# POLYGON_TRIANGULATE Triangulate a Polygon

POLYGON_TRIANGULATE is a FORTRAN77 library which triangulates a possibly nonconvex polygon in 2D, and which can use gnuplot to display the external edges and internal diagonals of the triangulation.

The polygon is defined by an input file which gives the coordinates of the vertices of the polygon, in counterclockwise order.

No consecutive pair of vertices should be equal; when describing a polygon, sometimes the first and last vertices are equal. For this program, that is not the case. To describe a square, your input file should contain four pairs of coordinates, for instance.

The vertices should be listed in counterclockwise order. If you list them in clockwise order, then the function will refuse to operate on the data.

It is possible to create a polygon that has zero area. The function will refuse to process such an object.

The polygon does not need to be convex. However, you should be careful not to specify a polygon which crosses itself, since this means the interior of the polygon is not well defined, and hence a triangulation is not well defined. As a simple example of such a problem, consider the four vertices of a square in counterclockwise order: V1, V2, V3, V4, and list them instead as V1, V3, V2, V4. This shape cannot be triangulated in the usual sense. However, the function may not realize this, in which case it will return what it thinks is a reasonable triangulation of the (unreasonable) data.

The output of the program is a list of the N-3 triples of nodes that form the triangles of the triangulation.

### Languages:

POLYGON_TRIANGULATE is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

POLYGON_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of a polygon in 2D.

POLYGON_MONTE_CARLO, a FORTRAN77 library which applies a Monte Carlo method to estimate the integral of a function over the interior of a polygon in 2D.

POLYGON_PROPERTIES, a FORTRAN77 library which computes properties of an arbitrary polygon in the plane, defined by a sequence of vertices, including interior angles, area, centroid, containment of a point, convexity, diameter, distance to a point, inradius, lattice area, nearest point in set, outradius, uniform sampling.

### Author:

Based on a C function by Joseph ORourke; FORTRAN77 version by John Burkardt.

### Reference:

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

### Examples and Tests:

COMB is an example of a "comb" polygon of 10 vertices

HAND outlines a hand using 59 vertices.

I18 is an example of a complicated nonconvex polygon, using 18 vertices.

### List of Routines:

• BETWEEN is TRUE if vertex C is between vertices A and B.
• 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.
• COLLINEAR returns a measure of collinearity for three points.
• DIAGONAL: VERTEX(IM1) to VERTEX(IP1) is a proper internal diagonal.
• DIAGONALIE is true if VERTEX(IM1):VERTEX(IP1) is a proper diagonal.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• GET_UNIT returns a free FORTRAN unit number.
• I4MAT_PRINT prints an I4MAT.
• I4MAT_PRINT_SOME prints some of an I4MAT.
• I4MAT_WRITE writes an I4MAT file.
• IN_CONE is TRUE if the diagonal VERTEX(IM1):VERTEX(IP1) is strictly internal.
• INTERSECT is true if lines VA:VB and VC:VD intersect.
• INTERSECT_PROP is TRUE if lines VA:VB and VC:VD have a proper intersection.
• L4_XOR returns the exclusive OR of two L4's.
• POLYGON_TRIANGULATE determines a triangulation of a polygon.