I used Fortune Algorithm to find the Voronoi point set diagram. I will return to the list of line segments, but I need to know which segments form closed polygons, and combine them into an object hashed by the starting point that they surround.
What could be the fastest way to find them? Should I save some important information from the algorithm? If so, then what?
Here is my implementation of the Fortune algorithm in Java ported from a C ++ implementation here
class Voronoi { // The set of points that control the centers of the cells private LinkedList<Point> pts; // A list of line segments that defines where the cells are divided private LinkedList<Edge> output; // The sites that have not yet been processed, in acending order of X coordinate private PriorityQueue sites; // Possible upcoming cirlce events in acending order of X coordinate private PriorityQueue events; // The root of the binary search tree of the parabolic wave front private Arc root; void runFortune(LinkedList pts) { sites.clear(); events.clear(); output.clear(); root = null; Point p; ListIterator i = pts.listIterator(0); while (i.hasNext()) { sites.offer(i.next()); } // Process the queues; select the top element with smaller x coordinate. while (sites.size() > 0) { if ((events.size() > 0) && ((((CircleEvent) events.peek()).xpos) <= (((Point) sites.peek()).x))) { processCircleEvent((CircleEvent) events.poll()); } else { //process a site event by adding a curve to the parabolic front frontInsert((Point) sites.poll()); } } // After all points are processed, do the remaining circle events. while (events.size() > 0) { processCircleEvent((CircleEvent) events.poll()); } // Clean up dangling edges. finishEdges(); } private void processCircleEvent(CircleEvent event) { if (event.valid) { //start a new edge Edge edgy = new Edge(event.p); // Remove the associated arc from the front. Arc parc = event.a; if (parc.prev != null) { parc.prev.next = parc.next; parc.prev.edge1 = edgy; } if (parc.next != null) { parc.next.prev = parc.prev; parc.next.edge0 = edgy; } // Finish the edges before and after this arc. if (parc.edge0 != null) { parc.edge0.finish(event.p); } if (parc.edge1 != null) { parc.edge1.finish(event.p); } // Recheck circle events on either side of p: if (parc.prev != null) { checkCircleEvent(parc.prev, event.xpos); } if (parc.next != null) { checkCircleEvent(parc.next, event.xpos); } } } void frontInsert(Point focus) { if (root == null) { root = new Arc(focus); return; } Arc parc = root; while (parc != null) { CircleResultPack rez = intersect(focus, parc); if (rez.valid) { // New parabola intersects parc. If necessary, duplicate parc. if (parc.next != null) { CircleResultPack rezz = intersect(focus, parc.next); if (!rezz.valid){ Arc bla = new Arc(parc.focus); bla.prev = parc; bla.next = parc.next; parc.next.prev = bla; parc.next = bla; } } else { parc.next = new Arc(parc.focus); parc.next.prev = parc; } parc.next.edge1 = parc.edge1; // Add new arc between parc and parc.next. Arc bla = new Arc(focus); bla.prev = parc; bla.next = parc.next; parc.next.prev = bla; parc.next = bla; parc = parc.next; // Now parc points to the new arc. // Add new half-edges connected to parc endpoints. parc.edge0 = new Edge(rez.center); parc.prev.edge1 = parc.edge0; parc.edge1 = new Edge(rez.center); parc.next.edge0 = parc.edge1; // Check for new circle events around the new arc: checkCircleEvent(parc, focus.x); checkCircleEvent(parc.prev, focus.x); checkCircleEvent(parc.next, focus.x); return; } //proceed to next arc parc = parc.next; } // Special case: If p never intersects an arc, append it to the list. parc = root; while (parc.next != null) { parc = parc.next; // Find the last node. } parc.next = new Arc(focus); parc.next.prev = parc; Point start = new Point(0, (parc.next.focus.y + parc.focus.y) / 2); parc.next.edge0 = new Edge(start); parc.edge1 = parc.next.edge0; } void checkCircleEvent(Arc parc, double xpos) { // Invalidate any old event. if ((parc.event != null) && (parc.event.xpos != xpos)) { parc.event.valid = false; } parc.event = null; if ((parc.prev == null) || (parc.next == null)) { return; } CircleResultPack result = circle(parc.prev.focus, parc.focus, parc.next.focus); if (result.valid && result.rightmostX > xpos) { // Create new event. parc.event = new CircleEvent(result.rightmostX, result.center, parc); events.offer(parc.event); } } // Find the rightmost point on the circle through a,b,c. CircleResultPack circle(Point a, Point b, Point c) { CircleResultPack result = new CircleResultPack(); // Check that bc is a "right turn" from ab. if ((bx - ax) * (cy - ay) - (cx - ax) * (by - ay) > 0) { result.valid = false; return result; } // Algorithm from O'Rourke 2ed p. 189. double A = bx - ax; double B = by - ay; double C = cx - ax; double D = cy - ay; double E = A * (ax + bx) + B * (ay + by); double F = C * (ax + cx) + D * (ay + cy); double G = 2 * (A * (cy - by) - B * (cx - bx)); if (G == 0) { // Points are co-linear. result.valid = false; return result; } // centerpoint of the circle. Point o = new Point((D * E - B * F) / G, (A * F - C * E) / G); result.center = o; // ox plus radius equals max x coordinate. result.rightmostX = ox + Math.sqrt(Math.pow(ax - ox, 2.0) + Math.pow(ay - oy, 2.0)); result.valid = true; return result; } // Will a new parabola at point p intersect with arc i? CircleResultPack intersect(Point p, Arc i) { CircleResultPack res = new CircleResultPack(); res.valid = false; if (i.focus.x == px) { return res; } double a = 0.0; double b = 0.0; if (i.prev != null) // Get the intersection of i->prev, i. { a = intersection(i.prev.focus, i.focus, px).y; } if (i.next != null) // Get the intersection of i->next, i. { b = intersection(i.focus, i.next.focus, px).y; } if ((i.prev == null || a <= py) && (i.next == null || py <= b)) { res.center = new Point(0, py); // Plug it back into the parabola equation to get the x coordinate res.center.x = (i.focus.x * i.focus.x + (i.focus.y - res.center.y) * (i.focus.y - res.center.y) - px * px) / (2 * i.focus.x - 2 * px); res.valid = true; return res; } return res; } // Where do two parabolas intersect? Point intersection(Point p0, Point p1, double l) { Point res = new Point(0, 0); Point p = p0; if (p0.x == p1.x) { res.y = (p0.y + p1.y) / 2; } else if (p1.x == l) { res.y = p1.y; } else if (p0.x == l) { res.y = p0.y; p = p1; } else { // Use the quadratic formula. double z0 = 2 * (p0.x - l); double z1 = 2 * (p1.x - l); double a = 1 / z0 - 1 / z1; double b = -2 * (p0.y / z0 - p1.y / z1); double c = (p0.y * p0.y + p0.x * p0.x - l * l) / z0 - (p1.y * p1.y + p1.x * p1.x - l * l) / z1; res.y = (-b - Math.sqrt((b * b - 4 * a * c))) / (2 * a); } // Plug back into one of the parabola equations. res.x = (px * px + (py - res.y) * (py - res.y) - l * l) / (2 * px - 2 * l); return res; } void finishEdges() { // Advance the sweep line so no parabolas can cross the bounding box. double l = gfx.width * 2 + gfx.height; // Extend each remaining segment to the new parabola intersections. Arc i = root; while (i != null) { if (i.edge1 != null) { i.edge1.finish(intersection(i.focus, i.next.focus, l * 2)); } i = i.next; } } class Point implements Comparable<Point> { public double x, y; //public Point goal; public Point(double X, double Y) { x = X; y = Y; } public int compareTo(Point foo) { return ((Double) this.x).compareTo((Double) foo.x); } } class CircleEvent implements Comparable<CircleEvent> { public double xpos; public Point p; public Arc a; public boolean valid; public CircleEvent(double X, Point P, Arc A) { xpos = X; a = A; p = P; valid = true; } public int compareTo(CircleEvent foo) { return ((Double) this.xpos).compareTo((Double) foo.xpos); } } class Edge { public Point start, end; public boolean done; public Edge(Point p) { start = p; end = new Point(0, 0); done = false; output.add(this); } public void finish(Point p) { if (done) { return; } end = p; done = true; } } class Arc { //parabolic arc is the set of points eqadistant from a focus point and the beach line public Point focus; //these object exsit in a linked list public Arc next, prev; // public CircleEvent event; // public Edge edge0, edge1; public Arc(Point p) { focus = p; next = null; prev = null; event = null; edge0 = null; edge1 = null; } } class CircleResultPack { // stupid Java doesnt let me return multiple variables without doing this public boolean valid; public Point center; public double rightmostX; } }
(I know that it will not compile, data structures must be initialized and missing import)
I want this:
LinkedList<Poly> polys; //contains all polygons created by Voronoi edges class Poly { //defines a single polygon public Point locus; public LinkedList<Points> verts; }
The most direct brute force method I can think of is to create an undirected graph of points on the diagram (end points of edges), with one record for each point and one connection for each edge between the point (without duplicates), then find all the loops in this graph, then for each set of loops that separate 3 or more points, discard everything except the shortest. However, this will be too slow.