/// <summary>
/// Sets to and from nodes as well as slope and intercept of line.
/// </summary>
/// <param name="fromNode"></param>
/// <param name="toNode"></param>
internal Line(Node fromNode, Node toNode)
{
FromNode = fromNode;
ToNode = toNode;
//Solve for slope and y intercept.
if (Math.Abs(ToNode.X - FromNode.X) < 1E-10) //If vertical line, set slope = inf.
{
m = double.PositiveInfinity;
b = double.PositiveInfinity;
}
else if (Math.Abs(ToNode.Y - FromNode.Y) < 1E-10) //If horizontal line, set slope = 0.
{
m = 0.0;
b = ToNode.Y;
}
else //Else y = mx + b
{
m = (ToNode.Y - FromNode.Y) / (ToNode.X - FromNode.X);
b = ToNode.Y - m * ToNode.X;
}
}
internal static void MergeSortedListsOfNodes(List<Node> sortedNodes, List<Node> negativeLoop, Node node)
{
//For each node in negativeLoop, minus the first node (which is already in the list)
int nodeID = sortedNodes.IndexOf(node);
for (int i = 1; i < negativeLoop.Count(); i++)
{
var isInserted = false;
//Starting from after the nodeID, search for an insertion location
for (int j = nodeID + 1; j < sortedNodes.Count; j++)
{
if (negativeLoop[i].Y > sortedNodes[j].Y)
{
sortedNodes.Insert(j, negativeLoop[i]);
isInserted = true;
break;
}
if (Math.Abs(negativeLoop[i].Y - sortedNodes[j].Y) < 0.000001 && negativeLoop[i].X < sortedNodes[j].X) //approaximately equal
{
sortedNodes.Insert(j, negativeLoop[i]);
isInserted = true;
break;
}
}
//If not greater than any elements in the list, ERROR. This should never happen since the negative loop is assumed
//to be fully encapsulated by the positive polygon.
if (isInserted == false)
{
throw new System.ArgumentException("Negative loop must be fully enclosed");
}
}
}
internal static double LinesToRight(Node node, IEnumerable<Line> lineList, out Line rightLine)
{
rightLine = null;
var xright = double.PositiveInfinity;
var counter = 0;
foreach (Line line in lineList)
{
var x = line.Xintercept(node.Y);
var xdif = x - node.X;
if (xdif > 0)
{
counter++;
if (xdif < xright)
{
xright = xdif;
rightLine = line;
}
}
}
return counter;
}
/// <summary>
/// Triangulates a Polygon into faces.
/// </summary>
/// <param name="points2D">The 2D points represented in loops.</param>
/// <param name="isPositive">Indicates whether the corresponding loop is positive or not.</param>
/// <returns>List<Point[]>, which represents vertices of new faces.</returns>
/// <exception cref="System.NotImplementedException"></exception>
public static List<PolygonalFace> Run(List<Point[]> points2D, Boolean[] isPositive)
{
//Return variable triangles
var triangles = new List<PolygonalFace>();
#region Preprocessing
//Preprocessing
// 1) For each loop in points2D
// 2) Create nodes and lines from points, and retain whether a point
// was in a positive or negative loop.
// 3) Add nodes to an ordered loop (same as points2D except now Nodes)
// and a sorted loop (used for sweeping).
var i = 0;
var orderedLoops = new List<List<Node>>();
var sortedLoops = new List<List<Node>>();
var listPositive = isPositive.ToList<bool>();
foreach (var loop in points2D)
{
var orderedLoop = new List<Node>();
//Create first node
var nodeType = GetNodeType(loop.Last(), loop[0], loop[1], isPositive[i]);
var firstNode = new Node(loop[0], nodeType, i);
var previousNode = firstNode;
orderedLoop.Add(firstNode);
//Create other nodes
for (var j = 1; j < loop.Count() - 1; j++)
{
//Create New Node
nodeType = GetNodeType(loop[j - 1], loop[j], loop[j + 1], isPositive[i]);
var node = new Node(loop[j], nodeType, i);
//Add node to the ordered loop
orderedLoop.Add(node);
//Create New Line
var line = new Line(previousNode, node);
previousNode.StartLine = line;
node.EndLine = line;
previousNode = node;
}
//Create last node
nodeType = GetNodeType(loop[loop.Count() - 2], loop[loop.Count() - 1], loop[0], isPositive[i]);
var lastNode = new Node(loop[loop.Count() - 1], nodeType, i);
orderedLoop.Add(lastNode);
//Create both missing lines
var line1 = new Line(previousNode, lastNode);
previousNode.StartLine = line1;
lastNode.EndLine = line1;
var line2 = new Line(lastNode, firstNode);
lastNode.StartLine = line2;
firstNode.EndLine = line2;
//Sort nodes by descending Y, ascending X
var sortedLoop = orderedLoop.OrderByDescending(node => node.Y).ThenBy(node => node.X).ToList<Node>();
orderedLoops.Add(orderedLoop);
sortedLoops.Add(sortedLoop);
i++;
}
#endregion
// 1) For each positive loop
// 2) Remove it from orderedLoops.
// 3) Create a new group
// 4) Insert the first node from all the negative loops remaining into the group list in the correct sorted order.
// 5) Use the red-black tree to determine if the first node from a negative loop is inside the polygon.
// Refer to http://alienryderflex.com/polygon/ for how to determine if a point is inside a polygon.
// 6) If not inside, remove that nodes from the group list.
// 7) else remove the negative loop from orderedLoops and merge the negative loop with the group list.
// 8) Continue with Trapezoidation
List<List<Node>> completeListSortedLoops = new List<List<Node>>(sortedLoops);
while (orderedLoops.Any())
{
//Get information about positive loop, remove from loops, and create new group
i = listPositive.FindIndex(true);
var posOrderedLoop = new List<Node>(orderedLoops[i]);
var sortedGroup = new List<Node>(sortedLoops[i]);
listPositive.RemoveAt(i);
orderedLoops.RemoveAt(i);
sortedLoops.RemoveAt(i);
//Insert the first node from all the negative loops remaining into the group list in the correct sorted order.
for (var j = 0; j < orderedLoops.Count(); j++)
{
if (listPositive[j] == false)
{
InsertNodeInSortedList(sortedGroup, sortedLoops[j][0]);
}
}
//inititallize lineList and sortedNodes
var lineList = new List<Line>();
var nodes = new List<Node>();
#region Trapezoidize Polygons
var trapTree = new List<PartialTrapezoid>();
var completedTrapezoids = new List<Trapezoid>();
//Use the red-black tree to determine if the first node from a negative loop is inside the polygon.
//for each node in sortedNodes, update the lineList. Note that at this point each node only has two edges.
foreach (var node in sortedGroup)
{
Line leftLine = null;
Line rightLine = null;
//Check if negative loop is inside polygon
//note that listPositive changes order /size , while isPositive is static like loopID.
//Similarly points2D is static.
if (node == completeListSortedLoops[node.LoopID][0] && isPositive[node.LoopID] == false) //if first point in the sorted loop and loop is negative
{
if (LinesToLeft(node, lineList, out leftLine) % 2 != 0) //If remainder is not equal to 0, then it is odd.
{
if (LinesToRight(node, lineList, out rightLine) % 2 == 0) //If remainder is not equal to 0, then it is odd.
{
//NOTE: This node must be a reflex upward point by observation
//leftLine and rightLine are set in the two previous call and are now not null.
//Add remaining points from loop into sortedGroup.
MergeSortedListsOfNodes(sortedGroup, completeListSortedLoops[node.LoopID], node);
}
else //Number of lines is even. Remove from group and go to next node
{
sortedGroup.Remove(node);
continue;
}
}
else //Number of lines is even. Remove from group and go to next node
{
sortedGroup.Remove(node);
continue;
}
}
//Add to or remove from Red-Black Tree
if (lineList.Contains(node.StartLine))
{
lineList.Remove(node.StartLine);
}
else
{
lineList.Add(node.StartLine);
}
if (lineList.Contains(node.EndLine))
{
lineList.Remove(node.EndLine);
}
else
{
lineList.Add(node.EndLine);
}
switch (node.Type)
{
case NodeType.DownwardReflex:
{
FindLeftLine(node, lineList, out leftLine);
FindRightLine(node, lineList, out rightLine);
//Close two trapezoids
//Left trapezoid:
InsertTrapezoid(node, leftLine, node.StartLine, ref trapTree, ref completedTrapezoids);
//Right trapezoid:
InsertTrapezoid(node, node.EndLine, rightLine, ref trapTree, ref completedTrapezoids);
//Create one new partial trapezoid
var newPartialTrapezoid = new PartialTrapezoid(node, leftLine, rightLine);
trapTree.Add(newPartialTrapezoid);
}
break;
case NodeType.UpwardReflex:
{
if (leftLine == null) //If from the first negative point, leftLine and rightLine will already be set.
{
FindLeftLine(node, lineList, out leftLine);
FindRightLine(node, lineList, out rightLine);
}
//Close one trapezoid
InsertTrapezoid(node, leftLine, rightLine, ref trapTree, ref completedTrapezoids);
//Create two new partial trapezoids
//Left Trapezoid
var newPartialTrapezoid1 = new PartialTrapezoid(node, leftLine, node.EndLine);
trapTree.Add(newPartialTrapezoid1);
//Right Trapezoid
var newPartialTrapezoid2 = new PartialTrapezoid(node, node.StartLine, rightLine);
trapTree.Add(newPartialTrapezoid2);
}
break;
case NodeType.Peak:
{
//Create one new partial trapezoid
var newPartialTrapezoid = new PartialTrapezoid(node, node.StartLine, node.EndLine);
trapTree.Add(newPartialTrapezoid);
}
break;
case NodeType.Root:
{
//Close one trapezoid
InsertTrapezoid(node, node.EndLine, node.StartLine, ref trapTree, ref completedTrapezoids);
}
break;
case NodeType.Left:
{
//Create one trapezoid
FindLeftLine(node, lineList, out leftLine);
rightLine = node.StartLine;
InsertTrapezoid(node, leftLine, rightLine, ref trapTree, ref completedTrapezoids);
//Create one new partial trapezoid
var newPartialTrapezoid = new PartialTrapezoid(node, leftLine, node.EndLine);
trapTree.Add(newPartialTrapezoid);
}
break;
case NodeType.Right:
{
//Create one trapezoid
FindRightLine(node, lineList, out rightLine);
leftLine = node.EndLine;
InsertTrapezoid(node, leftLine, rightLine, ref trapTree, ref completedTrapezoids);
//Create one new partial trapezoid
var newPartialTrapezoid = new PartialTrapezoid(node, node.StartLine, rightLine);
trapTree.Add(newPartialTrapezoid);
}
break;
}
}
#endregion
#region Create Monotone Polygons
//for each trapezoid with a reflex edge, split in two.
//Insert new trapezoids in correct position in list.
for (var j = 0; j < completedTrapezoids.Count; j++)
{
var trapezoid = completedTrapezoids[j];
if (trapezoid.TopNode.Type == NodeType.DownwardReflex) //If upper node is reflex down (bottom node could be reflex up, reflex down, or other)
{
var newLine = new Line(trapezoid.TopNode, trapezoid.BottomNode);
completedTrapezoids.RemoveAt(j);
var leftTrapezoid = new Trapezoid(trapezoid.TopNode, trapezoid.BottomNode, trapezoid.LeftLine, newLine);
var rightTrapezoid = new Trapezoid(trapezoid.TopNode, trapezoid.BottomNode, newLine, trapezoid.RightLine);
completedTrapezoids.Insert(j, rightTrapezoid); //right trapezoid will end up right below left trapezoid
completedTrapezoids.Insert(j, leftTrapezoid); //left trapezoid will end up were the original trapezoid was located
j++; //Extra counter to skip extra trapezoid
}
else if (trapezoid.BottomNode.Type == NodeType.UpwardReflex) //If bottom node is reflex up (if TopNode.Type = 0, this if statement will be skipped).
{
var newLine = new Line(trapezoid.TopNode, trapezoid.BottomNode);
completedTrapezoids.RemoveAt(j);
var leftTrapezoid = new Trapezoid(trapezoid.TopNode, trapezoid.BottomNode, trapezoid.LeftLine, newLine);
var rightTrapezoid = new Trapezoid(trapezoid.TopNode, trapezoid.BottomNode, newLine, trapezoid.RightLine);
completedTrapezoids.Insert(j, rightTrapezoid); //right trapezoid will end up right below left trapezoid
completedTrapezoids.Insert(j, leftTrapezoid); //left trapezoid will end up were the original trapezoid was located
j++; //Extra counter to skip extra trapezoid
}
}
//Create Monotone Polygons from Trapezoids
var currentTrap = completedTrapezoids[0];
var monotoneTrapPolygon = new List<Trapezoid> { currentTrap };
var monotoneTrapPolygons = new List<List<Trapezoid>> { monotoneTrapPolygon };
//for each trapezoid except the first one, which was added in the intitialization above.
for (var j = 1; j < completedTrapezoids.Count; j++)
{
//Check if next trapezoid can attach to any existing monotone polygon
var boolstatus = false;
for (var k = 0; k < monotoneTrapPolygons.Count; k++)
{
currentTrap = monotoneTrapPolygons[k].Last();
if (currentTrap.BottomNode == completedTrapezoids[j].TopNode)
{
if (currentTrap.LeftLine == completedTrapezoids[j].LeftLine ||
currentTrap.RightLine == completedTrapezoids[j].RightLine)
{
monotoneTrapPolygons[k].Add(completedTrapezoids[j]);
boolstatus = true;
break;
}
}
}
// If they cannot be attached to any existing monotone polygon, create a new monotone polygon
if (boolstatus == false)
{
var trapezoidList = new List<Trapezoid> { completedTrapezoids[j] };
monotoneTrapPolygons.Add(trapezoidList);
}
}
//Convert the lists of trapezoids that form monotone polygons into the monotone polygon class\
//This class includes a sorted list of all the points in the monotone polygon and two monotone chains.
//Both of these lists are used during traingulation.
var monotonePolygons = new List<MonotonePolygon>();
foreach (var monotoneTrapPoly in monotoneTrapPolygons)
{
//Biuld the right left chains and the sorted list of all nodes
var monotoneRightChain = new List<Node>();
var monotoneLeftChain = new List<Node>();
var sortedMonotonePolyNodes = new List<Node>();
//Add upper node to both chains and sorted list
monotoneRightChain.Add(monotoneTrapPoly[0].TopNode);
monotoneLeftChain.Add(monotoneTrapPoly[0].TopNode);
sortedMonotonePolyNodes.Add(monotoneTrapPoly[0].TopNode);
//Add all the middle nodes to one chain (right or left)
for (var j = 1; j < monotoneTrapPoly.Count; j++)
{
//Add the topNode of each trapezoid (minus the initial trapezoid) to the sorted list.
sortedMonotonePolyNodes.Add(monotoneTrapPoly[j].TopNode);
//If trapezoid upper node is on the right line, add it to the right chain
if (monotoneTrapPoly[j].RightLine.FromNode == monotoneTrapPoly[j].TopNode ||
monotoneTrapPoly[j].RightLine.ToNode == monotoneTrapPoly[j].TopNode)
{
monotoneRightChain.Add(monotoneTrapPoly[j].TopNode);
}
//Else add it to the left chain
else
{
monotoneLeftChain.Add(monotoneTrapPoly[j].TopNode);
}
}
//Add bottom node of last trapezoid to both chains and sorted list
monotoneRightChain.Add(monotoneTrapPoly.Last().BottomNode);
monotoneLeftChain.Add(monotoneTrapPoly.Last().BottomNode);
sortedMonotonePolyNodes.Add(monotoneTrapPoly.Last().BottomNode);
//Create new monotone polygon based on these two chains and sorted list.
var monotonePolygon = new MonotonePolygon(monotoneLeftChain, monotoneRightChain, sortedMonotonePolyNodes);
monotonePolygons.Add(monotonePolygon);
}
#endregion
#region Triangulate Monotone Polygons
//Triangulates the monotone polygons
foreach (var monotonePolygon2 in monotonePolygons)
triangles.AddRange(Triangulate(monotonePolygon2));
#endregion
}
return triangles;
}
internal static double LinesToLeft(Node node, IEnumerable<Line> lineList, out Line leftLine)
{
leftLine = null;
var xleft = double.NegativeInfinity;
var counter = 0;
foreach (Line line in lineList)
{
var x = line.Xintercept(node.Y);
var xdif = x - node.X;
if (xdif < 0)
{
counter++;
if (xdif > xleft)
{
xleft = xdif;
leftLine = line;
}
}
}
return counter;
}
internal static void InsertTrapezoid(Node node, Line leftLine, Line rightLine, ref List<PartialTrapezoid> trapTree, ref List<Trapezoid> completedTrapezoids)
{
var matchesTrap = false;
var i = 0;
while (matchesTrap == false && i < trapTree.Count)
{
var partialTrap = trapTree[i];
if (partialTrap.Contains(leftLine, rightLine))
{
var newTrapezoid = new Trapezoid(partialTrap.TopNode, node, partialTrap.LeftLine, partialTrap.RightLine);
completedTrapezoids.Add(newTrapezoid);
trapTree.Remove(partialTrap);
matchesTrap = true;
}
i++;
}
}
internal static int InsertNodeInSortedList(List<Node> sortedNodes, Node node)
{
//Search for insertion location starting from the first element in the list.
for (int i = 0; i < sortedNodes.Count(); i++)
{
if (node.Y > sortedNodes[i].Y)
{
sortedNodes.Insert(i, node);
return i;
}
if (Math.Abs(node.Y - sortedNodes[i].Y) < 0.000001 && node.X < sortedNodes[i].X) //approaximately equal
{
sortedNodes.Insert(i, node);
return i;
}
}
//If not greater than any elements in the list, add the element to the end of the list
sortedNodes.Add(node);
return sortedNodes.Count() - 1;
}
internal static void FindRightLine(Node node, IEnumerable<Line> lineList, out Line rightLine)
{
var counter = LinesToRight(node, lineList, out rightLine);
}
internal static void FindLeftLine(Node node, IEnumerable<Line> lineList, out Line leftLine)
{
var counter = LinesToLeft(node, lineList, out leftLine);
}
/// <summary>
/// Constructs a new trapezoid based on two nodes and two vertical lines.
/// </summary>
/// <param name="topNode"></param>
/// <param name="bottomNode"></param>
/// <param name="leftLine"></param>
/// <param name="rightLine"></param>
internal Trapezoid(Node topNode, Node bottomNode, Line leftLine, Line rightLine)
{
TopNode = topNode;
BottomNode = bottomNode;
LeftLine = leftLine;
RightLine = rightLine;
}
/// <summary>
/// Constructs a partial trapezoid
/// </summary>
/// <param name="topNode"></param>
/// <param name="leftLine"></param>
/// <param name="rightLine"></param>
internal PartialTrapezoid(Node topNode, Line leftLine, Line rightLine)
{
TopNode = topNode;
LeftLine = leftLine;
RightLine = rightLine;
}