGULATION_ORIENT, a C++ program which ensures that the triangles in an order 3 or order 6 triangulation have positive orientation;

TRIANGULATION_PLOT, a C++ program which makes a PostScript image of a triangulation of points.

TRIANGULATION_QUAD, a C++ program which estimates the integral of a function over a triangulated region.

TRIANGULATION_REFINE, a C++ program which can refine a triangulation.

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

### Reference:

1. Joseph ORourke,
Computational Geometry,
Second Edition,
Cambridge, 1998,
ISBN: 0521649765,
LC: QA448.D38.

### Source Code:

Last revised on 06 May 2020.