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@NateTG NateTG/a.scad
Created Mar 19, 2018

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Gone a little bit crazy with triangulation in Openscad
//Scraped from postscript using pathforall
a=[
43.9881,6.56189,"moveto",
42.2891,5.17072,41.0949,4.67827,39.6053,4.67827,"curveto",
37.3154,4.67827,36.626,6.06944,36.626,10.4522,"curveto",
36.626,29.8548,"lineto",
36.626,35.0255,36.1212,37.9186,34.6315,40.307,"curveto",
32.4401,43.8896,28.1681,45.7732,22.2957,45.7732,"curveto",
12.9391,45.7732,5.57699,40.898,5.57699,34.6315,"curveto",
5.57699,32.3416,7.46061,30.3472,9.849,30.3472,"curveto",
12.2374,30.3472,14.3303,32.3416,14.3303,34.533,"curveto",
14.3303,34.927,14.2318,35.4318,14.1333,36.1212,"curveto",
13.9363,37.0199,13.8378,37.8201,13.8378,38.5096,"curveto",
13.8378,41.1934,17.0141,43.3848,21.003,43.3848,"curveto",
25.8782,43.3848,28.5621,40.504,28.5621,35.124,"curveto",
28.5621,29.0545,"lineto",
13.3331,22.9851,11.5479,22.0864,7.26363,18.3068,"curveto",
5.07223,16.3247,3.68106,12.9391,3.68106,9.65202,"curveto",
3.68106,3.38559,8.06387,-0.997211,14.1333,-0.997211,"curveto",
18.5038,-0.997211,22.5911,1.0957,28.6606,6.26642,"curveto",
29.153,0.997211,30.9505,-0.997211,35.0255,-0.997211,"curveto",
38.4111,-0.997211,40.504,0.19698,43.9881,3.97653,"curveto",
43.9881,6.56189,"lineto",
"closepath",
28.5621,12.2374,"moveto",
28.5621,9.25806,28.0573,8.16236,25.9767,6.96816,"curveto",
23.4899,5.57699,20.7937,4.77676,18.7131,4.77676,"curveto",
15.229,4.77676,12.4344,8.16236,12.4344,12.4344,"curveto",
12.4344,12.8406,"lineto",
12.5328,18.8116,16.9156,22.5911,28.5621,26.6662,"curveto",
28.5621,12.2374,"lineto",
"closepath"
];
function splitpaths(script,i=0)=
("closepath"==script[i])?
(i==len(script)-1)?
[[for (j=[0:i-1]) script[j]]]
:
concat(
[[for (j=[0:i-1]) script[j]]],
splitpaths([for (j=[i+1:len(script)-1]) script[j]])
)
:
splitpaths(script,i+1)
;
function pathstep(path,i=0,curx,cury,steps=10)=
(i>=len(path))?
[]
:("moveto"==path[i])? //This should only happen when i==2
concat(
[[path[i-2],path[i-1]]],
pathstep(path,i+1,path[i-2],path[i-1],steps)
)
:("lineto"==path[i])? //This happens later...
concat(
[[path[i-2],path[i-1]]],
pathstep(path,i+1,path[i-2],path[i-1],steps)
)
:("curveto"==path[i])?
concat(
psbezier([curx,cury],[path[i-6],path[i-5]],[path[i-4],path[i-3]],[path[i-2],path[i-1]],steps),
pathstep(path,i+1,path[i-2],path[i-1],steps)
)
:
pathstep(path,i+1,curx,cury,steps);
function psbezier(a,b,c,d,steps=10) =
let(step=1/steps)
[for(i=[1:steps]) nbezier([a,b,c,d],step*i)]
;
function nbezier(points,t)=
1>=len(points)?
points[0]
:
nbezier(
[for(i=[0:len(points)-2]) points[i]*(1-t)+points[i+1]*t],
t
);
function stripnulls(path)=
let(pl=len(path))
[for (i=[0:pl-1]) if((path[i]-path[(pl+i-1)%pl])*(path[i]-path[(pl+i-1)%pl])>0) path[i]]
;
function mkpoly(paths,curpoints=[],curpaths=[],i=0)=
(i>=len(paths))?
[]
:(i==len(paths)-1)?
let(mypath=stripnulls(paths[i]))
[
concat(curpoints,mypath),
concat(curpaths,[[for (i=[0:len(mypath)-1]) i+len(curpoints)]])
]
:
let(mypath=stripnulls(paths[i]))
mkpoly(
paths,
concat(curpoints,mypath),
concat(curpaths,[[for (i=[0:len(mypath)-1]) i+len(curpoints)]]),
i+1
)
;
function makesegments(poly) =
[
for (i=[0:len(poly[1])-1]) [
for(j=[0:len(poly[1][i])-1])
[
poly[1][i][j],
poly[1][i][(j+1)%len(poly[1][i])]
]
]
]
;
apaths=splitpaths(a);
arender=[for (i=[0:len(apaths)-1]) pathstep(apaths[i])];
poly=mkpoly(arender);
translate([0,0,-1]) linear_extrude(height=1) {
polygon(points=poly[0],paths=poly[1]);
}
use<subdivision.scad>
use<FunctionalOpenSCAD/functional.scad>
//Sweep line triangulator using monotone polygons
//Requires:
// First path to be properly wound external face.
// Correctly wound paths (internal paths are CW, external CCW).
// No intesecting paths
// No vertices with more than 2 edges.
//include<a.scad>
use<a.scad>
include<text.scad>
ppaths=splitpaths(text);
prender=[for (i=[0:len(ppaths)-1]) pathstep(ppaths[i],steps=3)];
poly=mkpoly(prender);
/*
translate([0,0,-1]) linear_extrude(height=1) {
polygon(points=poly[0],paths=poly[1]);
}
*/
maxtest=0;
triangles=triangulate(poly[0],poly[1]);
//Testing stuff...
mypoints=mk3d(poly[0]);
//subdivided=[mypoints,triangles];
stest=sphere();
subdivided=subdivide_faces(2,[mypoints,triangles]);
echo(len(subdivided[0]));
//modified=[for (point=subdivided[0]) [point[0],point[1]+point[0]/10,((point[0]-325)*(point[0]-325)+point[1]*point[1])/1000]];
function toc(point)=
let(x=point[0],y=point[1],z=point[2])
[-50*cos(x),-50*sin(x),y-x/3]
;
modified=[for (point=subdivided[0]) toc(point)];
polyhedron(points=modified,faces=subdivided[1]);
color("black") cylinder(r=48,h=2000,$fn=36,center=true);
//showsweep(points=modified,closed=subdivided[1],open=[]);
//This one is often handy...
module line (a=[1,1,1],b=[21,2,2],r=0.125) {
v=b-a;
translate(a)
rotate(a=atan2(v[1],v[0]))
rotate(v=[0,1,0],a=atan2(sqrt(v[0]*v[0]+v[1]*v[1]),v[2]))
cylinder(h=sqrt(v*v),r1=r,r2=r);
}
module closedtest(pointlist,components) {
echo("testing components");
for(i=[0:len(components)-1]) {
for(j=[0:len(components[i])-1]) {
if(pointlist[components[i][j]][0]>pointlist[components[i][0]][0]) {
echo("no start on max x",i,j,pointlist[component[i][0]],pointlist[component[i][j]]);
}
}
}
}
module dot (p=[1,1,1],r=0.25) {
translate(p) sphere(r=r);
}
module showopen(points,component) {
list=component[0];
prev=points[component[1]];
next=points[component[2]];
color("white") translate(prev) sphere(r=0.5);
color("yellow") line(prev,points[list[0]]);
// color("white") translate(next) sphere(r=0.25);
color("white") line(points[list[len(list)-1]],next);
color("black") translate(points[list[0]]) sphere(r=0.25);
color("black") translate(points[list[len(list)-1]]) sphere(r=0.25);
//show box..
// color("white") line(points[list[0]],points[list[len(list)-1]]);
// color("white") line(prev,next);
//latest
color("green") dot(points[list[component[3]]],r=0.5);
if(len(list)>1) {
for(j=[0:len(list)-2]) {
color("black") line(points[list[j]],points[list[j+1]]);
}
}
}
module showsweep (points,closed,open,rz=0.1) {
if(len(closed)>0) {
for (i=[0:len(closed)-1]) {
for(j=[0:len(closed[i])-1]) {
color("blue") line(
points[closed[i][j]],points[closed[i][(j+1)%len(closed[i])]]);
}
}
}
if(len(open)>0) {
for (i=[0:len(open)-1]) {
translate([0,0,rz*(i+1)]) showopen(points,open[i]);
}
}
}
//End of test/debugging section
//Requires:
// First path to be properly wound.
// No intesecting paths
// No vertices with more than 2 edges.
//At this point I have monotone components. ...
function triangulate(points,paths)=
let(
mypoints=mk3d(points),
/*
localz=winding_direction(mypoints,paths[0]),
localx=nontrivialsegment(mypoints,paths[0]),
localy=cross(localz,localx),
*/
localx=[1,0,0],localy=[0,1,0],localz=[0,0,1],
neighborlist=mkneighborlist(mypoints,paths,localx,localy),
sweeplist=mksweeplist(neighborlist),
closed=sweeptriangulate(neighborlist,sweeplist,closed=[],open=[])
)
flatten([for (i=[0:len(closed)-1]) triangulatecomponent(neighborlist,closed[i])])
// flatten([for (i=[0:60]) triangulatecomponent(neighborlist,closed[i])])
;
//Use a "data structure" for components:
//vertices - component body
//prev - "left" dangler
//next - "right" dangler
//latest - index of the newest vertex (typically 0 or -1)
//pendant - undef or index of a 'roof cusp'
function flatten(list)=
[for (a=list) for (b=a) b]
;
//Fixme: Needs a component triangulation function.
//Trapezoid partition & uni-montone components, perhaps.
function newcomponent(pointlist,point) =
let(
index=2,
isexternal=5,
a=pointlist[point[3]],
b=pointlist[point[4]],
swap=!point[isexternal],
prev=swap?b[2]:a[2],
next=swap?a[2]:b[2]
)
[
[point[index]],
prev,
next,
0,
undef
]
;
//returns [ newclosed/undef, component ]
function componentnewprev(pointlist,component,point) =
let(
vertexlist=component[0],
next=component[2],
index=2,
pendant=component[4],
newprev=(vertexlist[0]==point[3])?point[4]:point[3]
)
(undef==pendant)? //No dangling roof pendant.
[
[],
[
concat([point[index]],vertexlist),
newprev,
next,
0,
undef
]
]
:
[
[
concat( [point[index]], sublist(vertexlist,0,pendant) )
],
[
concat( [point[index]], sublist(vertexlist,pendant,-1) ),
newprev,
next,
0,
undef
]
]
;
function componentnewnext(pointlist,component,point) =
let(
vertexlist=component[0],
prev=component[1],
index=2,
pendant=component[4],
newnext=(vertexlist[len(vertexlist)-1]==point[3])?point[4]:point[3]
)
(undef==pendant)? //No dangling roof pendant.
[
[],
[
concat(vertexlist,[point[index]]),
prev,
newnext,
len(vertexlist),
undef
]
]
:
[
[
concat( [point[index]], sublist(vertexlist,pendant,-1) )
],
[
concat( sublist(vertexlist,0,pendant), [point[index]] ),
prev,
newnext,
pendant+1,
undef
]
]
;
function componentfuse(pointlist,componentl,componentr,point)=
let(
pendant=4,
index=2,
lvertexlist=componentl[0],
rvertexlist=componentr[0],
newprev=componentl[1],
newnext=componentr[2],
lpendant=componentl[pendant],
rpendant=componentr[pendant]
)
(lpendant==undef && rpendant==undef)?
[
[],
[
concat(lvertexlist,[point[index]],rvertexlist),
newprev,
newnext,
len(lvertexlist),
len(lvertexlist)
]
]
:(rpendant==undef)?
[
[
//Make X-most point first in list for triangulation.
concat([point[index]],sublist(lvertexlist,lpendant,-1))
],
[
concat(sublist(lvertexlist,0,lpendant),[point[index]],rvertexlist),
newprev,
newnext,
lpendant+1,
lpendant+1
]
]
:(lpendant==undef)?
[
[
concat([point[index]],sublist(rvertexlist,0,rpendant))
],
[
concat(lvertexlist,[point[index]],sublist(rvertexlist,rpendant,-1)),
newprev,
newnext,
len(lvertexlist),
len(lvertexlist)
]
]
://both components have pendants
//fixme: I think this one still needs testing.
[
[
//Make x-most point first for triangulation.
concat([point[index]],sublist(lvertices,lpendant,-1)),
concat([point[index]],sublist(rvertexlist,0,rpendant))
],
[
concat(sublist(lvertices,0,lpendant),[point[index]],sublist(rvertexlist,rpendant,-1)),
newprev,
newnext,
lpendant+1,
lpendant+1
]
]
;
function componentsplit (pointlist,component,point)=
let(
latest=component[3],
vertexlist=component[0],
l=pointlist[vertexlist[latest]],
componentprev=component[1],
componentnext=component[2],
index=2,
isexternal=5,
a=pointlist[point[3]],
b=pointlist[point[4]],
swap=!point[isexternal],
//Reversed b/c this is inside -> outside
pointprev=swap?a[2]:b[2],
pointnext=swap?b[2]:a[2]
)
[
[
concat(sublist(vertexlist,0,latest),[point[index]]),
componentprev,
pointprev,
latest+1,
undef
],
[
concat([point[index]],sublist(vertexlist,latest,-1)),
pointnext,
componentnext,
0,
undef
]
]
;
function componentclose(open,point)=
let(
list=open[0],
pendant=open[4],
mypoint=point[2]
)
(undef==pendant)?
[
concat([mypoint],list)
]
:
//FIXME: This may be untested.
[
concat([mypoint],sublist(list,0,pendant)),
concat([mypoint],sublist(list,pendant,-1))
]
;
function r2ceil(x,n=1)=
(n>x)?
n
:
r2ceil(x,n=n*2)
;
function componentposition(point,neighborlist,open,i=0)=
let(
odd=(i%2==1),
path=odd?(i-1)/2:i/2,
prev=1,
next=2,
vlist=open[path][0],
ai=odd?open[path][next]:open[path][prev],
bi=odd?vlist[len(vlist)-1]:vlist[0],
a=[neighborlist[ai][0],neighborlist[ai][1]],
b=[neighborlist[bi][0],neighborlist[bi][1]],
p=[point[0],point[1]]
)
(i>=len(open)*2)?
i
:cross(b-a,p-b)>0?
i
//On the line case?
:(a[0]==p[0] && b[0]==p[0] && ((a[1]<=p[1] && p[1]<=b[1]) || (a[1]>=p[1] && p[1]>=b[1])))?
i
:
componentposition(point,neighborlist,open,i=i+1)
;
/*
FIXME: This doesn't work? - want binary search...
function componentposition(point,neighborlist,components)=
let(
r2c=r2ceil(2*len(components)+1) //radix 2 ceiling
)
componentpositionstep(point,neighborlist,components,pos=r2c-1,i=r2c/2)
;
function componentpositionstep(point,neighborlist,components,pos=0,i=1)=
let(
cur=floor(pos-i),
odd=(cur%2==1),
path=floor(cur/2),
prev=1,
next=2,
component=components[path],
vertexlist=component[0],
ai=odd?component[next]:component[prev],
bi=odd?vertexlist[len(vertexlist)-1]:vertexlist[0],
a=[neighborlist[ai][0],neighborlist[ai][1]],
b=[neighborlist[bi][0],neighborlist[bi][1]],
p=[point[0],point[1]]
)
(i<1)?
pos
:cur>=(len(components)*2)?
componentpositionstep(point,neighborlist,components,pos=pos-i,i=i/2)
:cross(b-a,p-b)>0?
componentpositionstep(point,neighborlist,components,pos=pos-i,i=i/2)
:
componentpositionstep(point,neighborlist,components,pos=pos,i=i/2)
;
*/
function checkstarts(point, open,incomp)=
let(
tocheck=[incomp,incomp-1,incomp+1],
mypoint=point[2],
prev=1
)
(0==len(open))?
[false]
:(mypoint==open[tocheck[0]][prev])?
[true,tocheck[0]]
:(tocheck[1]>=0 && mypoint==open[tocheck[1]][prev])?
[true,tocheck[1]]
:(tocheck[2]<len(open) && mypoint==open[tocheck[2]][prev])?
[true,tocheck[2]]
:
[false]
;
function checkends(point, open,incomp)=
let(
tocheck=[incomp,incomp-1,incomp+1],
mypoint=point[2],
next=2
)
(0==len(open))?
[false]
:(mypoint==open[tocheck[0]][next])?
[true,tocheck[0]]
:(tocheck[1]>=0 && mypoint==open[tocheck[1]][next])?
[true,tocheck[1]]
:(tocheck[2]<len(open) && mypoint==open[tocheck[2]][next])?
[true,tocheck[2]]
:
[false]
;
function sweeptriangulate (neighborlist,sweeplist,closed=[],open=[],i=0)=
let(
a=neighborlist[sweeplist[i][3]],
b=neighborlist[sweeplist[i][4]],
point=sweeplist[i],
//FIXME: These need to use binary search or something
//similar if this is to be O(N log N)
pos=componentposition(point,neighborlist,open),
incomp=floor(pos/2),
starts=checkstarts(point,open,incomp),
ends=checkends(point,open,incomp),
index=2 //Fake associative array?
)
//Testing stub
(maxtest>0 && i>=maxtest)?
[closed,open,"point",point,"incomp",incomp,"pos",pos,"starts",starts,"ends",ends]
:
(i>len(neighborlist)-1)?
//no more points - all the components should be closed.
closed
:(starts[0] && ends[0])?
(starts[1]==ends[1])? //Current point closes off a component
sweeptriangulate(
neighborlist,
sweeplist,
closed=concat(closed,componentclose(open[starts[1]],point)),
open=concat(
(starts[1]>0)?
sublist(open,0,starts[1]-1)
:
[]
,
(starts[1]<len(open)-1)?
sublist(open,starts[1]+1,len(open)-1)
:
[]
),
i=i+1
)
://Current point hooks together components.
//Should die if starts[1]!=ends[1]+1
let(
new=componentfuse(neighborlist,open[ends[1]],open[starts[1]],point),
newclosed=new[0],
newopen=new[1]
)
sweeptriangulate(
neighborlist,
sweeplist,
closed=concat(closed,newclosed),
open=concat(
(ends[1]>0)?sublist(open,0,ends[1]-1):[],
[newopen],
(starts[1]<len(open)-1)?sublist(open,starts[1]+1,len(open)-1):[]
),
i=i+1
)
:(starts[0])?//Current point extends the left end of a component
let(
new=componentnewprev(neighborlist,open[starts[1]],point),
newclosed=new[0],
newopen=new[1]
)
sweeptriangulate(
neighborlist,
sweeplist,
closed=concat(closed,newclosed),
open=concat(
(starts[1]>0)?sublist(open,0,starts[1]-1):[],
[newopen],
(starts[1]<len(open)-1)?sublist(open,starts[1]+1,-1):[]
),
i=i+1
)
:(ends[0])?//Current point extends the right end of a component
let(
new=componentnewnext(neighborlist,open[ends[1]],point),
newclosed=new[0],
newopen=new[1]
)
sweeptriangulate(
neighborlist,
sweeplist,
closed=concat(closed,newclosed),
open=concat(
(ends[1]>0)?sublist(open,0,ends[1]-1):[],
[newopen],
(ends[1]<len(open)-1)?sublist(open,ends[1]+1,-1):[]
),
i=i+1
)
:(1==pos%2)? // Current point is "in" a component
let(
newopenlist=componentsplit(neighborlist,open[incomp],point)
)
sweeptriangulate(
neighborlist,
sweeplist,
closed=closed,
open=concat(
(incomp>0)?sublist(open,0,incomp-1):[],
newopenlist,
(incomp<len(open)-1)?sublist(open,incomp+1,-1):[]
),
i=i+1
)
://Current point is 'between' components.
sweeptriangulate(
neighborlist,
sweeplist,
closed=closed,
open=concat(
(incomp>0)?sublist(open,0,incomp-1):[],
[
newcomponent(neighborlist,point)
],
//Since it's "between" the incompth open component is retained.
(incomp<=len(open)-1)?sublist(open,incomp,-1):[]
),
i=i+1
)
;
function sublist (list,a,b) =
let(
start=(a<0)?(len(list)+a)%len(list):a,
end=(b<0)?(len(list)+b)%len(list):b
)
(start<=end)?
[for (i=[start:end]) list[i]]
:
[]
;
function reverse(list)=
[for (i=[1:len(list)]) list[len(list)-i]]
;
function mkums (list,splitpoints) =
let(
ums=[for (i=[1:len(splitpoints)-1]) sublist(list,splitpoints[i-1]-1,splitpoints[i])],
isreturn=3
)
[for (i=[0:len(ums)-1]) ums[i][1][isreturn]?reverse(ums[i]):ums[i]]
;
function triangulateums (ums,stack,i=0) =
let(
length=len(ums),
next=ums[i],
last=len(stack)-1
)
i>=length-1?
[for (j=[1:len(stack)-1]) [stack[j-1],stack[j],ums[length-1]]]
:(len(stack)<2 || cross(
[stack[last-1][0],stack[last-1][1]]-[stack[last][0],stack[last][1]],[next[0],next[1]]-[stack[last][0],stack[last][1]])<0)?
triangulateums(ums,concat(stack,[next]),i+1)
:
concat(
[[stack[last-1],stack[last],next]],
triangulateums(ums,sublist(stack,0,-2),i=i)
)
;
function listgt (a,b,list,i=0) =
(i>=len(list) || i<0 || list[i]<0 || list[i]>=len(a) || list[i]>= len(b)) ?
false
:(a[list[i]]==b[list[i]])?
listgt(a,b,list,i+1)
:(a[list[i]]>b[list[i]])?
true
:(a[list[i]]<b[list[i]])?
false
:
false
;
function listeq (a,b,list,i=0) =
(i>=len(list) || i<0 || list[i]<0 || list[i]>=len(a) || list[i]>= len(b)) ?
false
:(a[list[i]]>b[list[i]])?
false
:(a[list[i]]<b[list[i]])?
false
:(i<len(list)-1)?
listeq(a,b,list,i+1)
:
true
;
function listquicksort(arr,list) =
!(len(arr)>0)?
[]
:
let(
pivot = arr[floor(len(arr)/2)],
lesser = [ for (y = arr) if(listgt(pivot,y,list)) y ],
equal = [ for (y = arr) if(listeq(pivot,y,list)) y ],
greater = [ for (y = arr) if(listgt(y,pivot,list)) y ]
)
concat(
listquicksort(lesser,list), equal, listquicksort(greater,list)
);
//Sorting by local x, then local y.
function mksweeplist(neighborlist)=
listquicksort(neighborlist,[0,1])
;
function patharea(points,path,localx,localy) =
let(
da=[for (i=[0:len(path)-1]) cross(points[path[i]]-points[path[0]],points[path[(i+1)%len(path)]]-points[path[i]])]
)
vsum(da)
;
//FIXME: Just making unimonotone components so far.
function triangulatecomponent(points,component)=
let(
l=len(component),
minxindex=minx(points,component),
//negating the x coordinates so that is a sort by decreasing x.
//second coordinate is negative on the return leg.
sortlist=[for (i=[0:l-1]) [-points[component[i]][0],-points[component[i]][1],i>minxindex?-i:i,i>minxindex,component[i]]],
sortedlist=listquicksort(sortlist,[0,2]),
switchpoints=[for (i=[1:l-1]) if(i==1 || i==l-1 || sortedlist[i][3]!=sortedlist[i-1][3]) i],
// splits=mkums(sortedlist,switchpoints),
// cleansplits=[for (split=splits) [for (a=split) a[4]]]
umss=mkums(sortedlist,switchpoints),
triangulated=flatten([for (ums=umss) triangulateums(ums,[],0)]),
cleantriangles=[for (triangle=triangulated) [for (a=triangle) a[4]]]
)
//["component",component,"sortedlist",sortedlist,"splitpoints",switchpoints,"splits",splits,"cleansplits",cleansplits]
// cleansplits
cleantriangles
;
function minx(points,component,i=0)=
(i>=len(component)-1 || points[component[i+1]][0]>points[component[i]][0])?
i
:
minx(points,component,i+1)
;
function mkneighborlist(points,paths,localx,localy)=
let(
localpoints=[for (i=[0:len(points)-1]) [points[i]*localx,points[i]*localy]],
areas=[for (i=[0:len(paths)-1]) patharea(localpoints,paths[i])],
list=[
for(i=[0:len(paths)-1])
for(j=[0:len(paths[i])-1])
[
//Sweep line sort values
localpoints[paths[i][j]][0],
localpoints[paths[i][j]][1],
//Index
paths[i][j],
//neighbors - use oriented area for consistent winding.
areas[i]>0? paths[i][(j+1)%len(paths[i])]: paths[i][(len(paths[i])+j-1)%len(paths[i])],
areas[i]>0? paths[i][(len(paths[i])+j-1)%len(paths[i])]:paths[i][(j+1)%len(paths[i])],
//path number
areas[i]>0
]
]
)
//sort by point index.
listquicksort(list,[2])
;
function nontrivialsegment(points,path,i=0)=
let(
count=len(path),
v=points[path[(i+1)%count]]-points[path[i]]
)
(i<count-1)?
v*v>0?
v
:
nontrivialsegment(points,path,i+1)
:
undef
;
function mk3d(list)=
(2==len(list[0]))?
[for (i=[0:len(list)-1]) concat(list[i],[0])]
:(3==len(list[0]))?
list
:
[]
;
function vsum(v,i=0) = len(v)-1 > i ? v[i] + vsum(v, i+1) : v[i];
function winding_direction(points,path)=
let(count=len(path))
vsum([
for(i=[0:count-1])
cross(
points[path[(i+1)%count]]-points[path[0]],
points[path[(i+2)%count]]-points[path[(i+1)%count]]
)
])
;
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