def as_box(quadrant):
""""Convert a quadrant of the form: ((x_min,y_min),width) to a box: ((x_min,y_min),(x_max,y_max))."""
width = quadrant[1]
minp = quadrant[0]
maxp = tuple(xy + width for xy in minp)
assert (x(minp) <= x(maxp) and y(minp) <= y(maxp))
return (minp, maxp)
def as_box( quadrant ):
""""Convert a quadrant of the form: ((x_min,y_min),width) to a box: ((x_min,y_min),(x_max,y_max))."""
width = quadrant[1]
minp = quadrant[0]
maxp = tuple(xy+width for xy in minp)
assert( x(minp) <= x(maxp) and y(minp) <= y(maxp) )
return (minp,maxp)
def bounds(vertices):
"""Return the iso-axis rectangle enclosing the given points"""
# find vertices set bounds
xmin = x(vertices[0])
ymin = y(vertices[0])
xmax = xmin
ymax = ymin
# we do not use min(vertices,key=x) because it would iterate 4 times over the list, instead of just one
for v in vertices:
xmin = min(x(v), xmin)
xmax = max(x(v), xmax)
ymin = min(y(v), ymin)
ymax = max(y(v), ymax)
return (xmin, ymin), (xmax, ymax)
def bounds( vertices ):
"""Return the iso-axis rectangle enclosing the given points"""
# find vertices set bounds
xmin = x(vertices[0])
ymin = y(vertices[0])
xmax = xmin
ymax = ymin
# we do not use min(vertices,key=x) because it would iterate 4 times over the list, instead of just one
for v in vertices:
xmin = min(x(v),xmin)
xmax = max(x(v),xmax)
ymin = min(y(v),ymin)
ymax = max(y(v),ymax)
return (xmin,ymin),(xmax,ymax)
def init(self, quadrant=None, box=None, points=None):
"""Initialize the root quadrant with the given quadrant, the given box or the given set of points."""
if len([k for k in (box, points, quadrant) if k]) > 1:
raise BaseException(
"ERROR: you should specify a box, a quadrant or points")
# Initialize the root quadrant as the given box
if box:
minp, maxp = box
width = max(x(maxp) - x(minp), y(maxp) - y(minp))
# Initialize the root quadrant as the box around the points
elif points:
minp, maxp = geometry.box(points)
width = max(x(maxp) - x(minp), y(maxp) - y(minp))
# Initialize the root quadrant as the given origin point and width
elif quadrant:
minp = quadrant[0]
width = quadrant[1]
assert (x(minp) <= x(minp) + width and y(minp) <= y(minp) + width)
# There is always the root quadrant in the list of available ones.
root = (minp, width)
quadrants = [root]
return root, quadrants
def init( self, quadrant = None, box = None, points = None ):
"""Initialize the root quadrant with the given quadrant, the given box or the given set of points."""
if len([k for k in (box,points,quadrant) if k]) > 1:
raise BaseException("ERROR: you should specify a box, a quadrant or points")
# Initialize the root quadrant as the given box
if box:
minp,maxp = box
width = max( x(maxp)-x(minp), y(maxp)-y(minp) )
# Initialize the root quadrant as the box around the points
elif points:
minp,maxp = geometry.box( points )
width = max( x(maxp)-x(minp), y(maxp)-y(minp) )
# Initialize the root quadrant as the given origin point and width
elif quadrant:
minp = quadrant[0]
width = quadrant[1]
assert( x(minp) <= x(minp)+width and y(minp) <= y(minp)+width )
# There is always the root quadrant in the list of available ones.
root = (minp,width)
quadrants = [ root ]
return root,quadrants
def in_circle(p, center, radius, epsilon=sys.float_info.epsilon):
"""Return True if the given point p is in the given circle"""
assert (len(p) == 2)
cx, cy = center
dxp = x(p) - cx
dyp = y(p) - cy
dr = math.sqrt(dxp**2 + dyp**2)
if (dr - radius) <= epsilon:
return True
else:
return False
def in_circle( p, center, radius, epsilon = sys.float_info.epsilon ):
"""Return True if the given point p is in the given circle"""
assert( len(p) == 2 )
cx,cy = center
dxp = x(p) - cx
dyp = y(p) - cy
dr = math.sqrt(dxp**2 + dyp**2)
if (dr - radius) <= epsilon:
return True
else:
return False
def circumcircle(triangle, epsilon=sys.float_info.epsilon):
"""Compute the circumscribed circle of a triangle and
Return a 2-tuple: ( (center_x, center_y), radius )"""
assert (len(triangle) == 3)
p0, p1, p2 = triangle
assert (len(p0) == 2)
assert (len(p1) == 2)
assert (len(p2) == 2)
dy01 = abs(y(p0) - y(p1))
dy12 = abs(y(p1) - y(p2))
if dy01 < epsilon and dy12 < epsilon:
# coincident points
raise CoincidentPointsError
elif dy01 < epsilon:
m12 = mtan(p2, p1)
mx12, my12 = middle(p1, p2)
cx = mid(x, p1, p0)
cy = m12 * (cx - mx12) + my12
elif dy12 < epsilon:
m01 = mtan(p1, p0)
mx01, my01 = middle(p0, p1)
cx = mid(x, p2, p1)
cy = m01 * (cx - mx01) + my01
else:
m01 = mtan(p1, p0)
m12 = mtan(p2, p1)
mx01, my01 = middle(p0, p1)
mx12, my12 = middle(p1, p2)
cx = (m01 * mx01 - m12 * mx12 + my12 - my01) / (m01 - m12)
if dy01 > dy12:
cy = m01 * (cx - mx01) + my01
else:
cy = m12 * (cx - mx12) + my12
dx1 = x(p1) - cx
dy1 = y(p1) - cy
r = math.sqrt(dx1**2 + dy1**2)
return (cx, cy), r
def circumcircle( triangle, epsilon = sys.float_info.epsilon ):
"""Compute the circumscribed circle of a triangle and
Return a 2-tuple: ( (center_x, center_y), radius )"""
assert( len(triangle) == 3 )
p0,p1,p2 = triangle
assert( len(p0) == 2 )
assert( len(p1) == 2 )
assert( len(p2) == 2 )
dy01 = abs( y(p0) - y(p1) )
dy12 = abs( y(p1) - y(p2) )
if dy01 < epsilon and dy12 < epsilon:
# coincident points
raise CoincidentPointsError
elif dy01 < epsilon:
m12 = mtan( p2,p1 )
mx12,my12 = middle( p1, p2 )
cx = mid( x, p1, p0 )
cy = m12 * (cx - mx12) + my12
elif dy12 < epsilon:
m01 = mtan( p1, p0 )
mx01,my01 = middle( p0, p1 )
cx = mid( x, p2, p1 )
cy = m01 * ( cx - mx01 ) + my01
else:
m01 = mtan( p1, p0 )
m12 = mtan( p2, p1 )
mx01,my01 = middle( p0, p1 )
mx12,my12 = middle( p1, p2 )
cx = ( m01 * mx01 - m12 * mx12 + my12 - my01 ) / ( m01 - m12 )
if dy01 > dy12:
cy = m01 * ( cx - mx01 ) + my01
else:
cy = m12 * ( cx - mx12 ) + my12
dx1 = x(p1) - cx
dy1 = y(p1) - cy
r = math.sqrt(dx1**2 + dy1**2)
return (cx,cy),r
def write( graph, stream ):
for k in graph:
stream.write( "%f,%f:" % (x(k),y(k)) )
for p in graph[k]:
stream.write( "%f,%f " % (x(p),y(p)) )
stream.write("\n")
def write_points( points, stream ):
for p in points:
stream.write( "%f,%f\n" % ( x(p),y(p) ) )
def write_segments( segments, stream ):
for seg in segments:
for p in seg:
stream.write( "%f,%f " % ( x(p),y(p) ) )
stream.write( "\n" )
def turn(p, q, r):
"""Returns -1, 0, 1 if the sequence of points (p,q,r) forms a right, straight, or left turn."""
qr = (x(q) - x(p)) * (y(r) - y(p))
rq = (x(r) - x(p)) * (y(q) - y(p))
# cmp(x,y) returns -1 if x<y, 0 if x==y, +1 if x>y
return cmp(qr - rq, 0)
def write_points(points, stream):
for p in points:
stream.write("%f,%f\n" % (x(p), y(p)))
def mtan( pa, pb ):
return -1 * ( x(pa) - x(pb) ) / ( y(pa) - y(pb) )
def delaunay_bowyer_watson(points,
supertri=None,
superdelta=0.1,
epsilon=sys.float_info.epsilon,
do_plot=None,
plot_filename="Bowyer-Watson_%i.png"):
"""Return the Delaunay triangulation of the given points
epsilon: used for floating point comparisons, two points are considered equals if their distance is < epsilon.
do_plot: if not None, plot intermediate steps on this matplotlib object and save them as images named: plot_filename % i
"""
if do_plot and len(points) > 10:
print "WARNING it is a bad idea to plot each steps of a triangulation of many points"
# Sort points first on the x-axis, then on the y-axis.
vertices = sorted(points)
# LOGN( "super-triangle",supertri )
if not supertri:
supertri = supertriangle(vertices, superdelta)
# It is the first triangle of the list.
triangles = [supertri]
completed = {supertri: False}
# The predicate returns true if at least one of the vertices
# is also found in the supertriangle.
def match_supertriangle(tri):
if tri[0] in supertri or \
tri[1] in supertri or \
tri[2] in supertri:
return True
# Returns the base of each plots, with points, current triangulation, super-triangle and bounding box.
def plot_base(ax, vi=len(vertices), vertex=None):
ax.set_aspect('equal')
# regular points
scatter_x = [p[0] for p in vertices[:vi]]
scatter_y = [p[1] for p in vertices[:vi]]
ax.scatter(scatter_x, scatter_y, s=30, marker='o', facecolor="black")
# super-triangle vertices
scatter_x = [p[0] for p in list(supertri)]
scatter_y = [p[1] for p in list(supertri)]
ax.scatter(scatter_x,
scatter_y,
s=30,
marker='o',
facecolor="lightgrey",
edgecolor="lightgrey")
# current vertex
if vertex:
ax.scatter(vertex[0],
vertex[1],
s=30,
marker='o',
facecolor="red",
edgecolor="red")
# current triangulation
uberplot.plot_segments(ax,
edges_of(triangles),
edgecolor="blue",
alpha=0.5,
linestyle='solid')
# bounding box
(xmin, ymin), (xmax, ymax) = bounds(vertices)
uberplot.plot_segments(ax,
tour([(xmin, ymin), (xmin, ymax), (xmax, ymax),
(xmax, ymin)]),
edgecolor="magenta",
alpha=0.2,
linestyle='dotted')
# Insert vertices one by one.
LOG("Insert vertices: ")
if do_plot:
it = 0
for vi, vertex in enumerate(vertices):
# LOGN( "\tvertex",vertex )
assert (len(vertex) == 2)
if do_plot:
ax = do_plot.add_subplot(111)
plot_base(ax, vi, vertex)
# All the triangles whose circumcircle encloses the point to be added are identified,
# the outside edges of those triangles form an enclosing polygon.
# Forget previous candidate polygon's edges.
enclosing = []
removed = []
for triangle in triangles:
# LOGN( "\t\ttriangle",triangle )
assert (len(triangle) == 3)
# Do not consider triangles already tested.
# If completed has a key, test it, else return False.
if completed.get(triangle, False):
# LOGN( "\t\t\tAlready completed" )
# if do_plot:
# uberplot.plot_segments( ax, tour(list(triangle)), edgecolor = "magenta", alpha=1, lw=1, linestyle='dotted' )
continue
# LOGN( "\t\t\tCircumcircle" )
assert (triangle[0] != triangle[1] and triangle[1] != triangle[2]
and triangle[2] != triangle[0])
center, radius = circumcircle(triangle, epsilon)
# If it match Delaunay's conditions.
if x(center) < x(vertex) and math.sqrt(
(x(vertex) - x(center))**2) > radius:
# LOGN( "\t\t\tMatch Delaunay, mark as completed" )
completed[triangle] = True
# If the current vertex is inside the circumscribe circle of the current triangle,
# add the current triangle's edges to the candidate polygon.
if in_circle(vertex, center, radius, epsilon):
# LOGN( "\t\t\tIn circumcircle, add to enclosing polygon",triangle )
if do_plot:
circ = plot.Circle(center,
radius,
facecolor='yellow',
edgecolor="orange",
alpha=0.2,
clip_on=False)
ax.add_patch(circ)
for p0, p1 in tour(list(triangle)):
# Then add this edge to the polygon enclosing the vertex,
enclosing.append((p0, p1))
# and remove the corresponding triangle from the current triangulation.
removed.append(triangle)
completed.pop(triangle, None)
elif do_plot:
circ = plot.Circle(center,
radius,
facecolor='lightgrey',
edgecolor="grey",
alpha=0.2,
clip_on=False)
ax.add_patch(circ)
# end for triangle in triangles
# The triangles in the enclosing polygon are deleted and
# new triangles are formed between the point to be added and
# each outside edge of the enclosing polygon.
# Actually remove triangles.
for triangle in removed:
triangles.remove(triangle)
# Remove duplicated edges.
# This leaves the edges of the enclosing polygon only,
# because enclosing edges are only in a single triangle,
# but edges inside the polygon are at least in two triangles.
hull = []
for i, (p0, p1) in enumerate(enclosing):
# Clockwise edges can only be in the remaining part of the list.
# Search for counter-clockwise edges as well.
if (p0, p1) not in enclosing[i + 1:] and (p1, p0) not in enclosing:
hull.append((p0, p1))
elif do_plot:
uberplot.plot_segments(ax, [(p0, p1)],
edgecolor="white",
alpha=1,
lw=1,
linestyle='dotted')
if do_plot:
uberplot.plot_segments(ax,
hull,
edgecolor="red",
alpha=1,
lw=1,
linestyle='solid')
# Create new triangles using the current vertex and the enclosing hull.
# LOGN( "\t\tCreate new triangles" )
for p0, p1 in hull:
assert (p0 != p1)
triangle = tuple([p0, p1, vertex])
# LOGN("\t\t\tNew triangle",triangle)
triangles.append(triangle)
completed[triangle] = False
if do_plot:
uberplot.plot_segments(ax, [(p0, vertex), (p1, vertex)],
edgecolor="green",
alpha=1,
linestyle='solid')
if do_plot:
plot.savefig(plot_filename % it, dpi=150)
plot.clf()
it += 1
LOG(".")
# end for vertex in vertices
LOGN(" done")
# Remove triangles that have at least one of the supertriangle vertices.
# LOGN( "\tRemove super-triangles" )
# Filter out elements for which the predicate is False,
# here: *keep* elements that *do not* have a common vertex.
# The filter is a generator, so we must make a list with it to actually get the data.
triangulation = list(filter_if_not(match_supertriangle, triangles))
if do_plot:
ax = do_plot.add_subplot(111)
plot_base(ax)
uberplot.plot_segments(ax,
edges_of(triangles),
edgecolor="red",
alpha=0.5,
linestyle='solid')
uberplot.plot_segments(ax,
edges_of(triangulation),
edgecolor="blue",
alpha=1,
linestyle='solid')
plot.savefig(plot_filename % it, dpi=150)
plot.clf()
return triangulation
def mtan(pa, pb):
return -1 * (x(pa) - x(pb)) / (y(pa) - y(pb))
def delaunay_bowyer_watson( points, supertri = None, superdelta = 0.1, epsilon = sys.float_info.epsilon,
do_plot = None, plot_filename = "Bowyer-Watson_%i.png" ):
"""Return the Delaunay triangulation of the given points
epsilon: used for floating point comparisons, two points are considered equals if their distance is < epsilon.
do_plot: if not None, plot intermediate steps on this matplotlib object and save them as images named: plot_filename % i
"""
if do_plot and len(points) > 10:
print "WARNING it is a bad idea to plot each steps of a triangulation of many points"
# Sort points first on the x-axis, then on the y-axis.
vertices = sorted( points )
# LOGN( "super-triangle",supertri )
if not supertri:
supertri = supertriangle( vertices, superdelta )
# It is the first triangle of the list.
triangles = [ supertri ]
completed = { supertri: False }
# The predicate returns true if at least one of the vertices
# is also found in the supertriangle.
def match_supertriangle( tri ):
if tri[0] in supertri or \
tri[1] in supertri or \
tri[2] in supertri:
return True
# Returns the base of each plots, with points, current triangulation, super-triangle and bounding box.
def plot_base(ax,vi = len(vertices), vertex = None):
ax.set_aspect('equal')
# regular points
scatter_x = [ p[0] for p in vertices[:vi]]
scatter_y = [ p[1] for p in vertices[:vi]]
ax.scatter( scatter_x,scatter_y, s=30, marker='o', facecolor="black")
# super-triangle vertices
scatter_x = [ p[0] for p in list(supertri)]
scatter_y = [ p[1] for p in list(supertri)]
ax.scatter( scatter_x,scatter_y, s=30, marker='o', facecolor="lightgrey", edgecolor="lightgrey")
# current vertex
if vertex:
ax.scatter( vertex[0],vertex[1], s=30, marker='o', facecolor="red", edgecolor="red")
# current triangulation
uberplot.plot_segments( ax, edges_of(triangles), edgecolor = "blue", alpha=0.5, linestyle='solid' )
# bounding box
(xmin,ymin),(xmax,ymax) = bounds(vertices)
uberplot.plot_segments( ax, tour([(xmin,ymin),(xmin,ymax),(xmax,ymax),(xmax,ymin)]), edgecolor = "magenta", alpha=0.2, linestyle='dotted' )
# Insert vertices one by one.
LOG("Insert vertices: ")
if do_plot:
it=0
for vi,vertex in enumerate(vertices):
# LOGN( "\tvertex",vertex )
assert( len(vertex) == 2 )
if do_plot:
ax = do_plot.add_subplot(111)
plot_base(ax,vi,vertex)
# All the triangles whose circumcircle encloses the point to be added are identified,
# the outside edges of those triangles form an enclosing polygon.
# Forget previous candidate polygon's edges.
enclosing = []
removed = []
for triangle in triangles:
# LOGN( "\t\ttriangle",triangle )
assert( len(triangle) == 3 )
# Do not consider triangles already tested.
# If completed has a key, test it, else return False.
if completed.get( triangle, False ):
# LOGN( "\t\t\tAlready completed" )
# if do_plot:
# uberplot.plot_segments( ax, tour(list(triangle)), edgecolor = "magenta", alpha=1, lw=1, linestyle='dotted' )
continue
# LOGN( "\t\t\tCircumcircle" )
assert( triangle[0] != triangle[1] and triangle[1] != triangle [2] and triangle[2] != triangle[0] )
center,radius = circumcircle( triangle, epsilon )
# If it match Delaunay's conditions.
if x(center) < x(vertex) and math.sqrt((x(vertex)-x(center))**2) > radius:
# LOGN( "\t\t\tMatch Delaunay, mark as completed" )
completed[triangle] = True
# If the current vertex is inside the circumscribe circle of the current triangle,
# add the current triangle's edges to the candidate polygon.
if in_circle( vertex, center, radius, epsilon ):
# LOGN( "\t\t\tIn circumcircle, add to enclosing polygon",triangle )
if do_plot:
circ = plot.Circle(center, radius, facecolor='yellow', edgecolor="orange", alpha=0.2, clip_on=False)
ax.add_patch(circ)
for p0,p1 in tour(list(triangle)):
# Then add this edge to the polygon enclosing the vertex,
enclosing.append( (p0,p1) )
# and remove the corresponding triangle from the current triangulation.
removed.append( triangle )
completed.pop(triangle,None)
elif do_plot:
circ = plot.Circle(center, radius, facecolor='lightgrey', edgecolor="grey", alpha=0.2, clip_on=False)
ax.add_patch(circ)
# end for triangle in triangles
# The triangles in the enclosing polygon are deleted and
# new triangles are formed between the point to be added and
# each outside edge of the enclosing polygon.
# Actually remove triangles.
for triangle in removed:
triangles.remove(triangle)
# Remove duplicated edges.
# This leaves the edges of the enclosing polygon only,
# because enclosing edges are only in a single triangle,
# but edges inside the polygon are at least in two triangles.
hull = []
for i,(p0,p1) in enumerate(enclosing):
# Clockwise edges can only be in the remaining part of the list.
# Search for counter-clockwise edges as well.
if (p0,p1) not in enclosing[i+1:] and (p1,p0) not in enclosing:
hull.append((p0,p1))
elif do_plot:
uberplot.plot_segments( ax, [(p0,p1)], edgecolor = "white", alpha=1, lw=1, linestyle='dotted' )
if do_plot:
uberplot.plot_segments( ax, hull, edgecolor = "red", alpha=1, lw=1, linestyle='solid' )
# Create new triangles using the current vertex and the enclosing hull.
# LOGN( "\t\tCreate new triangles" )
for p0,p1 in hull:
assert( p0 != p1 )
triangle = tuple([p0,p1,vertex])
# LOGN("\t\t\tNew triangle",triangle)
triangles.append( triangle )
completed[triangle] = False
if do_plot:
uberplot.plot_segments( ax, [(p0,vertex),(p1,vertex)], edgecolor = "green", alpha=1, linestyle='solid' )
if do_plot:
plot.savefig( plot_filename % it, dpi=150)
plot.clf()
it+=1
LOG(".")
# end for vertex in vertices
LOGN(" done")
# Remove triangles that have at least one of the supertriangle vertices.
# LOGN( "\tRemove super-triangles" )
# Filter out elements for which the predicate is False,
# here: *keep* elements that *do not* have a common vertex.
# The filter is a generator, so we must make a list with it to actually get the data.
triangulation = list(filter_if_not( match_supertriangle, triangles ))
if do_plot:
ax = do_plot.add_subplot(111)
plot_base(ax)
uberplot.plot_segments( ax, edges_of(triangles), edgecolor = "red", alpha=0.5, linestyle='solid' )
uberplot.plot_segments( ax, edges_of(triangulation), edgecolor = "blue", alpha=1, linestyle='solid' )
plot.savefig( plot_filename % it, dpi=150)
plot.clf()
return triangulation
def write(triangles, stream):
for tri in triangles:
assert (len(tri) == 3)
p, q, r = tri
stream.write("%f,%f %f,%f %f,%f\n" %
(x(p), y(p), x(q), y(q), x(r), y(r)))
def write( triangles, stream ):
for tri in triangles:
assert(len(tri)==3)
p,q,r = tri
stream.write("%f,%f %f,%f %f,%f\n" % ( x(p),y(p), x(q),y(q), x(r),y(r) ) )
def in_triangle(p0, triangle, exclude_edges=False):
"""Return True if the given point lies inside the given triangle"""
p1, p2, p3 = triangle
# Compute the barycentric coordinates
alpha = ( (y(p2) - y(p3)) * (x(p0) - x(p3)) + (x(p3) - x(p2)) * (y(p0) - y(p3)) ) \
/ ( (y(p2) - y(p3)) * (x(p1) - x(p3)) + (x(p3) - x(p2)) * (y(p1) - y(p3)) )
beta = ( (y(p3) - y(p1)) * (x(p0) - x(p3)) + (x(p1) - x(p3)) * (y(p0) - y(p3)) ) \
/ ( (y(p2) - y(p3)) * (x(p1) - x(p3)) + (x(p3) - x(p2)) * (y(p1) - y(p3)) )
gamma = 1.0 - alpha - beta
if exclude_edges:
# If all of alpha, beta, and gamma are strictly in ]0,1[,
# then the point p0 strictly lies within the triangle.
return all(0 < x < 1 for x in (alpha, beta, gamma))
else:
# If the inequality is not strict, then the point may lies on an edge.
return all(0 <= x <= 1 for x in (alpha, beta, gamma))
def in_triangle( p0, triangle, exclude_edges = False ):
"""Return True if the given point lies inside the given triangle"""
p1,p2,p3 = triangle
# Compute the barycentric coordinates
alpha = ( (y(p2) - y(p3)) * (x(p0) - x(p3)) + (x(p3) - x(p2)) * (y(p0) - y(p3)) ) \
/ ( (y(p2) - y(p3)) * (x(p1) - x(p3)) + (x(p3) - x(p2)) * (y(p1) - y(p3)) )
beta = ( (y(p3) - y(p1)) * (x(p0) - x(p3)) + (x(p1) - x(p3)) * (y(p0) - y(p3)) ) \
/ ( (y(p2) - y(p3)) * (x(p1) - x(p3)) + (x(p3) - x(p2)) * (y(p1) - y(p3)) )
gamma = 1.0 - alpha - beta
if exclude_edges:
# If all of alpha, beta, and gamma are strictly in ]0,1[,
# then the point p0 strictly lies within the triangle.
return all( 0 < x < 1 for x in (alpha, beta, gamma) )
else:
# If the inequality is not strict, then the point may lies on an edge.
return all( 0 <= x <= 1 for x in (alpha, beta, gamma) )
def turn(p, q, r):
"""Returns -1, 0, 1 if the sequence of points (p,q,r) forms a right, straight, or left turn."""
qr = ( x(q) - x(p) ) * ( y(r) - y(p) )
rq = ( x(r) - x(p) ) * ( y(q) - y(p) )
# cmp(x,y) returns -1 if x<y, 0 if x==y, +1 if x>y
return cmp( qr - rq, 0)
def write_segments(segments, stream):
for seg in segments:
for p in seg:
stream.write("%f,%f " % (x(p), y(p)))
stream.write("\n")
def write(graph, stream):
for k in graph:
stream.write("%f,%f:" % (x(k), y(k)))
for p in graph[k]:
stream.write("%f,%f " % (x(p), y(p)))
stream.write("\n")
