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 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 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 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 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 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_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 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")
