// From Wikipedia:
// One way to triangulate a simple polygon is by using the assertion that any simple polygon
// without holes has at least two so called 'ears'. An ear is a triangle with two sides on the edge
// of the polygon and the other one completely inside it. The algorithm then consists of finding
// such an ear, removing it from the polygon (which results in a new polygon that still meets
// the conditions) and repeating until there is only one triangle left.
// the algorithm here aims for simplicity over performance. there are other, more performant
// algorithms that are much more complex.
// convert a triangle to a list of triangles. each triangle is represented by a PointF array of length 3.
public static List<PointF[]> Triangulate(Polygon poly)
{
List<PointF[]> triangles = new List<PointF[]>(); // accumulate the triangles here
// keep clipping ears off of poly until only one triangle remains
while (poly.PtListOpen.Count > 3) // if only 3 points are left, we have the final triangle
{
int midvertex = FindEar(poly); // find the middle vertex of the next "ear"
triangles.Add(new PointF[] { poly.PtList[midvertex - 1], poly.PtList[midvertex], poly.PtList[midvertex + 1] });
// create a new polygon that clips off the ear; i.e., all vertices but midvertex
List<PointF> newPts = new List<PointF>(poly.PtList);
newPts.RemoveAt(midvertex); // clip off the ear
poly = new Polygon(newPts); // poly now has one less point
}
// only a single triangle remains, so add it to the triangle list
triangles.Add(poly.PtListOpen.ToArray());
return triangles;
}
// find the type of a specific vertex in a polygon, either Concave or Convex.
public PolygonType VertexType(int vertexNo)
{
Polygon triangle;
if (vertexNo == 0)
{
triangle = new Polygon(new List<PointF> { PtList[PtList.Count - 2], PtList[0], PtList[1] }); // the polygon is always closed so the last point is the same as the first
}
else
{
triangle = new Polygon(new List<PointF> { PtList[vertexNo - 1], PtList[vertexNo], PtList[vertexNo + 1] });
}
if (Math.Sign(triangle.Area) == Math.Sign(this.Area))
return PolygonType.Convex;
else
return PolygonType.Concave;
}
// find an ear (always a triangle) of the polygon and return the index of the middle (second) vertex in the ear
public static int FindEar(Polygon poly)
{
for (int i = 0; i < poly.PtList.Count - 2; i++)
{
if (poly.VertexType(i + 1) == PolygonType.Convex)
{
// get the three points of the triangle we are about to test
PointF a = poly.PtList[i];
PointF b = poly.PtList[i + 1];
PointF c = poly.PtList[i + 2];
bool foundAPointInTheTriangle = false; // see if any of the other points in the polygon are in this triangle
for (int j = 0; j < poly.PtListOpen.Count; j++) // don't check the last point, which is a duplicate of the first
{
if (j != i && j != i+1 && j != i+2 && PointInTriangle(poly.PtList[j], a, b, c)) foundAPointInTheTriangle = true;
}
if (!foundAPointInTheTriangle) // the middle point of this triangle is convex and none of the other points in the polygon are in this triangle, so it is an ear
return i + 1; // EXITING HERE!
}
}
throw new ApplicationException("Improperly formed polygon");
}