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####################################################################################################
#
# Patro - A Python library to make patterns for fashion design
# Copyright (C) 2017 Fabrice Salvaire
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
#
####################################################################################################
####################################################################################################
from math import log, sqrt # pow
from .BoundingBox import bounding_box_from_points
from .Interpolation import interpolate_two_points
from .Primitive import Primitive2D, ReversablePrimitiveMixin
from .Vector import Vector2D
####################################################################################################
class QuadraticBezier2D(Primitive2D, ReversablePrimitiveMixin):
"""Class to implements 2D Quadratic Bezier Curve."""
LineInterpolationPrecision = 0.05
##############################################
def __init__(self, p0, p1, p2):
self._p0 = Vector2D(p0)
self._p1 = Vector2D(p1)
self._p2 = Vector2D(p2)
##############################################
def clone(self):
return self.__class__(self._p0, self._p1, self._p2)
##############################################
def bounding_box(self):
return bounding_box_from_points((self._p0, self._p1, self._p2))
##############################################
def reverse(self):
return self.__class__(self._p2, self._p1, self._p0)
##############################################
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def __repr__(self):
return self.__class__.__name__ + '({0._p0}, {0._p1}, {0._p2})'.format(self)
##############################################
@property
def p0(self):
return self._p0
@p0.setter
def p0(self, value):
self._p0 = value
@property
def p1(self):
return self._p1
@p1.setter
def p1(self, value):
self._p1 = value
@property
def p2(self):
return self._p2
@p2.setter
def p2(self, value):
self._p2 = value
@property
def start_point(self):
return self._p0
@property
def end_point(self):
return self._p2
##############################################
@property
def length(self):
# Algorithm:
#
# http://www.gamedev.net/topic/551455-length-of-a-generalized-quadratic-bezier-curve-in-3d
# Dave Eberly Posted October 25, 2009
#
# The quadratic Bezier is
# (x(t),y(t)) = (1-t)^2*(x0,y0) + 2*t*(1-t)*(x1,y1) + t^2*(x2,y2)
#
# The derivative is
# (x'(t),y'(t)) = -2*(1-t)*(x0,y0) + (2-4*t)*(x1,y1) + 2*t*(x2,y2)
#
# The length of the curve for 0 <= t <= 1 is
# Integral[0,1] sqrt((x'(t))^2 + (y'(t))^2) dt
# The integrand is of the form sqrt(c*t^2 + b*t + a)
#
# You have three separate cases: c = 0, c > 0, or c < 0.
# * The case c = 0 is easy.
# * For the case c > 0, an antiderivative is
# (2*c*t+b)*sqrt(c*t^2+b*t+a)/(4*c) + (0.5*k)*log(2*sqrt(c*(c*t^2+b*t+a)) + 2*c*t + b)/sqrt(c)
# where k = 4*c/q with q = 4*a*c - b*b.
# * For the case c < 0, an antiderivative is
# (2*c*t+b)*sqrt(c*t^2+b*t+a)/(4*c) - (0.5*k)*arcsin((2*c*t+b)/sqrt(-q))/sqrt(-c)
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A0 = self._p1 - self._p0
A1 = self._p0 - self._p1 * 2 + self._p2
if A1.magnitude_square() != 0:
c = 4 * A1.dot(A1)
b = 8 * A0.dot(A1)
a = 4 * A0.dot(A0)
q = 4 * a * c - b * b
two_cb = 2 * c + b
sum_cba = c + b + a
m0 = 0.25 / c
m1 = q / (8 * c**1.5)
return (m0 * (two_cb * sqrt(sum_cba) - b * sqrt(a)) +
m1 * (log(2 * sqrt(c * sum_cba) + two_cb) - log(2 * sqrt(c * a) + b)))
else:
return 2 * A0.magnitude()
##############################################
def interpolated_length(self):
# Length of the curve obtained via line interpolation
dt = self.LineInterpolationPrecision / (self.end_point - self.start_point).magnitude()
length = 0
t = 0
while t < 1:
t0 = t
t = min(t + dt, 1)
length += (self.point_at_t(t) - self.point_at_t(t0)).magnitude()
return length
##############################################
def point_at_t(self, t):
# if 0 < t or 1 < t:
# raise ValueError()
u = 1 - t
return self._p0 * u**2 + self._p1 * 2 * t * u + self._p2 * t**2
##############################################
def split_at_t(self, t):
p01 = interpolate_two_points(self._p0, self._p1, t)
p12 = interpolate_two_points(self._p1, self._p2, t)
p = interpolate_two_points(p01, p12, t) # p = p012
# p = self.point_at_t(t)
return (QuadraticBezier2D(self._p0, p01, p), QuadraticBezier2D(p, p12, self._p2))
##############################################
@property
def tangent0(self):
return (self._p1 - self._p0).normalise()
##############################################
@property
def tangent1(self):
return (self._p2 - self._p1).normalise()
##############################################
@property
def normal0(self):
return self.tangent0.normal()
##############################################
@property
def tangent1(self):
return self.tangent1.normal()
##############################################
def tangent_at(self, t):
u = 1 - t
return (self._p1 - self._p0) * u + (self._p2 - self._p1) * t
####################################################################################################
_Sqrt3 = sqrt(3)
_Div18Sqrt3 = 18 / _Sqrt3
_OneThird = 1 / 3
_Sqrt3Div36 = _Sqrt3 / 36
class CubicBezier2D(QuadraticBezier2D):
"""Class to implements 2D Cubic Bezier Curve."""
InterpolationPrecision = 0.001
#######################################
def __init__(self, p0, p1, p2, p3):
QuadraticBezier2D.__init__(self, p0, p1, p2)
self._p3 = Vector2D(p3)
##############################################
def clone(self):
return self.__class__(self._p0, self._p1, self._p2, self._p3)
##############################################
def bounding_box(self):
return bounding_box_from_points((self._p0, self._p1, self._p2, self._p3))
##############################################
def reverse(self):
return self.__class__(self._p3, self._p2, self._p1, self._p0)
##############################################
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def __repr__(self):
return self.__class__.__name__ + '({0._p0}, {0._p1}, {0._p2}, {0._p3})'.format(self)
##############################################
@property
def p3(self):
return self._p3
@p3.setter
def p3(self, value):
self._p3 = value
@property
def end_point(self):
return self._p3
##############################################
@property
def length(self):
return self.adaptive_length_approximation()
##############################################
def point_at_t(self, t):
# if 0 < t or 1 < t:
# raise ValueError()
return (self._p0 +
(self._p1 - self._p0) * 3 * t +
(self._p2 - self._p1 * 2 + self._p0) * 3 * t**2 +
(self._p3 - self._p2 * 3 + self._p1 * 3 - self._p0) * t**3)
##############################################
def _q_point(self):
"""Return the control point for mid-point quadratic approximation"""
return (self._p2 * 3 - self._p3 + self._p1 * 3 - self._p0) / 4
##############################################
def mid_point_quadratic_approximation(self):
"""Return the mid-point quadratic approximation"""
p1 = self._q_point()
return QuadraticBezier2D(self._p0, p1, self._p3)
##############################################
def split_at_t(self, t):
p01 = interpolate_two_points(self._p0, self._p1, t)
p12 = interpolate_two_points(self._p1, self._p2, t)
p23 = interpolate_two_points(self._p2, self._p3, t)
p012 = interpolate_two_points(p01, p12, t)
p123 = interpolate_two_points(p12, p23, t)
p = interpolate_two_points(p012, p123, t) # p = p0123
# p = self.point_at_t(t)
return (CubicBezier2D(self._p0, p01, p012, p), CubicBezier2D(p, p123, p23, self._p3))
##############################################
def _d01(self):
"""Return the distance between 0 and 1 quadratic aproximations"""
return (self._p3 - self._p2 * 3 + self._p1 * 3 - self._p0).magnitude() / 2
##############################################
def _t_max(self):
"""Return the split point for adaptive quadratic approximation"""
return (_Div18Sqrt3 * self.InterpolationPrecision / self._d01())**_OneThird
##############################################
def q_length(self):
"""Return the length of the mid-point quadratic approximation"""
return self.mid_point_quadratic_approximation().length
##############################################
def adaptive_length_approximation(self):
"""Return the length of the adaptive quadratic approximation"""
segments = []
segment = self
t_max = segment._t_max()
while t_max < 1:
split = segment.split_at_t(t_max)
segments.append(split[0])
segment = split[1]
t_max = segment._t_max()
segments.append(segment)
return sum([segment.q_length() for segment in segments])
##############################################
@property
def tangent1(self):
return (self._p3 - self._p2).normalise()
##############################################
def tangent_at(self, t):
u = 1 - t
return (self._p1 - self._p0) * u**2 + (self._p2 - self._p1) * 2 * t * u + (self._p3 - self._p2) * t**2