286 lines
9.1 KiB
Python
286 lines
9.1 KiB
Python
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#!/usr/bin/env python3
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'''
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Copyright (C) 2010 Nick Drobchenko, nick@cnc-club.ru
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Copyright (C) 2005 Aaron Spike, aaron@ekips.org
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This program is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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This program is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with this program; if not, write to the Free Software
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Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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'''
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import math
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import cmath
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def rootWrapper(a,b,c,d):
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if a:
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# Monics formula see http://en.wikipedia.org/wiki/Cubic_function#Monic_formula_of_roots
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a,b,c = (b/a, c/a, d/a)
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m = 2.0*a**3 - 9.0*a*b + 27.0*c
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k = a**2 - 3.0*b
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n = m**2 - 4.0*k**3
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w1 = -.5 + .5*cmath.sqrt(-3.0)
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w2 = -.5 - .5*cmath.sqrt(-3.0)
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if n < 0:
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m1 = pow(complex((m+cmath.sqrt(n))/2),1./3)
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n1 = pow(complex((m-cmath.sqrt(n))/2),1./3)
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else:
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if m+math.sqrt(n) < 0:
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m1 = -pow(-(m+math.sqrt(n))/2,1./3)
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else:
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m1 = pow((m+math.sqrt(n))/2,1./3)
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if m-math.sqrt(n) < 0:
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n1 = -pow(-(m-math.sqrt(n))/2,1./3)
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else:
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n1 = pow((m-math.sqrt(n))/2,1./3)
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x1 = -1./3 * (a + m1 + n1)
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x2 = -1./3 * (a + w1*m1 + w2*n1)
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x3 = -1./3 * (a + w2*m1 + w1*n1)
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return (x1,x2,x3)
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elif b:
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det=c**2.0-4.0*b*d
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if det:
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return (-c+cmath.sqrt(det))/(2.0*b),(-c-cmath.sqrt(det))/(2.0*b)
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else:
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return -c/(2.0*b),
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elif c:
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return 1.0*(-d/c),
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return ()
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def bezierparameterize(xxx_todo_changeme):
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#parametric bezier
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((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)) = xxx_todo_changeme
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x0=bx0
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y0=by0
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cx=3*(bx1-x0)
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bx=3*(bx2-bx1)-cx
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ax=bx3-x0-cx-bx
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cy=3*(by1-y0)
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by=3*(by2-by1)-cy
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ay=by3-y0-cy-by
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return ax,ay,bx,by,cx,cy,x0,y0
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#ax,ay,bx,by,cx,cy,x0,y0=bezierparameterize(((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)))
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def linebezierintersect(xxx_todo_changeme1, xxx_todo_changeme2):
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#parametric line
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((lx1,ly1),(lx2,ly2)) = xxx_todo_changeme1
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((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)) = xxx_todo_changeme2
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dd=lx1
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cc=lx2-lx1
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bb=ly1
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aa=ly2-ly1
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if aa:
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coef1=cc/aa
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coef2=1
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else:
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coef1=1
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coef2=aa/cc
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ax,ay,bx,by,cx,cy,x0,y0=bezierparameterize(((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)))
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#cubic intersection coefficients
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a=coef1*ay-coef2*ax
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b=coef1*by-coef2*bx
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c=coef1*cy-coef2*cx
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d=coef1*(y0-bb)-coef2*(x0-dd)
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roots = rootWrapper(a,b,c,d)
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retval = []
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for i in roots:
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if type(i) is complex and i.imag==0:
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i = i.real
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if type(i) is not complex and 0<=i<=1:
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retval.append(bezierpointatt(((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)),i))
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return retval
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def bezierpointatt(xxx_todo_changeme3,t):
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((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)) = xxx_todo_changeme3
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ax,ay,bx,by,cx,cy,x0,y0=bezierparameterize(((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)))
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x=ax*(t**3)+bx*(t**2)+cx*t+x0
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y=ay*(t**3)+by*(t**2)+cy*t+y0
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return x,y
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def bezierslopeatt(xxx_todo_changeme4,t):
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((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)) = xxx_todo_changeme4
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ax,ay,bx,by,cx,cy,x0,y0=bezierparameterize(((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)))
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dx=3*ax*(t**2)+2*bx*t+cx
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dy=3*ay*(t**2)+2*by*t+cy
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return dx,dy
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def beziertatslope(xxx_todo_changeme5, xxx_todo_changeme6):
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((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)) = xxx_todo_changeme5
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(dy,dx) = xxx_todo_changeme6
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ax,ay,bx,by,cx,cy,x0,y0=bezierparameterize(((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)))
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#quadratic coefficents of slope formula
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if dx:
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slope = 1.0*(dy/dx)
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a=3*ay-3*ax*slope
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b=2*by-2*bx*slope
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c=cy-cx*slope
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elif dy:
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slope = 1.0*(dx/dy)
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a=3*ax-3*ay*slope
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b=2*bx-2*by*slope
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c=cx-cy*slope
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else:
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return []
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roots = rootWrapper(0,a,b,c)
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retval = []
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for i in roots:
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if type(i) is complex and i.imag==0:
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i = i.real
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if type(i) is not complex and 0<=i<=1:
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retval.append(i)
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return retval
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def tpoint(xxx_todo_changeme7, xxx_todo_changeme8,t):
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(x1,y1) = xxx_todo_changeme7
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(x2,y2) = xxx_todo_changeme8
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return x1+t*(x2-x1),y1+t*(y2-y1)
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def beziersplitatt(xxx_todo_changeme9,t):
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((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)) = xxx_todo_changeme9
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m1=tpoint((bx0,by0),(bx1,by1),t)
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m2=tpoint((bx1,by1),(bx2,by2),t)
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m3=tpoint((bx2,by2),(bx3,by3),t)
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m4=tpoint(m1,m2,t)
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m5=tpoint(m2,m3,t)
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m=tpoint(m4,m5,t)
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return ((bx0,by0),m1,m4,m),(m,m5,m3,(bx3,by3))
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'''
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Approximating the arc length of a bezier curve
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according to <http://www.cit.gu.edu.au/~anthony/info/graphics/bezier.curves>
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if:
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L1 = |P0 P1| +|P1 P2| +|P2 P3|
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L0 = |P0 P3|
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then:
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L = 1/2*L0 + 1/2*L1
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ERR = L1-L0
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ERR approaches 0 as the number of subdivisions (m) increases
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2^-4m
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Reference:
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Jens Gravesen <gravesen@mat.dth.dk>
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"Adaptive subdivision and the length of Bezier curves"
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mat-report no. 1992-10, Mathematical Institute, The Technical
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University of Denmark.
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'''
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def pointdistance(xxx_todo_changeme10, xxx_todo_changeme11):
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(x1,y1) = xxx_todo_changeme10
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(x2,y2) = xxx_todo_changeme11
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return math.sqrt(((x2 - x1) ** 2) + ((y2 - y1) ** 2))
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def Gravesen_addifclose(b, len, error = 0.001):
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box = 0
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for i in range(1,4):
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box += pointdistance(b[i-1], b[i])
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chord = pointdistance(b[0], b[3])
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if (box - chord) > error:
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first, second = beziersplitatt(b, 0.5)
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Gravesen_addifclose(first, len, error)
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Gravesen_addifclose(second, len, error)
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else:
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len[0] += (box / 2.0) + (chord / 2.0)
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def bezierlengthGravesen(b, error = 0.001):
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len = [0]
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Gravesen_addifclose(b, len, error)
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return len[0]
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# balf = Bezier Arc Length Function
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balfax,balfbx,balfcx,balfay,balfby,balfcy = 0,0,0,0,0,0
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def balf(t):
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retval = (balfax*(t**2) + balfbx*t + balfcx)**2 + (balfay*(t**2) + balfby*t + balfcy)**2
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return math.sqrt(retval)
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def Simpson(f, a, b, n_limit, tolerance):
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n = 2
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multiplier = (b - a)/6.0
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endsum = f(a) + f(b)
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interval = (b - a)/2.0
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asum = 0.0
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bsum = f(a + interval)
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est1 = multiplier * (endsum + (2.0 * asum) + (4.0 * bsum))
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est0 = 2.0 * est1
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#print multiplier, endsum, interval, asum, bsum, est1, est0
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while n < n_limit and abs(est1 - est0) > tolerance:
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n *= 2
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multiplier /= 2.0
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interval /= 2.0
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asum += bsum
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bsum = 0.0
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est0 = est1
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for i in range(1, n, 2):
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bsum += f(a + (i * interval))
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est1 = multiplier * (endsum + (2.0 * asum) + (4.0 * bsum))
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#print multiplier, endsum, interval, asum, bsum, est1, est0
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return est1
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def bezierlengthSimpson(xxx_todo_changeme12, tolerance = 0.001):
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((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)) = xxx_todo_changeme12
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global balfax,balfbx,balfcx,balfay,balfby,balfcy
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ax,ay,bx,by,cx,cy,x0,y0=bezierparameterize(((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)))
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balfax,balfbx,balfcx,balfay,balfby,balfcy = 3*ax,2*bx,cx,3*ay,2*by,cy
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return Simpson(balf, 0.0, 1.0, 4096, tolerance)
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def beziertatlength(xxx_todo_changeme13, l = 0.5, tolerance = 0.001):
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((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)) = xxx_todo_changeme13
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global balfax,balfbx,balfcx,balfay,balfby,balfcy
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ax,ay,bx,by,cx,cy,x0,y0=bezierparameterize(((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)))
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balfax,balfbx,balfcx,balfay,balfby,balfcy = 3*ax,2*bx,cx,3*ay,2*by,cy
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t = 1.0
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tdiv = t
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curlen = Simpson(balf, 0.0, t, 4096, tolerance)
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targetlen = l * curlen
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diff = curlen - targetlen
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while abs(diff) > tolerance:
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tdiv /= 2.0
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if diff < 0:
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t += tdiv
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else:
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t -= tdiv
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curlen = Simpson(balf, 0.0, t, 4096, tolerance)
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diff = curlen - targetlen
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return t
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#default bezier length method
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bezierlength = bezierlengthSimpson
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if __name__ == '__main__':
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import timing
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#print linebezierintersect(((,),(,)),((,),(,),(,),(,)))
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#print linebezierintersect(((0,1),(0,-1)),((-1,0),(-.5,0),(.5,0),(1,0)))
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tol = 0.00000001
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curves = [((0,0),(1,5),(4,5),(5,5)),
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((0,0),(0,0),(5,0),(10,0)),
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((0,0),(0,0),(5,1),(10,0)),
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((-10,0),(0,0),(10,0),(10,10)),
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((15,10),(0,0),(10,0),(-5,10))]
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'''
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for curve in curves:
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timing.start()
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g = bezierlengthGravesen(curve,tol)
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timing.finish()
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gt = timing.micro()
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timing.start()
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s = bezierlengthSimpson(curve,tol)
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timing.finish()
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st = timing.micro()
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print g, gt
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print s, st
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'''
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for curve in curves:
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print(beziertatlength(curve,0.5))
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