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#############################################################################
#
# Voronoi diagram calculator/ Delaunay triangulator
# Translated to Python by Bill Simons
# September, 2005
#
# 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.has_edge = {} # edge belongs to 2 vertices
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
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):
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)
try:
context.has_edge[bot.sitenum].append(edge.edgenum)
except:
context.has_edge[bot.sitenum]=[ edge.edgenum ]
try:
context.has_edge[newsite.sitenum].append(edge.edgenum)
except:
context.has_edge[newsite.sitenum]=[ edge.edgenum ]
# 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)
try:
context.has_edge[bot.sitenum].append(edge.edgenum)
except:
context.has_edge[bot.sitenum]=[edge.edgenum]
try:
context.has_edge[top.sitenum].append(edge.edgenum)
except:
context.has_edge[top.sitenum]=[edge.edgenum]
# 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 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)