//Orient triangles so they have the correct orientation
public static void OrientTrianglesClockwise(List <Triangle> triangles)
{
for (int i = 0; i < triangles.Count; i++)
{
Triangle tri = triangles[i];
Vector2 v1 = new Vector2(tri.v1.position.x, tri.v1.position.z);
Vector2 v2 = new Vector2(tri.v2.position.x, tri.v2.position.z);
Vector2 v3 = new Vector2(tri.v3.position.x, tri.v3.position.z);
if (!Delaunay.IsTriangleOrientedClockwise(v1, v2, v3))
{
tri.ChangeOrientation();
}
}
}
//Is a quadrilateral convex? Assume no 3 points are colinear and the shape doesnt look like an hourglass
public static bool IsQuadrilateralConvex(Vector2 a, Vector2 b, Vector2 c, Vector2 d)
{
bool isConvex = false;
bool abc = Delaunay.IsTriangleOrientedClockwise(a, b, c);
bool abd = Delaunay.IsTriangleOrientedClockwise(a, b, d);
bool bcd = Delaunay.IsTriangleOrientedClockwise(b, c, d);
bool cad = Delaunay.IsTriangleOrientedClockwise(c, a, d);
if (abc && abd && bcd & !cad)
{
isConvex = true;
}
else if (abc && abd && !bcd & cad)
{
isConvex = true;
}
else if (abc && !abd && bcd & cad)
{
isConvex = true;
}
//The opposite sign, which makes everything inverted
else if (!abc && !abd && !bcd & cad)
{
isConvex = true;
}
else if (!abc && !abd && bcd & !cad)
{
isConvex = true;
}
else if (!abc && abd && !bcd & !cad)
{
isConvex = true;
}
return(isConvex);
}
//Alternative 1. Triangulate with some algorithm - then flip edges until we have a delaunay triangulation
public static List <Triangle> TriangulateByFlippingEdges(List <Vector3> sites)
{
//Step 1. Triangulate the points with some algorithm
//Vector3 to vertex
List <Vertex> vertices = new List <Vertex>();
for (int i = 0; i < sites.Count; i++)
{
vertices.Add(new Vertex(sites[i]));
}
//Triangulate the convex hull of the sites
List <Triangle> triangles = IncrementalTriangulationAlgorithm.TriangulatePoints(vertices);
//List triangles = TriangulatePoints.TriangleSplitting(vertices);
//Step 2. Change the structure from triangle to half-edge to make it faster to flip edges
List <HalfEdge> halfEdges = Delaunay.TransformFromTriangleToHalfEdge(triangles);
//Step 3. Flip edges until we have a delaunay triangulation
int safety = 0;
int flippedEdges = 0;
while (true)
{
safety += 1;
if (safety > 100000)
{
Debug.WriteLine("Stuck in endless loop");
break;
}
bool hasFlippedEdge = false;
//Search through all edges to see if we can flip an edge
for (int i = 0; i < halfEdges.Count; i++)
{
HalfEdge thisEdge = halfEdges[i];
//Is this edge sharing an edge, otherwise its a border, and then we cant flip the edge
if (thisEdge.oppositeEdge == null)
{
continue;
}
//The vertices belonging to the two triangles, c-a are the edge vertices, b belongs to this triangle
Vertex a = thisEdge.v;
Vertex b = thisEdge.nextEdge.v;
Vertex c = thisEdge.prevEdge.v;
Vertex d = thisEdge.oppositeEdge.nextEdge.v;
Vector2 aPos = a.GetPos2D_XZ();
Vector2 bPos = b.GetPos2D_XZ();
Vector2 cPos = c.GetPos2D_XZ();
Vector2 dPos = d.GetPos2D_XZ();
//Use the circle test to test if we need to flip this edge
if (Delaunay.IsPointInsideOutsideOrOnCircle(aPos, bPos, cPos, dPos) < 0f)
{
//Are these the two triangles that share this edge forming a convex quadrilateral?
//Otherwise the edge cant be flipped
if (Delaunay.IsQuadrilateralConvex(aPos, bPos, cPos, dPos))
{
//If the new triangle after a flip is not better, then dont flip
//This will also stop the algoritm from ending up in an endless loop
if (Delaunay.IsPointInsideOutsideOrOnCircle(bPos, cPos, dPos, aPos) < 0f)
{
continue;
}
//Flip the edge
flippedEdges += 1;
hasFlippedEdge = true;
FlipEdge(thisEdge);
}
}
}
//We have searched through all edges and havent found an edge to flip, so we have a Delaunay triangulation!
if (!hasFlippedEdge)
{
//Debug.Log("Found a delaunay triangulation");
break;
}
}
//Debug.Log("Flipped edges: " + flippedEdges);
//Dont have to convert from half edge to triangle because the algorithm will modify the objects, which belongs to the
//original triangles, so the triangles have the data we need
return(triangles);
}
//From triangle where each triangle has one vertex to half edge
public static List <HalfEdge> TransformFromTriangleToHalfEdge(List <Triangle> triangles)
{
//Make sure the triangles have the same orientation
Delaunay.OrientTrianglesClockwise(triangles);
//First create a list with all possible half-edges
List <HalfEdge> halfEdges = new List <HalfEdge>(triangles.Count * 3);
for (int i = 0; i < triangles.Count; i++)
{
Triangle t = triangles[i];
HalfEdge he1 = new HalfEdge(t.v1);
HalfEdge he2 = new HalfEdge(t.v2);
HalfEdge he3 = new HalfEdge(t.v3);
he1.nextEdge = he2;
he2.nextEdge = he3;
he3.nextEdge = he1;
he1.prevEdge = he3;
he2.prevEdge = he1;
he3.prevEdge = he2;
//The vertex needs to know of an edge going from it
he1.v.halfEdge = he2;
he2.v.halfEdge = he3;
he3.v.halfEdge = he1;
//The face the half-edge is connected to
t.halfEdge = he1;
he1.t = t;
he2.t = t;
he3.t = t;
//Add the half-edges to the list
halfEdges.Add(he1);
halfEdges.Add(he2);
halfEdges.Add(he3);
}
//Find the half-edges going in the opposite direction
for (int i = 0; i < halfEdges.Count; i++)
{
HalfEdge he = halfEdges[i];
Vertex goingToVertex = he.v;
Vertex goingFromVertex = he.prevEdge.v;
for (int j = 0; j < halfEdges.Count; j++)
{
//Dont compare with itself
if (i == j)
{
continue;
}
HalfEdge heOpposite = halfEdges[j];
//Is this edge going between the vertices in the opposite direction
if (goingFromVertex.position == heOpposite.v.position && goingToVertex.position == heOpposite.prevEdge.v.position)
{
he.oppositeEdge = heOpposite;
break;
}
}
}
return(halfEdges);
}