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import math
from typing import List, Tuple
from beziers.line import Line
from beziers.point import Point
from beziers.quadraticbezier import QuadraticBezier
from beziers.segment import Segment
from beziers.utils import quadraticRoots
from beziers.utils.arclengthmixin import ArcLengthMixin
class CubicBezier(ArcLengthMixin, Segment):
"""A representation of a cubic bezier curve."""
def __init__(self, start: Point, c1: Point, c2: Point, end: Point):
"""Create a new cubic bezier curve.
Args:
start (Point): The starting point of the curve.
c1 (Point): The first control point.
c2 (Point): The second control point.
end (Point): The ending point of the curve.
"""
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: str):
"""Create a new cubic bezier curve from a string representation."""
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: float) -> Point:
"""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: Point) -> float:
"""Returns the time t (0->1) of a point on the curve."""
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: float) -> Tuple["CubicBezier", "CubicBezier"]:
"""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 NotImplementedError
def toQuadratic(self):
"""Not currently implemented: reduce this to a `QuadraticBezier`."""
raise NotImplementedError
def derivative(self) -> QuadraticBezier:
"""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) -> List[Line]:
"""Flattens the curve into a list of `Line` segments.
Args:
degree (int): The degree of flattening to perform.
"""
ss = []
if self.length < degree:
return [Line(self[0], self[3])]
samples = self.regularSample(self.length / degree)
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: str) -> List[float]:
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) -> List[float]:
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) -> List[float]:
"""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: float) -> float:
"""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) -> Point:
"""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) -> None:
"""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) -> bool:
"""Returns True if the curve has a loop."""
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) -> float:
"""Returns the signed area 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
) / 20.0
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