Fortune's Algorithm in C++

by Matt Brubeck

I wrote this code in 2002 for a Computational Geometry class taught by Greg Levin. Professor Levin did not grade on style, and portions of the code below are optimized for conciseness rather than clarity.

This visualization is useful for understanding my code:

The visualization shows a line that sweeps along the plane. (I call this the "sweep line," and use lowercase "L" to refer to its x-coordinate.) At any given time, there is a set of parabolic arcs defined by the line and each point it has passed. (I call this set the "parabolic front." The link above calls it the "beachline.")

Each time a new point is passed, a new parabola is added to the front. I call this a "site event." Whenever three parabolas intersect at the same point, this is where their Voronoi cell boundaries will intersect. I call this a "circle event" because if you draw a circle through all three points, the intersection happens at the center of the circle, and the sweep line will be at the rightmost edge of the circle. (Turn on "circles" in the visualization applet to see what I mean.)

The key difference between my implementation and the animated visualization above is that I don't calculate what happens at every location of the sweep line. Instead I jump ahead to the next event (whether it's a circle event or a site event), since those are the only places where new segments of the Voronoi diagram begin or end. I use a priority queue (sorted by x-coordinate) to efficiently keep track of pending events and process them in order.

For simplicity (because I had a limited time to finish this project) I used a linked list for the parabolic front. For optimal performance it should really be a binary search tree.

This code is a bit buggy; it produces incorrect output in certain cases. I have not taken the time to figure out these bugs.

P.S. Here's another great interactive version, this one in JavaScript.

#include "voronoi.hh"

priority_queue<point,  vector<point>,  gt> points; // site events
priority_queue<event*, vector<event*>, gt> events; // circle events

int main()
   // Read points from input.
   point p;
   while (cin >> p.x >> p.y) {

      // Keep track of bounding box size.
      if (p.x < X0) X0 = p.x;
      if (p.y < Y0) Y0 = p.y;
      if (p.x > X1) X1 = p.x;
      if (p.y > Y1) Y1 = p.y;
   // Add margins to the bounding box.
   double dx = (X1-X0+1)/5.0, dy = (Y1-Y0+1)/5.0;
   X0 -= dx;  X1 += dx;  Y0 -= dy;  Y1 += dy;

   // Process the queues; select the top element with smaller x coordinate.
   while (!points.empty())
      if (!events.empty() &&>x <=

   // After all points are processed, do the remaining circle events.
   while (!events.empty())

   finish_edges(); // Clean up dangling edges.
   print_output(); // Output the voronoi diagram.

void process_point()
   // Get the next point from the queue.
   point p =;

   // Add a new arc to the parabolic front.

void process_event()
   // Get the next event from the queue.
   event *e =;

   if (e->valid) {
      // Start a new edge.
      seg *s = new seg(e->p);

      // Remove the associated arc from the front.
      arc *a = e->a;
      if (a->prev) {
         a->prev->next = a->next;
         a->prev->s1 = s;
      if (a->next) {
         a->next->prev = a->prev;
         a->next->s0 = s;

      // Finish the edges before and after a.
      if (a->s0) a->s0->finish(e->p);
      if (a->s1) a->s1->finish(e->p);

      // Recheck circle events on either side of p:
      if (a->prev) check_circle_event(a->prev, e->x);
      if (a->next) check_circle_event(a->next, e->x);
   delete e;

void front_insert(point p)
   if (!root) {
      root = new arc(p);

   // Find the current arc(s) at height p.y (if there are any).
   for (arc *i = root; i; i = i->next) {
      point z, zz;
      if (intersect(p,i,&z)) {
         // New parabola intersects arc i.  If necessary, duplicate i.
         if (i->next && !intersect(p,i->next, &zz)) {
            i->next->prev = new arc(i->p,i,i->next);
            i->next = i->next->prev;
         else i->next = new arc(i->p,i);
         i->next->s1 = i->s1;

         // Add p between i and i->next.
         i->next->prev = new arc(p,i,i->next);
         i->next = i->next->prev;

         i = i->next; // Now i points to the new arc.

         // Add new half-edges connected to i's endpoints.
         i->prev->s1 = i->s0 = new seg(z);
         i->next->s0 = i->s1 = new seg(z);

         // Check for new circle events around the new arc:
         check_circle_event(i, p.x);
         check_circle_event(i->prev, p.x);
         check_circle_event(i->next, p.x);


   // Special case: If p never intersects an arc, append it to the list.
   arc *i;
   for (i = root; i->next; i=i->next) ; // Find the last node.

   i->next = new arc(p,i);
   // Insert segment between p and i
   point start;
   start.x = X0;
   start.y = (i->next->p.y + i->p.y) / 2;
   i->s1 = i->next->s0 = new seg(start);

// Look for a new circle event for arc i.
void check_circle_event(arc *i, double x0)
   // Invalidate any old event.
   if (i->e && i->e->x != x0)
      i->e->valid = false;
   i->e = NULL;

   if (!i->prev || !i->next)

   double x;
   point o;

   if (circle(i->prev->p, i->p, i->next->p, &x,&o) && x > x0) {
      // Create new event.
      i->e = new event(x, o, i);

// Find the rightmost point on the circle through a,b,c.
bool circle(point a, point b, point c, double *x, point *o)
   // Check that bc is a "right turn" from ab.
   if ((b.x-a.x)*(c.y-a.y) - (c.x-a.x)*(b.y-a.y) > 0)
      return false;

   // Algorithm from O'Rourke 2ed p. 189.
   double A = b.x - a.x,  B = b.y - a.y,
          C = c.x - a.x,  D = c.y - a.y,
          E = A*(a.x+b.x) + B*(a.y+b.y),
          F = C*(a.x+c.x) + D*(a.y+c.y),
          G = 2*(A*(c.y-b.y) - B*(c.x-b.x));

   if (G == 0) return false;  // Points are co-linear.

   // Point o is the center of the circle.
   o->x = (D*E-B*F)/G;
   o->y = (A*F-C*E)/G;

   // o.x plus radius equals max x coordinate.
   *x = o->x + sqrt( pow(a.x - o->x, 2) + pow(a.y - o->y, 2) );
   return true;

// Will a new parabola at point p intersect with arc i?
bool intersect(point p, arc *i, point *res)
   if (i->p.x == p.x) return false;

   double a,b;
   if (i->prev) // Get the intersection of i->prev, i.
      a = intersection(i->prev->p, i->p, p.x).y;
   if (i->next) // Get the intersection of i->next, i.
      b = intersection(i->p, i->next->p, p.x).y;

   if ((!i->prev || a <= p.y) && (!i->next || p.y <= b)) {
      res->y = p.y;

      // Plug it back into the parabola equation.
      res->x = (i->p.x*i->p.x + (i->p.y-res->y)*(i->p.y-res->y) - p.x*p.x)
                / (2*i->p.x - 2*p.x);

      return true;
   return false;

// Where do two parabolas intersect?
point intersection(point p0, point p1, double l)
   point res, 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 - sqrt(b*b - 4*a*c) ) / (2*a);
   // Plug back into one of the parabola equations.
   res.x = (p.x*p.x + (p.y-res.y)*(p.y-res.y) - l*l)/(2*p.x-2*l);
   return res;

void finish_edges()
   // Advance the sweep line so no parabolas can cross the bounding box.
   double l = X1 + (X1-X0) + (Y1-Y0);

   // Extend each remaining segment to the new parabola intersections.
   for (arc *i = root; i->next; i = i->next)
      if (i->s1)
         i->s1->finish(intersection(i->p, i->next->p, l*2));

void print_output()
   // Bounding box coordinates.
   cout << X0 << " "<< X1 << " " << Y0 << " " << Y1 << endl;

   // Each output segment in four-column format.
   vector<seg*>::iterator i;
   for (i = output.begin(); i != output.end(); i++) {
      point p0 = (*i)->start;
      point p1 = (*i)->end;
      cout << p0.x << " " << p0.y << " " << p1.x << " " << p1.y << endl;


#include <iostream>
#include <queue>
#include <set>
#include <math.h>

using namespace std;

// Notation for working with points
typedef pair<double, double> point;
#define x first
#define y second

// Arc, event, and segment datatypes
struct arc;
struct seg;

struct event {
   double x;
   point p;
   arc *a;
   bool valid;

   event(double xx, point pp, arc *aa)
      : x(xx), p(pp), a(aa), valid(true) {}

struct arc {
   point p;
   arc *prev, *next;
   event *e;

   seg *s0, *s1;

   arc(point pp, arc *a=0, arc *b=0)
    : p(pp), prev(a), next(b), e(0), s0(0), s1(0) {}

vector<seg*> output;  // Array of output segments.

struct seg {
   point start, end;
   bool done;

   seg(point p)
      : start(p), end(0,0), done(false)
   { output.push_back(this); }

   // Set the end point and mark as "done."
   void finish(point p) { if (done) return; end = p; done = true; }

arc *root = 0; // First item in the parabolic front linked list.

// Function declarations
void process_point();
void process_event();
void front_insert(point  p);

bool circle(point a, point b, point c, double *x, point *o);
void check_circle_event(arc *i, double x0);

bool intersect(point p, arc *i, point *result = 0);
point intersection(point p0, point p1, double l);

void finish_edges();
void print_output();

// "Greater than" comparison, for reverse sorting in priority queue.
struct gt {
   bool operator()(point a, point b) {return a.x==b.x ? a.y>b.y : a.x>b.x;}
   bool operator()(event *a, event *b) {return a->x>b->x;}

// Bounding box coordinates.
double X0 = 0, X1 = 0, Y0 = 0, Y1 = 0;