############################################################################# # # Voronoi diagram calculator/ Delaunay triangulator # Translated to Python by Bill Simons # September, 2005 # # Additional changes by Carson Farmer added November 2010 # # Calculate Delaunay triangulation or the Voronoi polygons for a set of # 2D input points. # # Derived from code bearing the following notice: # # The author of this software is Steven Fortune. Copyright (c) 1994 by AT&T # Bell Laboratories. # Permission to use, copy, modify, and distribute this software for any # purpose without fee is hereby granted, provided that this entire notice # is included in all copies of any software which is or includes a copy # or modification of this software and in all copies of the supporting # documentation for such software. # THIS SOFTWARE IS BEING PROVIDED "AS IS", WITHOUT ANY EXPRESS OR IMPLIED # WARRANTY. IN PARTICULAR, NEITHER THE AUTHORS NOR AT&T MAKE ANY # REPRESENTATION OR WARRANTY OF ANY KIND CONCERNING THE MERCHANTABILITY # OF THIS SOFTWARE OR ITS FITNESS FOR ANY PARTICULAR PURPOSE. # # Comments were incorporated from Shane O'Sullivan's translation of the # original code into C++ (http://mapviewer.skynet.ie/voronoi.html) # # Steve Fortune's homepage: http://netlib.bell-labs.com/cm/cs/who/sjf/index.html # ############################################################################# def usage(): print """ voronoi - compute Voronoi diagram or Delaunay triangulation voronoi [-t -p -d] [filename] Voronoi reads from filename (or standard input if no filename given) for a set of points in the plane and writes either the Voronoi diagram or the Delaunay triangulation to the standard output. Each input line should consist of two real numbers, separated by white space. If option -t is present, the Delaunay triangulation is produced. Each output line is a triple i j k, which are the indices of the three points in a Delaunay triangle. Points are numbered starting at 0. If option -t is not present, the Voronoi diagram is produced. There are four output record types. s a b indicates that an input point at coordinates a b was seen. l a b c indicates a line with equation ax + by = c. v a b indicates a vertex at a b. e l v1 v2 indicates a Voronoi segment which is a subsegment of line number l with endpoints numbered v1 and v2. If v1 or v2 is -1, the line extends to infinity. Other options include: d Print debugging info p Produce output suitable for input to plot (1), rather than the forms described above. On unsorted data uniformly distributed in the unit square, voronoi uses about 20n+140 bytes of storage. AUTHOR Steve J. Fortune (1987) A Sweepline Algorithm for Voronoi Diagrams, Algorithmica 2, 153-174. """ ############################################################################# # # For programmatic use two functions are available: # # computeVoronoiDiagram(points) # # Takes a list of point objects (which must have x and y fields). # Returns a 3-tuple of: # # (1) a list of 2-tuples, which are the x,y coordinates of the # Voronoi diagram vertices # (2) a list of 3-tuples (a,b,c) which are the equations of the # lines in the Voronoi diagram: a*x + b*y = c # (3) a list of 3-tuples, (l, v1, v2) representing edges of the # Voronoi diagram. l is the index of the line, v1 and v2 are # the indices of the vetices at the end of the edge. If # v1 or v2 is -1, the line extends to infinity. # # computeDelaunayTriangulation(points): # # Takes a list of point objects (which must have x and y fields). # Returns a list of 3-tuples: the indices of the points that form a # Delaunay triangle. # ############################################################################# import math import sys import getopt TOLERANCE = 1e-9 BIG_FLOAT = 1e38 #------------------------------------------------------------------ class Context(object): def __init__(self): self.doPrint = 0 self.debug = 0 self.plot = 0 self.triangulate = False self.vertices = [] # list of vertex 2-tuples: (x,y) self.lines = [] # equation of line 3-tuple (a b c), for the equation of the line a*x+b*y = c self.edges = [] # edge 3-tuple: (line index, vertex 1 index, vertex 2 index) if either vertex index is -1, the edge extends to infiinity self.triangles = [] # 3-tuple of vertex indices self.polygons = {} # a dict of site:[edges] pairs def circle(self,x,y,rad): pass def clip_line(self,edge): pass def line(self,x0,y0,x1,y1): pass def outSite(self,s): if(self.debug): print "site (%d) at %f %f" % (s.sitenum, s.x, s.y) elif(self.triangulate): pass elif(self.plot): self.circle (s.x, s.y, cradius) elif(self.doPrint): print "s %f %f" % (s.x, s.y) def outVertex(self,s): self.vertices.append((s.x,s.y)) if(self.debug): print "vertex(%d) at %f %f" % (s.sitenum, s.x, s.y) elif(self.triangulate): pass elif(self.doPrint and not self.plot): print "v %f %f" % (s.x,s.y) def outTriple(self,s1,s2,s3): self.triangles.append((s1.sitenum, s2.sitenum, s3.sitenum)) if(self.debug): print "circle through left=%d right=%d bottom=%d" % (s1.sitenum, s2.sitenum, s3.sitenum) elif(self.triangulate and self.doPrint and not self.plot): print "%d %d %d" % (s1.sitenum, s2.sitenum, s3.sitenum) def outBisector(self,edge): self.lines.append((edge.a, edge.b, edge.c)) if(self.debug): print "line(%d) %gx+%gy=%g, bisecting %d %d" % (edge.edgenum, edge.a, edge.b, edge.c, edge.reg[0].sitenum, edge.reg[1].sitenum) elif(self.triangulate): if(self.plot): self.line(edge.reg[0].x, edge.reg[0].y, edge.reg[1].x, edge.reg[1].y) elif(self.doPrint and not self.plot): print "l %f %f %f" % (edge.a, edge.b, edge.c) def outEdge(self,edge): sitenumL = -1 if edge.ep[Edge.LE] is not None: sitenumL = edge.ep[Edge.LE].sitenum sitenumR = -1 if edge.ep[Edge.RE] is not None: sitenumR = edge.ep[Edge.RE].sitenum if edge.reg[0].sitenum not in self.polygons: self.polygons[edge.reg[0].sitenum] = [] if edge.reg[1].sitenum not in self.polygons: self.polygons[edge.reg[1].sitenum] = [] self.polygons[edge.reg[0].sitenum].append((edge.edgenum,sitenumL,sitenumR)) self.polygons[edge.reg[1].sitenum].append((edge.edgenum,sitenumL,sitenumR)) self.edges.append((edge.edgenum,sitenumL,sitenumR)) if(not self.triangulate): if self.plot: self.clip_line(edge) elif(self.doPrint): print "e %d" % edge.edgenum, print " %d " % sitenumL, print "%d" % sitenumR #------------------------------------------------------------------ def voronoi(siteList,context): try: edgeList = EdgeList(siteList.xmin,siteList.xmax,len(siteList)) priorityQ = PriorityQueue(siteList.ymin,siteList.ymax,len(siteList)) siteIter = siteList.iterator() bottomsite = siteIter.next() context.outSite(bottomsite) newsite = siteIter.next() minpt = Site(-BIG_FLOAT,-BIG_FLOAT) while True: if not priorityQ.isEmpty(): minpt = priorityQ.getMinPt() if (newsite and (priorityQ.isEmpty() or cmp(newsite,minpt) < 0)): # newsite is smallest - this is a site event context.outSite(newsite) # get first Halfedge to the LEFT and RIGHT of the new site lbnd = edgeList.leftbnd(newsite) rbnd = lbnd.right # if this halfedge has no edge, bot = bottom site (whatever that is) # create a new edge that bisects bot = lbnd.rightreg(bottomsite) edge = Edge.bisect(bot,newsite) context.outBisector(edge) # create a new Halfedge, setting its pm field to 0 and insert # this new bisector edge between the left and right vectors in # a linked list bisector = Halfedge(edge,Edge.LE) edgeList.insert(lbnd,bisector) # if the new bisector intersects with the left edge, remove # the left edge's vertex, and put in the new one p = lbnd.intersect(bisector) if p is not None: priorityQ.delete(lbnd) priorityQ.insert(lbnd,p,newsite.distance(p)) # create a new Halfedge, setting its pm field to 1 # insert the new Halfedge to the right of the original bisector lbnd = bisector bisector = Halfedge(edge,Edge.RE) edgeList.insert(lbnd,bisector) # if this new bisector intersects with the right Halfedge p = bisector.intersect(rbnd) if p is not None: # push the Halfedge into the ordered linked list of vertices priorityQ.insert(bisector,p,newsite.distance(p)) newsite = siteIter.next() elif not priorityQ.isEmpty(): # intersection is smallest - this is a vector (circle) event # pop the Halfedge with the lowest vector off the ordered list of # vectors. Get the Halfedge to the left and right of the above HE # and also the Halfedge to the right of the right HE lbnd = priorityQ.popMinHalfedge() llbnd = lbnd.left rbnd = lbnd.right rrbnd = rbnd.right # get the Site to the left of the left HE and to the right of # the right HE which it bisects bot = lbnd.leftreg(bottomsite) top = rbnd.rightreg(bottomsite) # output the triple of sites, stating that a circle goes through them mid = lbnd.rightreg(bottomsite) context.outTriple(bot,top,mid) # get the vertex that caused this event and set the vertex number # couldn't do this earlier since we didn't know when it would be processed v = lbnd.vertex siteList.setSiteNumber(v) context.outVertex(v) # set the endpoint of the left and right Halfedge to be this vector if lbnd.edge.setEndpoint(lbnd.pm,v): context.outEdge(lbnd.edge) if rbnd.edge.setEndpoint(rbnd.pm,v): context.outEdge(rbnd.edge) # delete the lowest HE, remove all vertex events to do with the # right HE and delete the right HE edgeList.delete(lbnd) priorityQ.delete(rbnd) edgeList.delete(rbnd) # if the site to the left of the event is higher than the Site # to the right of it, then swap them and set 'pm' to RIGHT pm = Edge.LE if bot.y > top.y: bot,top = top,bot pm = Edge.RE # Create an Edge (or line) that is between the two Sites. This # creates the formula of the line, and assigns a line number to it edge = Edge.bisect(bot, top) context.outBisector(edge) # create a HE from the edge bisector = Halfedge(edge, pm) # insert the new bisector to the right of the left HE # set one endpoint to the new edge to be the vector point 'v' # If the site to the left of this bisector is higher than the right # Site, then this endpoint is put in position 0; otherwise in pos 1 edgeList.insert(llbnd, bisector) if edge.setEndpoint(Edge.RE - pm, v): context.outEdge(edge) # if left HE and the new bisector don't intersect, then delete # the left HE, and reinsert it p = llbnd.intersect(bisector) if p is not None: priorityQ.delete(llbnd); priorityQ.insert(llbnd, p, bot.distance(p)) # if right HE and the new bisector don't intersect, then reinsert it p = bisector.intersect(rrbnd) if p is not None: priorityQ.insert(bisector, p, bot.distance(p)) else: break he = edgeList.leftend.right while he is not edgeList.rightend: context.outEdge(he.edge) he = he.right Edge.EDGE_NUM = 0 except Exception, err: print "######################################################" print str(err) #------------------------------------------------------------------ def isEqual(a,b,relativeError=TOLERANCE): # is nearly equal to within the allowed relative error norm = max(abs(a),abs(b)) return (norm < relativeError) or (abs(a - b) < (relativeError * norm)) #------------------------------------------------------------------ class Site(object): def __init__(self,x=0.0,y=0.0,sitenum=0): self.x = x self.y = y self.sitenum = sitenum def dump(self): print "Site #%d (%g, %g)" % (self.sitenum,self.x,self.y) def __cmp__(self,other): if self.y < other.y: return -1 elif self.y > other.y: return 1 elif self.x < other.x: return -1 elif self.x > other.x: return 1 else: return 0 def distance(self,other): dx = self.x - other.x dy = self.y - other.y return math.sqrt(dx*dx + dy*dy) #------------------------------------------------------------------ class Edge(object): LE = 0 RE = 1 EDGE_NUM = 0 DELETED = {} # marker value def __init__(self): self.a = 0.0 self.b = 0.0 self.c = 0.0 self.ep = [None,None] self.reg = [None,None] self.edgenum = 0 def dump(self): print "(#%d a=%g, b=%g, c=%g)" % (self.edgenum,self.a,self.b,self.c) print "ep",self.ep print "reg",self.reg def setEndpoint(self, lrFlag, site): self.ep[lrFlag] = site if self.ep[Edge.RE - lrFlag] is None: return False return True @staticmethod def bisect(s1,s2): newedge = Edge() newedge.reg[0] = s1 # store the sites that this edge is bisecting newedge.reg[1] = s2 # to begin with, there are no endpoints on the bisector - it goes to infinity # ep[0] and ep[1] are None # get the difference in x dist between the sites dx = float(s2.x - s1.x) dy = float(s2.y - s1.y) adx = abs(dx) # make sure that the difference in positive ady = abs(dy) # get the slope of the line newedge.c = float(s1.x * dx + s1.y * dy + (dx*dx + dy*dy)*0.5) if adx > ady : # set formula of line, with x fixed to 1 newedge.a = 1.0 newedge.b = dy/dx newedge.c /= dx else: # set formula of line, with y fixed to 1 newedge.b = 1.0 newedge.a = dx/dy newedge.c /= dy newedge.edgenum = Edge.EDGE_NUM Edge.EDGE_NUM += 1 return newedge #------------------------------------------------------------------ class Halfedge(object): def __init__(self,edge=None,pm=Edge.LE): self.left = None # left Halfedge in the edge list self.right = None # right Halfedge in the edge list self.qnext = None # priority queue linked list pointer self.edge = edge # edge list Edge self.pm = pm self.vertex = None # Site() self.ystar = BIG_FLOAT def dump(self): print "Halfedge--------------------------" print "left: ", self.left print "right: ", self.right print "edge: ", self.edge print "pm: ", self.pm print "vertex: ", if self.vertex: self.vertex.dump() else: print "None" print "ystar: ", self.ystar def __cmp__(self,other): if self.ystar > other.ystar: return 1 elif self.ystar < other.ystar: return -1 elif self.vertex.x > other.vertex.x: return 1 elif self.vertex.x < other.vertex.x: return -1 else: return 0 def leftreg(self,default): if not self.edge: return default elif self.pm == Edge.LE: return self.edge.reg[Edge.LE] else: return self.edge.reg[Edge.RE] def rightreg(self,default): if not self.edge: return default elif self.pm == Edge.LE: return self.edge.reg[Edge.RE] else: return self.edge.reg[Edge.LE] # returns True if p is to right of halfedge self def isPointRightOf(self,pt): e = self.edge topsite = e.reg[1] right_of_site = pt.x > topsite.x if(right_of_site and self.pm == Edge.LE): return True if(not right_of_site and self.pm == Edge.RE): return False if(e.a == 1.0): dyp = pt.y - topsite.y dxp = pt.x - topsite.x fast = 0; if ((not right_of_site and e.b < 0.0) or (right_of_site and e.b >= 0.0)): above = dyp >= e.b * dxp fast = above else: above = pt.x + pt.y * e.b > e.c if(e.b < 0.0): above = not above if (not above): fast = 1 if (not fast): dxs = topsite.x - (e.reg[0]).x above = e.b * (dxp*dxp - dyp*dyp) < dxs*dyp*(1.0+2.0*dxp/dxs + e.b*e.b) if(e.b < 0.0): above = not above else: # e.b == 1.0 yl = e.c - e.a * pt.x t1 = pt.y - yl t2 = pt.x - topsite.x t3 = yl - topsite.y above = t1*t1 > t2*t2 + t3*t3 if(self.pm==Edge.LE): return above else: return not above #-------------------------- # create a new site where the Halfedges el1 and el2 intersect def intersect(self,other): e1 = self.edge e2 = other.edge if (e1 is None) or (e2 is None): return None # if the two edges bisect the same parent return None if e1.reg[1] is e2.reg[1]: return None d = e1.a * e2.b - e1.b * e2.a if isEqual(d,0.0): return None xint = (e1.c*e2.b - e2.c*e1.b) / d yint = (e2.c*e1.a - e1.c*e2.a) / d if(cmp(e1.reg[1],e2.reg[1]) < 0): he = self e = e1 else: he = other e = e2 rightOfSite = xint >= e.reg[1].x if((rightOfSite and he.pm == Edge.LE) or (not rightOfSite and he.pm == Edge.RE)): return None # create a new site at the point of intersection - this is a new # vector event waiting to happen return Site(xint,yint) #------------------------------------------------------------------ class EdgeList(object): def __init__(self,xmin,xmax,nsites): if xmin > xmax: xmin,xmax = xmax,xmin self.hashsize = int(2*math.sqrt(nsites+4)) self.xmin = xmin self.deltax = float(xmax - xmin) self.hash = [None]*self.hashsize self.leftend = Halfedge() self.rightend = Halfedge() self.leftend.right = self.rightend self.rightend.left = self.leftend self.hash[0] = self.leftend self.hash[-1] = self.rightend def insert(self,left,he): he.left = left he.right = left.right left.right.left = he left.right = he def delete(self,he): he.left.right = he.right he.right.left = he.left he.edge = Edge.DELETED # Get entry from hash table, pruning any deleted nodes def gethash(self,b): if(b < 0 or b >= self.hashsize): return None he = self.hash[b] if he is None or he.edge is not Edge.DELETED: return he # Hash table points to deleted half edge. Patch as necessary. self.hash[b] = None return None def leftbnd(self,pt): # Use hash table to get close to desired halfedge bucket = int(((pt.x - self.xmin)/self.deltax * self.hashsize)) if(bucket < 0): bucket =0; if(bucket >=self.hashsize): bucket = self.hashsize-1 he = self.gethash(bucket) if(he is None): i = 1 while True: he = self.gethash(bucket-i) if (he is not None): break; he = self.gethash(bucket+i) if (he is not None): break; i += 1 # Now search linear list of halfedges for the corect one if (he is self.leftend) or (he is not self.rightend and he.isPointRightOf(pt)): he = he.right while he is not self.rightend and he.isPointRightOf(pt): he = he.right he = he.left; else: he = he.left while (he is not self.leftend and not he.isPointRightOf(pt)): he = he.left # Update hash table and reference counts if(bucket > 0 and bucket < self.hashsize-1): self.hash[bucket] = he return he #------------------------------------------------------------------ class PriorityQueue(object): def __init__(self,ymin,ymax,nsites): self.ymin = ymin self.deltay = ymax - ymin self.hashsize = int(4 * math.sqrt(nsites)) self.count = 0 self.minidx = 0 self.hash = [] for i in range(self.hashsize): self.hash.append(Halfedge()) def __len__(self): return self.count def isEmpty(self): return self.count == 0 def insert(self,he,site,offset): he.vertex = site he.ystar = site.y + offset last = self.hash[self.getBucket(he)] next = last.qnext while((next is not None) and cmp(he,next) > 0): last = next next = last.qnext he.qnext = last.qnext last.qnext = he self.count += 1 def delete(self,he): if (he.vertex is not None): last = self.hash[self.getBucket(he)] while last.qnext is not he: last = last.qnext last.qnext = he.qnext self.count -= 1 he.vertex = None def getBucket(self,he): bucket = int(((he.ystar - self.ymin) / self.deltay) * self.hashsize) if bucket < 0: bucket = 0 if bucket >= self.hashsize: bucket = self.hashsize-1 if bucket < self.minidx: self.minidx = bucket return bucket def getMinPt(self): while(self.hash[self.minidx].qnext is None): self.minidx += 1 he = self.hash[self.minidx].qnext x = he.vertex.x y = he.ystar return Site(x,y) def popMinHalfedge(self): curr = self.hash[self.minidx].qnext self.hash[self.minidx].qnext = curr.qnext self.count -= 1 return curr #------------------------------------------------------------------ class SiteList(object): def __init__(self,pointList): self.__sites = [] self.__sitenum = 0 self.__xmin = pointList[0].x self.__ymin = pointList[0].y self.__xmax = pointList[0].x self.__ymax = pointList[0].y for i,pt in enumerate(pointList): self.__sites.append(Site(pt.x,pt.y,i)) if pt.x < self.__xmin: self.__xmin = pt.x if pt.y < self.__ymin: self.__ymin = pt.y if pt.x > self.__xmax: self.__xmax = pt.x if pt.y > self.__ymax: self.__ymax = pt.y self.__sites.sort() def setSiteNumber(self,site): site.sitenum = self.__sitenum self.__sitenum += 1 class Iterator(object): def __init__(this,lst): this.generator = (s for s in lst) def __iter__(this): return this def next(this): try: return this.generator.next() except StopIteration: return None def iterator(self): return SiteList.Iterator(self.__sites) def __iter__(self): return SiteList.Iterator(self.__sites) def __len__(self): return len(self.__sites) def _getxmin(self): return self.__xmin def _getymin(self): return self.__ymin def _getxmax(self): return self.__xmax def _getymax(self): return self.__ymax xmin = property(_getxmin) ymin = property(_getymin) xmax = property(_getxmax) ymax = property(_getymax) #------------------------------------------------------------------ def computeVoronoiDiagram(points): """ Takes a list of point objects (which must have x and y fields). Returns a 3-tuple of: (1) a list of 2-tuples, which are the x,y coordinates of the Voronoi diagram vertices (2) a list of 3-tuples (a,b,c) which are the equations of the lines in the Voronoi diagram: a*x + b*y = c (3) a list of 3-tuples, (l, v1, v2) representing edges of the Voronoi diagram. l is the index of the line, v1 and v2 are the indices of the vetices at the end of the edge. If v1 or v2 is -1, the line extends to infinity. """ siteList = SiteList(points) context = Context() voronoi(siteList,context) return (context.vertices,context.lines,context.edges) #------------------------------------------------------------------ def computeDelaunayTriangulation(points): """ Takes a list of point objects (which must have x and y fields). Returns a list of 3-tuples: the indices of the points that form a Delaunay triangle. """ siteList = SiteList(points) context = Context() context.triangulate = true voronoi(siteList,context) return context.triangles #----------------------------------------------------------------------------- if __name__=="__main__": try: optlist,args = getopt.getopt(sys.argv[1:],"thdp") except getopt.GetoptError: usage() sys.exit(2) doHelp = 0 c = Context() c.doPrint = 1 for opt in optlist: if opt[0] == "-d": c.debug = 1 if opt[0] == "-p": c.plot = 1 if opt[0] == "-t": c.triangulate = 1 if opt[0] == "-h": doHelp = 1 if not doHelp: pts = [] fp = sys.stdin if len(args) > 0: fp = open(args[0],'r') for line in fp: fld = line.split() x = float(fld[0]) y = float(fld[1]) pts.append(Site(x,y)) if len(args) > 0: fp.close() if doHelp or len(pts) == 0: usage() sys.exit(2) sl = SiteList(pts) voronoi(sl,c)