voronoi.py

#############################################################################
#
# Voronoi diagram calculator/ Delaunay triangulator
# Translated to Python by Bill Simons
# September, 2005
#
# Ported to Python 3 by Joel Lawhead (GeospatialPython.com)
# August 2015
#
# 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

def cmp(a, b):
#http://python3porting.com/problems.html#comparisons
    return (a > b) - (a < b)

#------------------------------------------------------------------
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.extra_edges = []  # list of additional vertex 2-tubles (x,y) based on bounded voronoi tesselation
#        self.set_bounds(None)
#        self.use_bound = False
        self.xmin = self.ymin = self.xmax = self.ymax = None

    def circle(self,x,y,rad):
        pass

    def clip_line(self,edge,lid,rid):
        pass
        # here is where I will create false verticies if
        # the voronoi line extends to infinity...
        # the extra verticies will be added to the
        # extra edges list as 2-tuples
#        a,b,c = edge.a,edge.b,edge.c
#        if lid == -1:
#            x = self.xMin
#            y = (c-a*x) / b
#            if y < self.yMin or y > self.yMax:
#                if y < self.yMin: y = self.yMin
#                elif y > self.yMax: y = self.yMax
#                x = (c-b*y) / a
#            self.extra_edges.append((x,y))
#            lid = -(len(self.extra_edges)-1)
#        if rid == -1:
#            x = self.xMax
#            y = (c-a*x) / b
#            if y < self.yMin or y > self.yMax:
#                if y < self.yMin: y = self.yMin
#                elif y > self.yMax: y = self.yMax
#                x = (c-b*y) / a
#            self.extra_edges.append((x,y))
#            rid = -(len(self.extra_edges)-1)
#        print lid,rid
#        return (lid,rid)

    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, 3) #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.xmin is None or s.x < self.xmin:
            self.xmin = s.x
        elif self.xmax is None or s.x > self.xmax:
            self.xmax = s.x
        if self.ymin is None or s.y < self.ymin:
            self.ymin = s.y
        elif self.ymax is None or s.y > self.ymax:
            self.ymax = 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 sitenumL == -1 or sitenumR == -1 and self.use_bound:
#            sitenumL,sitenumR = self.clip_line(edge,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, end=' ')
                print(" %d " % sitenumL, end=' ')
                print("%d" % sitenumR)

    def set_bounds(self,bounds):
        if not bounds == None:
            self.xmin = bounds.xmin
            self.ymin = bounds.ymin
            self.xmax = bounds.xmax
            self.ymax = bounds.ymax
        else:
            self.xmin = self.ymin = self.xmax = self.ymax = None
        

#------------------------------------------------------------------
def voronoi(siteList,context):
    edgeList  = EdgeList(siteList.xmin,siteList.xmax,len(siteList))
    priorityQ = PriorityQueue(siteList.ymin,siteList.ymax,len(siteList))
    siteIter = siteList.iterator()
    
    bottomsite = next(siteIter)
    context.outSite(bottomsite)
    newsite = next(siteIter)
    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 = next(siteIter)

        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

#------------------------------------------------------------------
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 __lt__(self, other):
        if self.y < other.y:
            return True
        elif self.x < other.x:
            return True
        else:
            return False
            
    def __eq__(self, other):
        if self.y == other.y:
            return True
        elif self.x == other.x:
            return True
        else:
            return False

    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
#        print(s1.x, s1.y, s2.x, s2.y)
        # 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: ", end=' ')
        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 __lt__(self, other):
        if self.ystar < other.ystar:
            return True
        elif self.vertex.x < other.vertex.x:
            return True
        else:
            return False

    def __eq__(self, other):
        if self.ystar == other.ystar:
            return True
        elif self.vertex.x == other.vertex.x:
            return True
        else:
            return False

    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(key=lambda s: s.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 next(this.generator)
            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()
    context.set_bounds( siteList )
    voronoi( siteList, context )
    return ( context.vertices, context.lines, context.edges, (context.xmin,context.ymin,context.xmax,context.ymax))

#------------------------------------------------------------------
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)