# TRIANGULATION Triangulation of 2D data

TRIANGULATION, a FORTRAN90 code which computes a triangulation of a set of points in 2D, and carries out various other related operations on triangulations of order 3 or 6.

The mesh is the collection of triangles. Each triangle is termed an "element". The points used to define the shape of the triangle (the corners, and sometimes a few more points) are called the "nodes".

Routines are available to:

• evaluate "quality measures" for the mesh;
• create a "node neighbor array" for each node;
• create an "element neighbor array" for each element;
• estimate the integral of a function over the region covered by the mesh;
• plot the nodes and elements of a mesh;
• determine the parts of the mesh that lie on the boundary;
• sample points at random from the region covered by the mesh;
• search a mesh to determine which element contains a point.

Since triangulations are often used to define a finite element mesh, which in turn defines a sparse matrix, there are routines available which can define the sparse compressed column arrays needed for a sparse matrix associated with a mesh of order 3 or 6. The special case of the Taylor-Hood mixed element is also handled, which is essentially an order 6 grid counted twice and an order 3 grid that only uses the vertices of the order 6 grid.

### Languages:

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

### Related Data and Programs:

CVT_TRIANGULATION, a FORTRAN90 code which uses routines from the test_triangulation library to create a CVT-based triangularization.

MESH_BANDWIDTH, a FORTRAN90 code which returns the geometric bandwidth associated with a mesh of elements of any order and in a space of arbitrary dimension.

MESH_TO_XML, a FORTRAN90 code which reads information defining a 1D, 2D or 3D mesh, namely a file of node coordinates and a file of elements defined by node indices, and creates a corresponding XML file for input to DOLFIN or FENICS.

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

TEST_TRIANGULATION, a FORTRAN90 code which sets up a number of triangulation test problems.

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

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

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

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

### 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. Paul Bratley, Bennett Fox, Linus Schrage,
A Guide to Simulation,
Second Edition,
Springer, 1987,
ISBN: 0387964673,
LC: QA76.9.C65.B73.
3. Marc deBerg, Marc Krevald, Mark Overmars, Otfried Schwarzkopf,
Computational Geometry,
Springer, 2000,
ISBN: 3-540-65620-0,
LC: QA448.D38.C65.
4. Barry Joe,
GEOMPACK - a software package for the generation of meshes using geometric algorithms,
Volume 13, Number 5, 1991, pages 325-331.
5. Albert Nijenhuis, Herbert Wilf,
Combinatorial Algorithms for Computers and Calculators,
Second Edition,
ISBN: 0-12-519260-6,
LC: QA164.N54.
6. Atsuyuki Okabe, Barry Boots, Kokichi Sugihara, Sung Nok Chiu,
Spatial Tessellations: Concepts and Applications of Voronoi Diagrams,
Second Edition,
Wiley, 2000,
ISBN: 0-471-98635-6,
LC: QA278.2.O36.
7. Joseph ORourke,
Computational Geometry,
Second Edition,
Cambridge, 1998,
ISBN: 0521649765,
LC: QA448.D38.