private static double RightBreakpoint(RBTreeNode <BeachSection> node, double directrix)
{
var rightNode = node.Next;
//degenerate parabola
if ((node.Data.Site.Y - directrix).ApproxEqual(0))
{
return(node.Data.Site.X);
}
//node is the last piece of the beach line
if (rightNode == null)
{
return(double.PositiveInfinity);
}
//left node is degenerate
if ((rightNode.Data.Site.Y - directrix).ApproxEqual(0))
{
return(rightNode.Data.Site.X);
}
var site = node.Data.Site;
var rightSite = rightNode.Data.Site;
return(ParabolaMath.IntersectParabolaX(site.X, site.Y, rightSite.X, rightSite.Y, directrix));
}
private static double LeftBreakpoint(RBTreeNode <BeachSection> node, double directrix)
{
var leftNode = node.Previous;
//degenerate parabola
if ((node.Data.Site.Y - directrix).ApproxEqual(0))
{
return(node.Data.Site.X);
}
//node is the first piece of the beach line
if (leftNode == null)
{
return(double.NegativeInfinity);
}
//left node is degenerate
if ((leftNode.Data.Site.Y - directrix).ApproxEqual(0))
{
return(leftNode.Data.Site.X);
}
var site = node.Data.Site;
var leftSite = leftNode.Data.Site;
return(ParabolaMath.IntersectParabolaX(leftSite.X, leftSite.Y, site.X, site.Y, directrix));
}
private static void CheckCircle(RBTreeNode <BeachSection> section, MinHeap <FortuneEvent> eventQueue)
{
//if (section == null)
// return;
var left = section.Previous;
var right = section.Next;
if (left == null || right == null)
{
return;
}
var leftSite = left.Data.Site;
var centerSite = section.Data.Site;
var rightSite = right.Data.Site;
//if the left arc and right arc are defined by the same
//focus, the two arcs cannot converge
if (leftSite == rightSite)
{
return;
}
// http://mathforum.org/library/drmath/view/55002.html
// because every piece of this program needs to be demoed in maple >.<
//MATH HACKS: place center at origin and
//draw vectors a and c to
//left and right respectively
double bx = centerSite.X,
by = centerSite.Y,
ax = leftSite.X - bx,
ay = leftSite.Y - by,
cx = rightSite.X - bx,
cy = rightSite.Y - by;
//The center beach section can only dissapear when
//the angle between a and c is negative
var d = ax * cy - ay * cx;
if (d.ApproxGreaterThanOrEqualTo(0))
{
return;
}
var magnitudeA = ax * ax + ay * ay;
var magnitudeC = cx * cx + cy * cy;
var x = (cy * magnitudeA - ay * magnitudeC) / (2 * d);
var y = (ax * magnitudeC - cx * magnitudeA) / (2 * d);
//add back offset
var ycenter = y + by;
//y center is off
var circleEvent = new FortuneCircleEvent(
new VPoint(x + bx, ycenter + Math.Sqrt(x * x + y * y)),
ycenter, section
);
section.Data.CircleEvent = circleEvent;
eventQueue.Insert(circleEvent);
}
internal void AddBeachSection(FortuneSiteEvent siteEvent, MinHeap <FortuneEvent> eventQueue, HashSet <FortuneCircleEvent> deleted, LinkedList <VEdge> edges)
{
var site = siteEvent.Site;
var x = site.X;
var directrix = site.Y;
RBTreeNode <BeachSection> leftSection = null;
RBTreeNode <BeachSection> rightSection = null;
var node = beachLine.Root;
//find the parabola(s) above this site
while (node != null && leftSection == null && rightSection == null)
{
var distanceLeft = LeftBreakpoint(node, directrix) - x;
if (distanceLeft > 0)
{
//the new site is before the left breakpoint
if (node.Left == null)
{
rightSection = node;
}
else
{
node = node.Left;
}
continue;
}
var distanceRight = x - RightBreakpoint(node, directrix);
if (distanceRight > 0)
{
//the new site is after the right breakpoint
if (node.Right == null)
{
leftSection = node;
}
else
{
node = node.Right;
}
continue;
}
//the point lies below the left breakpoint
if (distanceLeft.ApproxEqual(0))
{
leftSection = node.Previous;
rightSection = node;
continue;
}
//the point lies below the right breakpoint
if (distanceRight.ApproxEqual(0))
{
leftSection = node;
rightSection = node.Next;
continue;
}
// distance Right < 0 and distance Left < 0
// this section is above the new site
leftSection = rightSection = node;
}
//our goal is to insert the new node between the
//left and right sections
var section = new BeachSection(site);
//left section could be null, in which case this node is the first
//in the tree
var newSection = beachLine.InsertSuccessor(leftSection, section);
//new beach section is the first beach section to be added
if (leftSection == null && rightSection == null)
{
return;
}
//main case:
//if both left section and right section point to the same valid arc
//we need to split the arc into a left arc and a right arc with our
//new arc sitting in the middle
if (leftSection != null && leftSection == rightSection)
{
//if the arc has a circle event, it was a false alarm.
//remove it
if (leftSection.Data.CircleEvent != null)
{
deleted.Add(leftSection.Data.CircleEvent);
leftSection.Data.CircleEvent = null;
}
//we leave the existing arc as the left section in the tree
//however we need to insert the right section defined by the arc
var copy = new BeachSection(leftSection.Data.Site);
rightSection = beachLine.InsertSuccessor(newSection, copy);
//grab the projection of this site onto the parabola
var y = ParabolaMath.EvalParabola(leftSection.Data.Site.X, leftSection.Data.Site.Y, directrix, x);
var intersection = new VPoint(x, y);
//create the two half edges corresponding to this intersection
var leftEdge = new VEdge(intersection, site, leftSection.Data.Site);
var rightEdge = new VEdge(intersection, leftSection.Data.Site, site);
leftEdge.Neighbor = rightEdge;
//put the edge in the list
edges.AddFirst(leftEdge);
//store the left edge on each arc section
newSection.Data.Edge = leftEdge;
rightSection.Data.Edge = rightEdge;
//store neighbors for delaunay
leftSection.Data.Site.Neighbors.Add(newSection.Data.Site);
newSection.Data.Site.Neighbors.Add(leftSection.Data.Site);
//create circle events
CheckCircle(leftSection, eventQueue);
CheckCircle(rightSection, eventQueue);
}
//site is the last beach section on the beach line
//this can only happen if all previous sites
//had the same y value
else if (leftSection != null && rightSection == null)
{
var start = new VPoint((leftSection.Data.Site.X + site.X) / 2, double.MinValue);
var infEdge = new VEdge(start, leftSection.Data.Site, site);
var newEdge = new VEdge(start, site, leftSection.Data.Site);
newEdge.Neighbor = infEdge;
edges.AddFirst(newEdge);
leftSection.Data.Site.Neighbors.Add(newSection.Data.Site);
newSection.Data.Site.Neighbors.Add(leftSection.Data.Site);
newSection.Data.Edge = newEdge;
//cant check circles since they are colinear
}
//site is directly above a break point
else if (leftSection != null && leftSection != rightSection)
{
//remove false alarms
if (leftSection.Data.CircleEvent != null)
{
deleted.Add(leftSection.Data.CircleEvent);
leftSection.Data.CircleEvent = null;
}
if (rightSection.Data.CircleEvent != null)
{
deleted.Add(rightSection.Data.CircleEvent);
rightSection.Data.CircleEvent = null;
}
//the breakpoint will dissapear if we add this site
//which means we will create an edge
//we treat this very similar to a circle event since
//an edge is finishing at the center of the circle
//created by circumscribing the left center and right
//sites
//bring a to the origin
var leftSite = leftSection.Data.Site;
var ax = leftSite.X;
var ay = leftSite.Y;
var bx = site.X - ax;
var by = site.Y - ay;
var rightSite = rightSection.Data.Site;
var cx = rightSite.X - ax;
var cy = rightSite.Y - ay;
var d = bx * cy - by * cx;
var magnitudeB = bx * bx + by * by;
var magnitudeC = cx * cx + cy * cy;
var vertex = new VPoint(
(cy * magnitudeB - by * magnitudeC) / (2 * d) + ax,
(bx * magnitudeC - cx * magnitudeB) / (2 * d) + ay);
rightSection.Data.Edge.End = vertex;
//next we create a two new edges
newSection.Data.Edge = new VEdge(vertex, site, leftSection.Data.Site);
rightSection.Data.Edge = new VEdge(vertex, rightSection.Data.Site, site);
edges.AddFirst(newSection.Data.Edge);
edges.AddFirst(rightSection.Data.Edge);
//add neighbors for delaunay
newSection.Data.Site.Neighbors.Add(leftSection.Data.Site);
leftSection.Data.Site.Neighbors.Add(newSection.Data.Site);
newSection.Data.Site.Neighbors.Add(rightSection.Data.Site);
rightSection.Data.Site.Neighbors.Add(newSection.Data.Site);
CheckCircle(leftSection, eventQueue);
CheckCircle(rightSection, eventQueue);
}
}