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from beziers.segment import Segment
from beziers.line import Line
from beziers.point import Point
from beziers.quadraticbezier import QuadraticBezier
from beziers.utils.arclengthmixin import ArcLengthMixin
import math
from beziers.utils.legendregauss import Tvalues, Cvalues
from beziers.utils import quadraticRoots
class CubicBezier(ArcLengthMixin,Segment):
def __init__(self, start, c1,c2,end):
self.points = [start,c1,c2,end]
self._range = [0,1]
def __repr__(self):
return "B<%s-%s-%s-%s>" % (self[0],self[1],self[2],self[3])
@classmethod
def fromRepr(klass,text):
import re
p = re.compile("^B<(<.*?>)-(<.*?>)-(<.*?>)-(<.*?>)>$")
m = p.match(text)
points = [ Point.fromRepr(m.group(t)) for t in range(1,5) ]
return klass(*points)
def pointAtTime(self,t):
"""Returns the point at time t (0->1) along the curve."""
x = (1 - t) * (1 - t) * (1 - t) * self[0].x + 3 * (1 - t) * (1 - t) * t * self[1].x + 3 * (1 - t) * t * t * self[2].x + t * t * t * self[3].x;
y = (1 - t) * (1 - t) * (1 - t) * self[0].y + 3 * (1 - t) * (1 - t) * t * self[1].y + 3 * (1 - t) * t * t * self[2].y + t * t * t * self[3].y;
return Point(x,y)
def tOfPoint(self,p):
precision = 1.0/50.0
bestDist = float("inf")
bestT = -1
samples = self.regularSampleTValue(50)
for t in samples:
dist = self.pointAtTime(t).distanceFrom(p)
if dist < bestDist:
bestDist = dist
bestT = t
while precision > 1e-5:
precision = precision / 2
lower = bestT - precision
if lower < 0: lower = 0
upper = bestT + precision
if upper > 1: upper = 1
ldist = self.pointAtTime(lower).distanceFrom(p)
rdist = self.pointAtTime(lower).distanceFrom(p)
if ldist < bestDist:
bestT = lower
bestDist = ldist
if rdist < bestDist:
bestT = upper
bestDist = rdist
return bestT
def splitAtTime(self,t):
"""Returns two segments, dividing the given segment at a point t (0->1) along the curve."""
p4 = self[0].lerp(self[1],t)
p5 = self[1].lerp(self[2],t)
p6 = self[2].lerp(self[3],t)
p7 = p4.lerp(p5,t)
p8 = p5.lerp(p6,t)
p9 = p7.lerp(p8,t)
return (CubicBezier(self[0],p4,p7,p9), CubicBezier(p9,p8,p6,self[3]))
def join(self,other):
"""Not currently implemented: join two `CubicBezier` together."""
raise "Not implemented"
def toQuadratic(self):
"""Not currently implemented: reduce this to a `QuadraticBezier`."""
raise "Not implemented"
def derivative(self):
"""Returns a `QuadraticBezier` representing the derivative of this curve."""
return QuadraticBezier(
(self[1]-self[0])*3,
(self[2]-self[1])*3,
(self[3]-self[2])*3
)
def flatten(self, degree=8):
samples = self.regularSample(self.length/degree)
ss = []
for i in range(1,len(samples)):
l = Line(samples[i-1], samples[i])
l._orig = self
ss.append(l)
return ss
def _findRoots(self,dimension):
def cuberoot(v):
if v<0: return -math.pow(-v,1/3.0)
return math.pow(v,1/3.0)
if dimension == "x":
pa,pb,pc,pd = self[0].x,self[1].x,self[2].x,self[3].x
elif dimension == "y":
pa,pb,pc,pd = self[0].y,self[1].y,self[2].y,self[3].y
else:
raise SyntaxError("Meh.")
a = (3*pa - 6*pb + 3*pc)
b = (-3*pa + 3*pb)
c = pa
d = (-pa + 3*pb - 3*pc + pd)
if d == 0: return []
a = a/d
b = b/d
c = c/d
p = (3*b - a*a)/3
p3 = p/3
q = (2*a*a*a - 9*a*b + 27*c)/27.0
q2 = q/2
discriminant = q2*q2 + p3*p3*p3
if discriminant < 0:
mp3 = -p/3
mp33 = mp3*mp3*mp3
r = math.sqrt( mp33 )
t = -q / (2*r)
cosphi = max(min(t,1),-1)
phi = math.acos(cosphi)
crtr = cuberoot(r)
t1 = 2*crtr
root1 = t1 * math.cos(phi/3) - a/3
root2 = t1 * math.cos((phi+2*math.pi)/3) - a/3
root3 = t1 * math.cos((phi+4*math.pi)/3) - a/3
roots = [root1, root2, root3]
return sorted([x for x in roots if x >= 0 and x <= 1])
if discriminant == 0:
if q2 < 0:
u1 = cuberoot(-q2)
else:
u1 =-cuberoot(q2);
root1 = 2*u1 - a/3.0;
root2 = -u1 - a/3.0;
roots = [root1,root2]
return sorted([x for x in roots if x >= 0 and x <= 1])
sd = math.sqrt(discriminant);
u1 = cuberoot(sd - q2);
v1 = cuberoot(sd + q2);
root1 = u1 - v1 - a/3;
return [x for x in [root1] if x >= 0 and x <= 1]
def _findDRoots(self):
d = self.derivative()
roots = []
# We have f(t) = w1 (1-t)^2 + 2 w2 (1-t) t + w3 t^2
# We want f(t) = a t^2 + b^t + c to solve with the quadratic formula
roots.extend(quadraticRoots(d[0].x - 2*d[1].x + d[2].x, 2 * (d[1].x-d[0].x), d[0].x))
roots.extend(quadraticRoots(d[0].y - 2*d[1].y + d[2].y, 2 * (d[1].y-d[0].y), d[0].y))
return roots
def findExtremes(self, inflections = False):
"""Returns a list of time `t` values for extremes of the curve."""
r = self._findDRoots()
if inflections:
r.extend(self.derivative()._findDRoots())
r.sort()
return [ root for root in r if root >= 0.01 and root <= 0.99 ]
def curvatureAtTime(self,t):
"""Returns the C curvature at time `t`.."""
d = self.derivative()
d2 = d.derivative()
return d.pointAtTime(t).x * d2.pointAtTime(t).y - d.pointAtTime(t).y * d2.pointAtTime(t).x
@property
def tunniPoint(self):
"""Returns the Tunni point of this Bezier (the intersection of
the handles)."""
h1 = Line(self[0], self[1])
h2 = Line(self[2], self[3])
i = h1.intersections(h2, limited = False)
if len(i)<1: return
i = i[0].point
if i.distanceFrom(self[0]) > 5 * self.length:
return
else:
return i
def balance(self):
"""Perform Tunni balancing on this Bezier."""
p = self.tunniPoint
if not p: return
if self[0].distanceFrom(p) == 0.0:
fraction1 = 0.43
else:
fraction1 = self[0].distanceFrom(self[1]) / self[0].distanceFrom(p)
if self[3].distanceFrom(p) == 0.0:
fraction2 = 0.73
else:
fraction2 = self[3].distanceFrom(self[2]) / self[3].distanceFrom(p)
avg = (fraction2 + fraction1) / 2.0
if avg > 0 and avg < 1:
self[1] = self[0].lerp(p, avg)
self[2] = self[3].lerp(p, avg)
@property
def hasLoop(self):
a1 = self[0].x * (self[3].y - self[2].y) + self[0].y * (self[2].x - self[3].x) + self[3].x * self[2].y - self[3].y * self[2].x
a2 = self[1].x * (self[0].y - self[3].y) + self[1].y * (self[3].x - self[0].x) + self[0].x * self[3].y - self[0].y * self[3].x
a3 = self[2].x * (self[1].y - self[0].y) + self[2].y * (self[0].x - self[1].x) + self[1].x * self[0].y - self[1].y * self[0].x
d3 = 3 * a3
d2 = d3 - a2
d1 = d2 - a2 + a1
l = math.sqrt(d1 * d1 + d2 * d2 + d3 * d3)
s = 0
if l != 0: s = 1 / l
d1 *= s
d2 *= s
d3 *= s
d = 3 * d2 * d2 - 4 * d1 * d3
if d >= 0: return False
f1 = math.sqrt(-d)
f2 = 2 * d1
return ((d2 + f1) / f2, (d2 - f1) / f2)
@property
def area(self):
"""Returns the signed rea between the curve and the y-axis"""
return (10 * (self[3].x*self[3].y - self[0].x*self[0].y) +
6 * (self[1].x*self[0].y - self[0].x*self[1].y + self[3].x*self[2].y - self[2].x*self[3].y) +
3 * (self[2].x*self[0].y - self[0].x*self[2].y + self[2].x*self[1].y - self[1].x*self[2].y +
self[3].x*self[1].y - self[1].x*self[3].y) +
self[3].x*self[0].y - self[0].x*self[3].y) / 20.
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