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laser.py
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laser.py
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#!/usr/bin/env python
"""
Modified by Jay Johnson 2015, J Tech Photonics, Inc., jtechphotonics.com
modified by Adam Polak 2014, polakiumengineering.org
based on Copyright (C) 2009 Nick Drobchenko, nick@cnc-club.ru
based on gcode.py (C) 2007 hugomatic...
based on addnodes.py (C) 2005,2007 Aaron Spike, aaron@ekips.org
based on dots.py (C) 2005 Aaron Spike, aaron@ekips.org
based on interp.py (C) 2005 Aaron Spike, aaron@ekips.org
based on bezmisc.py (C) 2005 Aaron Spike, aaron@ekips.org
based on cubicsuperpath.py (C) 2005 Aaron Spike, aaron@ekips.org
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 2 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, write to the Free Software
Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
"""
import inkex, simplestyle, simplepath
import cubicsuperpath, simpletransform, bezmisc
import os
import math
import bezmisc
import re
import copy
import sys
import time
import cmath
import numpy
import codecs
import random
import gettext
_ = gettext.gettext
### Check if inkex has errormsg (0.46 version doesnot have one.) Could be removed later.
if "errormsg" not in dir(inkex):
inkex.errormsg = lambda msg: sys.stderr.write((unicode(msg) + "\n").encode("UTF-8"))
def bezierslopeatt(((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)),t):
ax,ay,bx,by,cx,cy,x0,y0=bezmisc.bezierparameterize(((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)))
dx=3*ax*(t**2)+2*bx*t+cx
dy=3*ay*(t**2)+2*by*t+cy
if dx==dy==0 :
dx = 6*ax*t+2*bx
dy = 6*ay*t+2*by
if dx==dy==0 :
dx = 6*ax
dy = 6*ay
if dx==dy==0 :
print_("Slope error x = %s*t^3+%s*t^2+%s*t+%s, y = %s*t^3+%s*t^2+%s*t+%s, t = %s, dx==dy==0" % (ax,bx,cx,dx,ay,by,cy,dy,t))
print_(((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3)))
dx, dy = 1, 1
return dx,dy
bezmisc.bezierslopeatt = bezierslopeatt
def ireplace(self,old,new,count=0):
pattern = re.compile(re.escape(old),re.I)
return re.sub(pattern,new,self,count)
################################################################################
###
### Styles and additional parameters
###
################################################################################
math.pi2 = math.pi*2
straight_tolerance = 0.0001
straight_distance_tolerance = 0.0001
engraving_tolerance = 0.0001
loft_lengths_tolerance = 0.0000001
options = {}
defaults = {
'header': """
G90
""",
'footer': """G0 X0 Y0
M18
"""
}
intersection_recursion_depth = 10
intersection_tolerance = 0.00001
styles = {
"loft_style" : {
'main curve': simplestyle.formatStyle({ 'stroke': '#88f', 'fill': 'none', 'stroke-width':'1', 'marker-end':'url(#Arrow2Mend)' }),
},
"biarc_style" : {
'biarc0': simplestyle.formatStyle({ 'stroke': '#88f', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'1' }),
'biarc1': simplestyle.formatStyle({ 'stroke': '#8f8', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'1' }),
'line': simplestyle.formatStyle({ 'stroke': '#f88', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'1' }),
'area': simplestyle.formatStyle({ 'stroke': '#777', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'0.1' }),
},
"biarc_style_dark" : {
'biarc0': simplestyle.formatStyle({ 'stroke': '#33a', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'1' }),
'biarc1': simplestyle.formatStyle({ 'stroke': '#3a3', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'1' }),
'line': simplestyle.formatStyle({ 'stroke': '#a33', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'1' }),
'area': simplestyle.formatStyle({ 'stroke': '#222', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'0.3' }),
},
"biarc_style_dark_area" : {
'biarc0': simplestyle.formatStyle({ 'stroke': '#33a', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'0.1' }),
'biarc1': simplestyle.formatStyle({ 'stroke': '#3a3', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'0.1' }),
'line': simplestyle.formatStyle({ 'stroke': '#a33', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'0.1' }),
'area': simplestyle.formatStyle({ 'stroke': '#222', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'0.3' }),
},
"biarc_style_i" : {
'biarc0': simplestyle.formatStyle({ 'stroke': '#880', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'1' }),
'biarc1': simplestyle.formatStyle({ 'stroke': '#808', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'1' }),
'line': simplestyle.formatStyle({ 'stroke': '#088', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'1' }),
'area': simplestyle.formatStyle({ 'stroke': '#999', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'0.3' }),
},
"biarc_style_dark_i" : {
'biarc0': simplestyle.formatStyle({ 'stroke': '#dd5', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'1' }),
'biarc1': simplestyle.formatStyle({ 'stroke': '#d5d', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'1' }),
'line': simplestyle.formatStyle({ 'stroke': '#5dd', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'1' }),
'area': simplestyle.formatStyle({ 'stroke': '#aaa', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'0.3' }),
},
"biarc_style_lathe_feed" : {
'biarc0': simplestyle.formatStyle({ 'stroke': '#07f', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'.4' }),
'biarc1': simplestyle.formatStyle({ 'stroke': '#0f7', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'.4' }),
'line': simplestyle.formatStyle({ 'stroke': '#f44', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'.4' }),
'area': simplestyle.formatStyle({ 'stroke': '#aaa', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'0.3' }),
},
"biarc_style_lathe_passing feed" : {
'biarc0': simplestyle.formatStyle({ 'stroke': '#07f', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'.4' }),
'biarc1': simplestyle.formatStyle({ 'stroke': '#0f7', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'.4' }),
'line': simplestyle.formatStyle({ 'stroke': '#f44', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'.4' }),
'area': simplestyle.formatStyle({ 'stroke': '#aaa', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'0.3' }),
},
"biarc_style_lathe_fine feed" : {
'biarc0': simplestyle.formatStyle({ 'stroke': '#7f0', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'.4' }),
'biarc1': simplestyle.formatStyle({ 'stroke': '#f70', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'.4' }),
'line': simplestyle.formatStyle({ 'stroke': '#744', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'.4' }),
'area': simplestyle.formatStyle({ 'stroke': '#aaa', 'fill': 'none', "marker-end":"url(#DrawCurveMarker)", 'stroke-width':'0.3' }),
},
"area artefact": simplestyle.formatStyle({ 'stroke': '#ff0000', 'fill': '#ffff00', 'stroke-width':'1' }),
"area artefact arrow": simplestyle.formatStyle({ 'stroke': '#ff0000', 'fill': '#ffff00', 'stroke-width':'1' }),
"dxf_points": simplestyle.formatStyle({ "stroke": "#ff0000", "fill": "#ff0000"}),
}
################################################################################
### Cubic Super Path additional functions
################################################################################
def csp_simple_bound(csp):
minx,miny,maxx,maxy = None,None,None,None
for subpath in csp:
for sp in subpath :
for p in sp:
minx = min(minx,p[0]) if minx!=None else p[0]
miny = min(miny,p[1]) if miny!=None else p[1]
maxx = max(maxx,p[0]) if maxx!=None else p[0]
maxy = max(maxy,p[1]) if maxy!=None else p[1]
return minx,miny,maxx,maxy
def csp_segment_to_bez(sp1,sp2) :
return sp1[1:]+sp2[:2]
def bound_to_bound_distance(sp1,sp2,sp3,sp4) :
min_dist = 1e100
max_dist = 0
points1 = csp_segment_to_bez(sp1,sp2)
points2 = csp_segment_to_bez(sp3,sp4)
for i in range(4) :
for j in range(4) :
min_, max_ = line_to_line_min_max_distance_2(points1[i-1], points1[i], points2[j-1], points2[j])
min_dist = min(min_dist,min_)
max_dist = max(max_dist,max_)
print_("bound_to_bound", min_dist, max_dist)
return min_dist, max_dist
def csp_to_point_distance(csp, p, dist_bounds = [0,1e100], tolerance=.01) :
min_dist = [1e100,0,0,0]
for j in range(len(csp)) :
for i in range(1,len(csp[j])) :
d = csp_seg_to_point_distance(csp[j][i-1],csp[j][i],p,sample_points = 5, tolerance = .01)
if d[0] < dist_bounds[0] :
# draw_pointer( list(csp_at_t(subpath[dist[2]-1],subpath[dist[2]],dist[3]))
# +list(csp_at_t(csp[dist[4]][dist[5]-1],csp[dist[4]][dist[5]],dist[6])),"red","line", comment = math.sqrt(dist[0]))
return [d[0],j,i,d[1]]
else :
if d[0] < min_dist[0] : min_dist = [d[0],j,i,d[1]]
return min_dist
def csp_seg_to_point_distance(sp1,sp2,p,sample_points = 5, tolerance = .01) :
ax,ay,bx,by,cx,cy,dx,dy = csp_parameterize(sp1,sp2)
dx, dy = dx-p[0], dy-p[1]
if sample_points < 2 : sample_points = 2
d = min( [(p[0]-sp1[1][0])**2 + (p[1]-sp1[1][1])**2,0.], [(p[0]-sp2[1][0])**2 + (p[1]-sp2[1][1])**2,1.] )
for k in range(sample_points) :
t = float(k)/(sample_points-1)
i = 0
while i==0 or abs(f)>0.000001 and i<20 :
t2,t3 = t**2,t**3
f = (ax*t3+bx*t2+cx*t+dx)*(3*ax*t2+2*bx*t+cx) + (ay*t3+by*t2+cy*t+dy)*(3*ay*t2+2*by*t+cy)
df = (6*ax*t+2*bx)*(ax*t3+bx*t2+cx*t+dx) + (3*ax*t2+2*bx*t+cx)**2 + (6*ay*t+2*by)*(ay*t3+by*t2+cy*t+dy) + (3*ay*t2+2*by*t+cy)**2
if df!=0 :
t = t - f/df
else :
break
i += 1
if 0<=t<=1 :
p1 = csp_at_t(sp1,sp2,t)
d1 = (p1[0]-p[0])**2 + (p1[1]-p[1])**2
if d1 < d[0] :
d = [d1,t]
return d
def csp_seg_to_csp_seg_distance(sp1,sp2,sp3,sp4, dist_bounds = [0,1e100], sample_points = 5, tolerance=.01) :
# check the ending points first
dist = csp_seg_to_point_distance(sp1,sp2,sp3[1],sample_points, tolerance)
dist += [0.]
if dist[0] <= dist_bounds[0] : return dist
d = csp_seg_to_point_distance(sp1,sp2,sp4[1],sample_points, tolerance)
if d[0]<dist[0] :
dist = d+[1.]
if dist[0] <= dist_bounds[0] : return dist
d = csp_seg_to_point_distance(sp3,sp4,sp1[1],sample_points, tolerance)
if d[0]<dist[0] :
dist = [d[0],0.,d[1]]
if dist[0] <= dist_bounds[0] : return dist
d = csp_seg_to_point_distance(sp3,sp4,sp2[1],sample_points, tolerance)
if d[0]<dist[0] :
dist = [d[0],1.,d[1]]
if dist[0] <= dist_bounds[0] : return dist
sample_points -= 2
if sample_points < 1 : sample_points = 1
ax1,ay1,bx1,by1,cx1,cy1,dx1,dy1 = csp_parameterize(sp1,sp2)
ax2,ay2,bx2,by2,cx2,cy2,dx2,dy2 = csp_parameterize(sp3,sp4)
# try to find closes points using Newtons method
for k in range(sample_points) :
for j in range(sample_points) :
t1,t2 = float(k+1)/(sample_points+1), float(j)/(sample_points+1)
t12, t13, t22, t23 = t1*t1, t1*t1*t1, t2*t2, t2*t2*t2
i = 0
F1, F2, F = [0,0], [[0,0],[0,0]], 1e100
x,y = ax1*t13+bx1*t12+cx1*t1+dx1 - (ax2*t23+bx2*t22+cx2*t2+dx2), ay1*t13+by1*t12+cy1*t1+dy1 - (ay2*t23+by2*t22+cy2*t2+dy2)
while i<2 or abs(F-Flast)>tolerance and i<30 :
#draw_pointer(csp_at_t(sp1,sp2,t1))
f1x = 3*ax1*t12+2*bx1*t1+cx1
f1y = 3*ay1*t12+2*by1*t1+cy1
f2x = 3*ax2*t22+2*bx2*t2+cx2
f2y = 3*ay2*t22+2*by2*t2+cy2
F1[0] = 2*f1x*x + 2*f1y*y
F1[1] = -2*f2x*x - 2*f2y*y
F2[0][0] = 2*(6*ax1*t1+2*bx1)*x + 2*f1x*f1x + 2*(6*ay1*t1+2*by1)*y +2*f1y*f1y
F2[0][1] = -2*f1x*f2x - 2*f1y*f2y
F2[1][0] = -2*f2x*f1x - 2*f2y*f1y
F2[1][1] = -2*(6*ax2*t2+2*bx2)*x + 2*f2x*f2x - 2*(6*ay2*t2+2*by2)*y + 2*f2y*f2y
F2 = inv_2x2(F2)
if F2!=None :
t1 -= ( F2[0][0]*F1[0] + F2[0][1]*F1[1] )
t2 -= ( F2[1][0]*F1[0] + F2[1][1]*F1[1] )
t12, t13, t22, t23 = t1*t1, t1*t1*t1, t2*t2, t2*t2*t2
x,y = ax1*t13+bx1*t12+cx1*t1+dx1 - (ax2*t23+bx2*t22+cx2*t2+dx2), ay1*t13+by1*t12+cy1*t1+dy1 - (ay2*t23+by2*t22+cy2*t2+dy2)
Flast = F
F = x*x+y*y
else :
break
i += 1
if F < dist[0] and 0<=t1<=1 and 0<=t2<=1:
dist = [F,t1,t2]
if dist[0] <= dist_bounds[0] :
return dist
return dist
def csp_to_csp_distance(csp1,csp2, dist_bounds = [0,1e100], tolerance=.01) :
dist = [1e100,0,0,0,0,0,0]
for i1 in range(len(csp1)) :
for j1 in range(1,len(csp1[i1])) :
for i2 in range(len(csp2)) :
for j2 in range(1,len(csp2[i2])) :
d = csp_seg_bound_to_csp_seg_bound_max_min_distance(csp1[i1][j1-1],csp1[i1][j1],csp2[i2][j2-1],csp2[i2][j2])
if d[0] >= dist_bounds[1] : continue
if d[1] < dist_bounds[0] : return [d[1],i1,j1,1,i2,j2,1]
d = csp_seg_to_csp_seg_distance(csp1[i1][j1-1],csp1[i1][j1],csp2[i2][j2-1],csp2[i2][j2], dist_bounds, tolerance=tolerance)
if d[0] < dist[0] :
dist = [d[0], i1,j1,d[1], i2,j2,d[2]]
if dist[0] <= dist_bounds[0] :
return dist
if dist[0] >= dist_bounds[1] :
return dist
return dist
# draw_pointer( list(csp_at_t(csp1[dist[1]][dist[2]-1],csp1[dist[1]][dist[2]],dist[3]))
# + list(csp_at_t(csp2[dist[4]][dist[5]-1],csp2[dist[4]][dist[5]],dist[6])), "#507","line")
def csp_split(sp1,sp2,t=.5) :
[x1,y1],[x2,y2],[x3,y3],[x4,y4] = sp1[1], sp1[2], sp2[0], sp2[1]
x12 = x1+(x2-x1)*t
y12 = y1+(y2-y1)*t
x23 = x2+(x3-x2)*t
y23 = y2+(y3-y2)*t
x34 = x3+(x4-x3)*t
y34 = y3+(y4-y3)*t
x1223 = x12+(x23-x12)*t
y1223 = y12+(y23-y12)*t
x2334 = x23+(x34-x23)*t
y2334 = y23+(y34-y23)*t
x = x1223+(x2334-x1223)*t
y = y1223+(y2334-y1223)*t
return [sp1[0],sp1[1],[x12,y12]], [[x1223,y1223],[x,y],[x2334,y2334]], [[x34,y34],sp2[1],sp2[2]]
def csp_true_bounds(csp) :
# Finds minx,miny,maxx,maxy of the csp and return their (x,y,i,j,t)
minx = [float("inf"), 0, 0, 0]
maxx = [float("-inf"), 0, 0, 0]
miny = [float("inf"), 0, 0, 0]
maxy = [float("-inf"), 0, 0, 0]
for i in range(len(csp)):
for j in range(1,len(csp[i])):
ax,ay,bx,by,cx,cy,x0,y0 = bezmisc.bezierparameterize((csp[i][j-1][1],csp[i][j-1][2],csp[i][j][0],csp[i][j][1]))
roots = cubic_solver(0, 3*ax, 2*bx, cx) + [0,1]
for root in roots :
if type(root) is complex and abs(root.imag)<1e-10:
root = root.real
if type(root) is not complex and 0<=root<=1:
y = ay*(root**3)+by*(root**2)+cy*root+y0
x = ax*(root**3)+bx*(root**2)+cx*root+x0
maxx = max([x,y,i,j,root],maxx)
minx = min([x,y,i,j,root],minx)
roots = cubic_solver(0, 3*ay, 2*by, cy) + [0,1]
for root in roots :
if type(root) is complex and root.imag==0:
root = root.real
if type(root) is not complex and 0<=root<=1:
y = ay*(root**3)+by*(root**2)+cy*root+y0
x = ax*(root**3)+bx*(root**2)+cx*root+x0
maxy = max([y,x,i,j,root],maxy)
miny = min([y,x,i,j,root],miny)
maxy[0],maxy[1] = maxy[1],maxy[0]
miny[0],miny[1] = miny[1],miny[0]
return minx,miny,maxx,maxy
############################################################################
### csp_segments_intersection(sp1,sp2,sp3,sp4)
###
### Returns array containig all intersections between two segmets of cubic
### super path. Results are [ta,tb], or [ta0, ta1, tb0, tb1, "Overlap"]
### where ta, tb are values of t for the intersection point.
############################################################################
def csp_segments_intersection(sp1,sp2,sp3,sp4) :
a, b = csp_segment_to_bez(sp1,sp2), csp_segment_to_bez(sp3,sp4)
def polish_intersection(a,b,ta,tb, tolerance = intersection_tolerance) :
ax,ay,bx,by,cx,cy,dx,dy = bezmisc.bezierparameterize(a)
ax1,ay1,bx1,by1,cx1,cy1,dx1,dy1 = bezmisc.bezierparameterize(b)
i = 0
F, F1 = [.0,.0], [[.0,.0],[.0,.0]]
while i==0 or (abs(F[0])**2+abs(F[1])**2 > tolerance and i<10):
ta3, ta2, tb3, tb2 = ta**3, ta**2, tb**3, tb**2
F[0] = ax*ta3+bx*ta2+cx*ta+dx-ax1*tb3-bx1*tb2-cx1*tb-dx1
F[1] = ay*ta3+by*ta2+cy*ta+dy-ay1*tb3-by1*tb2-cy1*tb-dy1
F1[0][0] = 3*ax *ta2 + 2*bx *ta + cx
F1[0][1] = -3*ax1*tb2 - 2*bx1*tb - cx1
F1[1][0] = 3*ay *ta2 + 2*by *ta + cy
F1[1][1] = -3*ay1*tb2 - 2*by1*tb - cy1
det = F1[0][0]*F1[1][1] - F1[0][1]*F1[1][0]
if det!=0 :
F1 = [ [ F1[1][1]/det, -F1[0][1]/det], [-F1[1][0]/det, F1[0][0]/det] ]
ta = ta - ( F1[0][0]*F[0] + F1[0][1]*F[1] )
tb = tb - ( F1[1][0]*F[0] + F1[1][1]*F[1] )
else: break
i += 1
return ta, tb
def recursion(a,b, ta0,ta1,tb0,tb1, depth_a,depth_b) :
global bezier_intersection_recursive_result
if a==b :
bezier_intersection_recursive_result += [[ta0,tb0,ta1,tb1,"Overlap"]]
return
tam, tbm = (ta0+ta1)/2, (tb0+tb1)/2
if depth_a>0 and depth_b>0 :
a1,a2 = bez_split(a,0.5)
b1,b2 = bez_split(b,0.5)
if bez_bounds_intersect(a1,b1) : recursion(a1,b1, ta0,tam,tb0,tbm, depth_a-1,depth_b-1)
if bez_bounds_intersect(a2,b1) : recursion(a2,b1, tam,ta1,tb0,tbm, depth_a-1,depth_b-1)
if bez_bounds_intersect(a1,b2) : recursion(a1,b2, ta0,tam,tbm,tb1, depth_a-1,depth_b-1)
if bez_bounds_intersect(a2,b2) : recursion(a2,b2, tam,ta1,tbm,tb1, depth_a-1,depth_b-1)
elif depth_a>0 :
a1,a2 = bez_split(a,0.5)
if bez_bounds_intersect(a1,b) : recursion(a1,b, ta0,tam,tb0,tb1, depth_a-1,depth_b)
if bez_bounds_intersect(a2,b) : recursion(a2,b, tam,ta1,tb0,tb1, depth_a-1,depth_b)
elif depth_b>0 :
b1,b2 = bez_split(b,0.5)
if bez_bounds_intersect(a,b1) : recursion(a,b1, ta0,ta1,tb0,tbm, depth_a,depth_b-1)
if bez_bounds_intersect(a,b2) : recursion(a,b2, ta0,ta1,tbm,tb1, depth_a,depth_b-1)
else : # Both segments have been subdevided enougth. Let's get some intersections :).
intersection, t1, t2 = straight_segments_intersection([a[0]]+[a[3]],[b[0]]+[b[3]])
if intersection :
if intersection == "Overlap" :
t1 = ( max(0,min(1,t1[0]))+max(0,min(1,t1[1])) )/2
t2 = ( max(0,min(1,t2[0]))+max(0,min(1,t2[1])) )/2
bezier_intersection_recursive_result += [[ta0+t1*(ta1-ta0),tb0+t2*(tb1-tb0)]]
global bezier_intersection_recursive_result
bezier_intersection_recursive_result = []
recursion(a,b,0.,1.,0.,1.,intersection_recursion_depth,intersection_recursion_depth)
intersections = bezier_intersection_recursive_result
for i in range(len(intersections)) :
if len(intersections[i])<5 or intersections[i][4] != "Overlap" :
intersections[i] = polish_intersection(a,b,intersections[i][0],intersections[i][1])
return intersections
def csp_segments_true_intersection(sp1,sp2,sp3,sp4) :
intersections = csp_segments_intersection(sp1,sp2,sp3,sp4)
res = []
for intersection in intersections :
if (
(len(intersection)==5 and intersection[4] == "Overlap" and (0<=intersection[0]<=1 or 0<=intersection[1]<=1) and (0<=intersection[2]<=1 or 0<=intersection[3]<=1) )
or ( 0<=intersection[0]<=1 and 0<=intersection[1]<=1 )
) :
res += [intersection]
return res
def csp_get_t_at_curvature(sp1,sp2,c, sample_points = 16):
# returns a list containning [t1,t2,t3,...,tn], 0<=ti<=1...
if sample_points < 2 : sample_points = 2
tolerance = .0000000001
res = []
ax,ay,bx,by,cx,cy,dx,dy = csp_parameterize(sp1,sp2)
for k in range(sample_points) :
t = float(k)/(sample_points-1)
i, F = 0, 1e100
while i<2 or abs(F)>tolerance and i<17 :
try : # some numerical calculation could exceed the limits
t2 = t*t
#slopes...
f1x = 3*ax*t2+2*bx*t+cx
f1y = 3*ay*t2+2*by*t+cy
f2x = 6*ax*t+2*bx
f2y = 6*ay*t+2*by
f3x = 6*ax
f3y = 6*ay
d = (f1x**2+f1y**2)**1.5
F1 = (
( (f1x*f3y-f3x*f1y)*d - (f1x*f2y-f2x*f1y)*3.*(f2x*f1x+f2y*f1y)*((f1x**2+f1y**2)**.5) ) /
((f1x**2+f1y**2)**3)
)
F = (f1x*f2y-f1y*f2x)/d - c
t -= F/F1
except:
break
i += 1
if 0<=t<=1 and F<=tolerance:
if len(res) == 0 :
res.append(t)
for i in res :
if abs(t-i)<=0.001 :
break
if not abs(t-i)<=0.001 :
res.append(t)
return res
def csp_max_curvature(sp1,sp2):
ax,ay,bx,by,cx,cy,dx,dy = csp_parameterize(sp1,sp2)
tolerance = .0001
F = 0.
i = 0
while i<2 or F-Flast<tolerance and i<10 :
t = .5
f1x = 3*ax*t**2 + 2*bx*t + cx
f1y = 3*ay*t**2 + 2*by*t + cy
f2x = 6*ax*t + 2*bx
f2y = 6*ay*t + 2*by
f3x = 6*ax
f3y = 6*ay
d = pow(f1x**2+f1y**2,1.5)
if d != 0 :
Flast = F
F = (f1x*f2y-f1y*f2x)/d
F1 = (
( d*(f1x*f3y-f3x*f1y) - (f1x*f2y-f2x*f1y)*3.*(f2x*f1x+f2y*f1y)*pow(f1x**2+f1y**2,.5) ) /
(f1x**2+f1y**2)**3
)
i+=1
if F1!=0:
t -= F/F1
else:
break
else: break
return t
def csp_curvature_at_t(sp1,sp2,t, depth = 3) :
ax,ay,bx,by,cx,cy,dx,dy = bezmisc.bezierparameterize(csp_segment_to_bez(sp1,sp2))
#curvature = (x'y''-y'x'') / (x'^2+y'^2)^1.5
f1x = 3*ax*t**2 + 2*bx*t + cx
f1y = 3*ay*t**2 + 2*by*t + cy
f2x = 6*ax*t + 2*bx
f2y = 6*ay*t + 2*by
d = (f1x**2+f1y**2)**1.5
if d != 0 :
return (f1x*f2y-f1y*f2x)/d
else :
t1 = f1x*f2y-f1y*f2x
if t1 > 0 : return 1e100
if t1 < 0 : return -1e100
# Use the Lapitals rule to solve 0/0 problem for 2 times...
t1 = 2*(bx*ay-ax*by)*t+(ay*cx-ax*cy)
if t1 > 0 : return 1e100
if t1 < 0 : return -1e100
t1 = bx*ay-ax*by
if t1 > 0 : return 1e100
if t1 < 0 : return -1e100
if depth>0 :
# little hack ;^) hope it wont influence anything...
return csp_curvature_at_t(sp1,sp2,t*1.004, depth-1)
return 1e100
def csp_curvature_radius_at_t(sp1,sp2,t) :
c = csp_curvature_at_t(sp1,sp2,t)
if c == 0 : return 1e100
else: return 1/c
def csp_special_points(sp1,sp2) :
# special points = curvature == 0
ax,ay,bx,by,cx,cy,dx,dy = bezmisc.bezierparameterize((sp1[1],sp1[2],sp2[0],sp2[1]))
a = 3*ax*by-3*ay*bx
b = 3*ax*cy-3*cx*ay
c = bx*cy-cx*by
roots = cubic_solver(0, a, b, c)
res = []
for i in roots :
if type(i) is complex and i.imag==0:
i = i.real
if type(i) is not complex and 0<=i<=1:
res.append(i)
return res
def csp_subpath_ccw(subpath):
# Remove all zerro length segments
s = 0
#subpath = subpath[:]
if (P(subpath[-1][1])-P(subpath[0][1])).l2() > 1e-10 :
subpath[-1][2] = subpath[-1][1]
subpath[0][0] = subpath[0][1]
subpath += [ [subpath[0][1],subpath[0][1],subpath[0][1]] ]
pl = subpath[-1][2]
for sp1 in subpath:
for p in sp1 :
s += (p[0]-pl[0])*(p[1]+pl[1])
pl = p
return s<0
def csp_at_t(sp1,sp2,t):
ax,bx,cx,dx = sp1[1][0], sp1[2][0], sp2[0][0], sp2[1][0]
ay,by,cy,dy = sp1[1][1], sp1[2][1], sp2[0][1], sp2[1][1]
x1, y1 = ax+(bx-ax)*t, ay+(by-ay)*t
x2, y2 = bx+(cx-bx)*t, by+(cy-by)*t
x3, y3 = cx+(dx-cx)*t, cy+(dy-cy)*t
x4,y4 = x1+(x2-x1)*t, y1+(y2-y1)*t
x5,y5 = x2+(x3-x2)*t, y2+(y3-y2)*t
x,y = x4+(x5-x4)*t, y4+(y5-y4)*t
return [x,y]
def csp_splitatlength(sp1, sp2, l = 0.5, tolerance = 0.01):
bez = (sp1[1][:],sp1[2][:],sp2[0][:],sp2[1][:])
t = bezmisc.beziertatlength(bez, l, tolerance)
return csp_split(sp1, sp2, t)
def cspseglength(sp1,sp2, tolerance = 0.001):
bez = (sp1[1][:],sp1[2][:],sp2[0][:],sp2[1][:])
return bezmisc.bezierlength(bez, tolerance)
def csplength(csp):
total = 0
lengths = []
for sp in csp:
for i in xrange(1,len(sp)):
l = cspseglength(sp[i-1],sp[i])
lengths.append(l)
total += l
return lengths, total
def csp_segments(csp):
l, seg = 0, [0]
for sp in csp:
for i in xrange(1,len(sp)):
l += cspseglength(sp[i-1],sp[i])
seg += [ l ]
if l>0 :
seg = [seg[i]/l for i in xrange(len(seg))]
return seg,l
def rebuild_csp (csp, segs, s=None):
# rebuild_csp() adds to csp control points making it's segments looks like segs
if s==None : s, l = csp_segments(csp)
if len(s)>len(segs) : return None
segs = segs[:]
segs.sort()
for i in xrange(len(s)):
d = None
for j in xrange(len(segs)):
d = min( [abs(s[i]-segs[j]),j], d) if d!=None else [abs(s[i]-segs[j]),j]
del segs[d[1]]
for i in xrange(len(segs)):
for j in xrange(0,len(s)):
if segs[i]<s[j] : break
if s[j]-s[j-1] != 0 :
t = (segs[i] - s[j-1])/(s[j]-s[j-1])
sp1,sp2,sp3 = csp_split(csp[j-1],csp[j], t)
csp = csp[:j-1] + [sp1,sp2,sp3] + csp[j+1:]
s = s[:j] + [ s[j-1]*(1-t)+s[j]*t ] + s[j:]
return csp, s
def csp_slope(sp1,sp2,t):
bez = (sp1[1][:],sp1[2][:],sp2[0][:],sp2[1][:])
return bezmisc.bezierslopeatt(bez,t)
def csp_line_intersection(l1,l2,sp1,sp2):
dd=l1[0]
cc=l2[0]-l1[0]
bb=l1[1]
aa=l2[1]-l1[1]
if aa==cc==0 : return []
if aa:
coef1=cc/aa
coef2=1
else:
coef1=1
coef2=aa/cc
bez = (sp1[1][:],sp1[2][:],sp2[0][:],sp2[1][:])
ax,ay,bx,by,cx,cy,x0,y0=bezmisc.bezierparameterize(bez)
a=coef1*ay-coef2*ax
b=coef1*by-coef2*bx
c=coef1*cy-coef2*cx
d=coef1*(y0-bb)-coef2*(x0-dd)
roots = cubic_solver(a,b,c,d)
retval = []
for i in roots :
if type(i) is complex and abs(i.imag)<1e-7:
i = i.real
if type(i) is not complex and -1e-10<=i<=1.+1e-10:
retval.append(i)
return retval
def csp_split_by_two_points(sp1,sp2,t1,t2) :
if t1>t2 : t1, t2 = t2, t1
if t1 == t2 :
sp1,sp2,sp3 = csp_split(sp1,sp2,t)
return [sp1,sp2,sp2,sp3]
elif t1 <= 1e-10 and t2 >= 1.-1e-10 :
return [sp1,sp1,sp2,sp2]
elif t1 <= 1e-10:
sp1,sp2,sp3 = csp_split(sp1,sp2,t2)
return [sp1,sp1,sp2,sp3]
elif t2 >= 1.-1e-10 :
sp1,sp2,sp3 = csp_split(sp1,sp2,t1)
return [sp1,sp2,sp3,sp3]
else:
sp1,sp2,sp3 = csp_split(sp1,sp2,t1)
sp2,sp3,sp4 = csp_split(sp2,sp3,(t2-t1)/(1-t1) )
return [sp1,sp2,sp3,sp4]
def csp_subpath_split_by_points(subpath, points) :
# points are [[i,t]...] where i-segment's number
points.sort()
points = [[1,0.]] + points + [[len(subpath)-1,1.]]
parts = []
for int1,int2 in zip(points,points[1:]) :
if int1==int2 :
continue
if int1[1] == 1. :
int1[0] += 1
int1[1] = 0.
if int1==int2 :
continue
if int2[1] == 0. :
int2[0] -= 1
int2[1] = 1.
if int1[0] == 0 and int2[0]==len(subpath)-1:# and small(int1[1]) and small(int2[1]-1) :
continue
if int1[0]==int2[0] : # same segment
sp = csp_split_by_two_points(subpath[int1[0]-1],subpath[int1[0]],int1[1], int2[1])
if sp[1]!=sp[2] :
parts += [ [sp[1],sp[2]] ]
else :
sp5,sp1,sp2 = csp_split(subpath[int1[0]-1],subpath[int1[0]],int1[1])
sp3,sp4,sp5 = csp_split(subpath[int2[0]-1],subpath[int2[0]],int2[1])
if int1[0]==int2[0]-1 :
parts += [ [sp1, [sp2[0],sp2[1],sp3[2]], sp4] ]
else :
parts += [ [sp1,sp2]+subpath[int1[0]+1:int2[0]-1]+[sp3,sp4] ]
return parts
def csp_from_arc(start, end, center, r, slope_st) :
# Creates csp that approximise specified arc
r = abs(r)
alpha = (atan2(end[0]-center[0],end[1]-center[1]) - atan2(start[0]-center[0],start[1]-center[1])) % math.pi2
sectors = int(abs(alpha)*2/math.pi)+1
alpha_start = atan2(start[0]-center[0],start[1]-center[1])
cos_,sin_ = math.cos(alpha_start), math.sin(alpha_start)
k = (4.*math.tan(alpha/sectors/4.)/3.)
if dot(slope_st , [- sin_*k*r, cos_*k*r]) < 0 :
if alpha>0 : alpha -= math.pi2
else: alpha += math.pi2
if abs(alpha*r)<0.001 :
return []
sectors = int(abs(alpha)*2/math.pi)+1
k = (4.*math.tan(alpha/sectors/4.)/3.)
result = []
for i in range(sectors+1) :
cos_,sin_ = math.cos(alpha_start + alpha*i/sectors), math.sin(alpha_start + alpha*i/sectors)
sp = [ [], [center[0] + cos_*r, center[1] + sin_*r], [] ]
sp[0] = [sp[1][0] + sin_*k*r, sp[1][1] - cos_*k*r ]
sp[2] = [sp[1][0] - sin_*k*r, sp[1][1] + cos_*k*r ]
result += [sp]
result[0][0] = result[0][1][:]
result[-1][2] = result[-1][1]
return result
def point_to_arc_distance(p, arc):
### Distance calculattion from point to arc
P0,P2,c,a = arc
dist = None
p = P(p)
r = (P0-c).mag()
if r>0 :
i = c + (p-c).unit()*r
alpha = ((i-c).angle() - (P0-c).angle())
if a*alpha<0:
if alpha>0: alpha = alpha-math.pi2
else: alpha = math.pi2+alpha
if between(alpha,0,a) or min(abs(alpha),abs(alpha-a))<straight_tolerance :
return (p-i).mag(), [i.x, i.y]
else :
d1, d2 = (p-P0).mag(), (p-P2).mag()
if d1<d2 :
return (d1, [P0.x,P0.y])
else :
return (d2, [P2.x,P2.y])
def csp_to_arc_distance(sp1,sp2, arc1, arc2, tolerance = 0.01 ): # arc = [start,end,center,alpha]
n, i = 10, 0
d, d1, dl = (0,(0,0)), (0,(0,0)), 0
while i<1 or (abs(d1[0]-dl[0])>tolerance and i<4):
i += 1
dl = d1*1
for j in range(n+1):
t = float(j)/n
p = csp_at_t(sp1,sp2,t)
d = min(point_to_arc_distance(p,arc1), point_to_arc_distance(p,arc2))
d1 = max(d1,d)
n=n*2
return d1[0]
def csp_simple_bound_to_point_distance(p, csp):
minx,miny,maxx,maxy = None,None,None,None
for subpath in csp:
for sp in subpath:
for p_ in sp:
minx = min(minx,p_[0]) if minx!=None else p_[0]
miny = min(miny,p_[1]) if miny!=None else p_[1]
maxx = max(maxx,p_[0]) if maxx!=None else p_[0]
maxy = max(maxy,p_[1]) if maxy!=None else p_[1]
return math.sqrt(max(minx-p[0],p[0]-maxx,0)**2+max(miny-p[1],p[1]-maxy,0)**2)
def csp_point_inside_bound(sp1, sp2, p):
bez = [sp1[1],sp1[2],sp2[0],sp2[1]]
x,y = p
c = 0
for i in range(4):
[x0,y0], [x1,y1] = bez[i-1], bez[i]
if x0-x1!=0 and (y-y0)*(x1-x0)>=(x-x0)*(y1-y0) and x>min(x0,x1) and x<=max(x0,x1) :
c +=1
return c%2==0
def csp_bound_to_point_distance(sp1, sp2, p):
if csp_point_inside_bound(sp1, sp2, p) :
return 0.
bez = csp_segment_to_bez(sp1,sp2)
min_dist = 1e100
for i in range(0,4):
d = point_to_line_segment_distance_2(p, bez[i-1],bez[i])
if d <= min_dist : min_dist = d
return min_dist
def line_line_intersect(p1,p2,p3,p4) : # Return only true intersection.
if (p1[0]==p2[0] and p1[1]==p2[1]) or (p3[0]==p4[0] and p3[1]==p4[1]) : return False
x = (p2[0]-p1[0])*(p4[1]-p3[1]) - (p2[1]-p1[1])*(p4[0]-p3[0])
if x==0 : # Lines are parallel
if (p3[0]-p1[0])*(p2[1]-p1[1]) == (p3[1]-p1[1])*(p2[0]-p1[0]) :
if p3[0]!=p4[0] :
t11 = (p1[0]-p3[0])/(p4[0]-p3[0])
t12 = (p2[0]-p3[0])/(p4[0]-p3[0])
t21 = (p3[0]-p1[0])/(p2[0]-p1[0])
t22 = (p4[0]-p1[0])/(p2[0]-p1[0])
else:
t11 = (p1[1]-p3[1])/(p4[1]-p3[1])
t12 = (p2[1]-p3[1])/(p4[1]-p3[1])
t21 = (p3[1]-p1[1])/(p2[1]-p1[1])
t22 = (p4[1]-p1[1])/(p2[1]-p1[1])
return ("Overlap" if (0<=t11<=1 or 0<=t12<=1) and (0<=t21<=1 or 0<=t22<=1) else False)
else: return False
else :
return (
0<=((p4[0]-p3[0])*(p1[1]-p3[1]) - (p4[1]-p3[1])*(p1[0]-p3[0]))/x<=1 and
0<=((p2[0]-p1[0])*(p1[1]-p3[1]) - (p2[1]-p1[1])*(p1[0]-p3[0]))/x<=1 )
def line_line_intersection_points(p1,p2,p3,p4) : # Return only points [ (x,y) ]
if (p1[0]==p2[0] and p1[1]==p2[1]) or (p3[0]==p4[0] and p3[1]==p4[1]) : return []
x = (p2[0]-p1[0])*(p4[1]-p3[1]) - (p2[1]-p1[1])*(p4[0]-p3[0])
if x==0 : # Lines are parallel
if (p3[0]-p1[0])*(p2[1]-p1[1]) == (p3[1]-p1[1])*(p2[0]-p1[0]) :
if p3[0]!=p4[0] :
t11 = (p1[0]-p3[0])/(p4[0]-p3[0])
t12 = (p2[0]-p3[0])/(p4[0]-p3[0])
t21 = (p3[0]-p1[0])/(p2[0]-p1[0])
t22 = (p4[0]-p1[0])/(p2[0]-p1[0])
else:
t11 = (p1[1]-p3[1])/(p4[1]-p3[1])
t12 = (p2[1]-p3[1])/(p4[1]-p3[1])
t21 = (p3[1]-p1[1])/(p2[1]-p1[1])
t22 = (p4[1]-p1[1])/(p2[1]-p1[1])
res = []
if (0<=t11<=1 or 0<=t12<=1) and (0<=t21<=1 or 0<=t22<=1) :
if 0<=t11<=1 : res += [p1]
if 0<=t12<=1 : res += [p2]
if 0<=t21<=1 : res += [p3]
if 0<=t22<=1 : res += [p4]
return res
else: return []
else :
t1 = ((p4[0]-p3[0])*(p1[1]-p3[1]) - (p4[1]-p3[1])*(p1[0]-p3[0]))/x
t2 = ((p2[0]-p1[0])*(p1[1]-p3[1]) - (p2[1]-p1[1])*(p1[0]-p3[0]))/x
if 0<=t1<=1 and 0<=t2<=1 : return [ [p1[0]*(1-t1)+p2[0]*t1, p1[1]*(1-t1)+p2[1]*t1] ]
else : return []
def point_to_point_d2(a,b):
return (a[0]-b[0])**2 + (a[1]-b[1])**2
def point_to_point_d(a,b):
return math.sqrt((a[0]-b[0])**2 + (a[1]-b[1])**2)
def point_to_line_segment_distance_2(p1, p2,p3) :
# p1 - point, p2,p3 - line segment
#draw_pointer(p1)
w0 = [p1[0]-p2[0], p1[1]-p2[1]]
v = [p3[0]-p2[0], p3[1]-p2[1]]
c1 = w0[0]*v[0] + w0[1]*v[1]
if c1 <= 0 :
return w0[0]*w0[0]+w0[1]*w0[1]
c2 = v[0]*v[0] + v[1]*v[1]
if c2 <= c1 :
return (p1[0]-p3[0])**2 + (p1[1]-p3[1])**2
return (p1[0]- p2[0]-v[0]*c1/c2)**2 + (p1[1]- p2[1]-v[1]*c1/c2)
def line_to_line_distance_2(p1,p2,p3,p4):
if line_line_intersect(p1,p2,p3,p4) : return 0
return min(
point_to_line_segment_distance_2(p1,p3,p4),
point_to_line_segment_distance_2(p2,p3,p4),
point_to_line_segment_distance_2(p3,p1,p2),
point_to_line_segment_distance_2(p4,p1,p2))
def csp_seg_bound_to_csp_seg_bound_max_min_distance(sp1,sp2,sp3,sp4) :
bez1 = csp_segment_to_bez(sp1,sp2)
bez2 = csp_segment_to_bez(sp3,sp4)
min_dist = 1e100
max_dist = 0.
for i in range(4) :
if csp_point_inside_bound(sp1, sp2, bez2[i]) or csp_point_inside_bound(sp3, sp4, bez1[i]) :
min_dist = 0.
break
for i in range(4) :
for j in range(4) :
d = line_to_line_distance_2(bez1[i-1],bez1[i],bez2[j-1],bez2[j])
if d < min_dist : min_dist = d
d = (bez2[j][0]-bez1[i][0])**2 + (bez2[j][1]-bez1[i][1])**2
if max_dist < d : max_dist = d
return min_dist, max_dist
def csp_reverse(csp) :
for i in range(len(csp)) :
n = []
for j in csp[i] :
n = [ [j[2][:],j[1][:],j[0][:]] ] + n
csp[i] = n[:]
return csp
def csp_normalized_slope(sp1,sp2,t) :
ax,ay,bx,by,cx,cy,dx,dy=bezmisc.bezierparameterize((sp1[1][:],sp1[2][:],sp2[0][:],sp2[1][:]))
if sp1[1]==sp2[1]==sp1[2]==sp2[0] : return [1.,0.]
f1x = 3*ax*t*t+2*bx*t+cx
f1y = 3*ay*t*t+2*by*t+cy
if abs(f1x*f1x+f1y*f1y) > 1e-20 :
l = math.sqrt(f1x*f1x+f1y*f1y)
return [f1x/l, f1y/l]
if t == 0 :
f1x = sp2[0][0]-sp1[1][0]
f1y = sp2[0][1]-sp1[1][1]
if abs(f1x*f1x+f1y*f1y) > 1e-20 :
l = math.sqrt(f1x*f1x+f1y*f1y)
return [f1x/l, f1y/l]
else :
f1x = sp2[1][0]-sp1[1][0]
f1y = sp2[1][1]-sp1[1][1]
if f1x*f1x+f1y*f1y != 0 :
l = math.sqrt(f1x*f1x+f1y*f1y)
return [f1x/l, f1y/l]
elif t == 1 :
f1x = sp2[1][0]-sp1[2][0]
f1y = sp2[1][1]-sp1[2][1]
if abs(f1x*f1x+f1y*f1y) > 1e-20 :
l = math.sqrt(f1x*f1x+f1y*f1y)
return [f1x/l, f1y/l]
else :
f1x = sp2[1][0]-sp1[1][0]
f1y = sp2[1][1]-sp1[1][1]
if f1x*f1x+f1y*f1y != 0 :
l = math.sqrt(f1x*f1x+f1y*f1y)
return [f1x/l, f1y/l]
else :
return [1.,0.]
def csp_normalized_normal(sp1,sp2,t) :
nx,ny = csp_normalized_slope(sp1,sp2,t)
return [-ny, nx]
def csp_parameterize(sp1,sp2):
return bezmisc.bezierparameterize(csp_segment_to_bez(sp1,sp2))
def csp_concat_subpaths(*s):
def concat(s1,s2) :
if s1 == [] : return s2
if s2 == [] : return s1
if (s1[-1][1][0]-s2[0][1][0])**2 + (s1[-1][1][1]-s2[0][1][1])**2 > 0.00001 :
return s1[:-1]+[ [s1[-1][0],s1[-1][1],s1[-1][1]], [s2[0][1],s2[0][1],s2[0][2]] ] + s2[1:]