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// Bezier triangulation routines written by Orest Shardt, 2008.
private real fuzz=sqrtEpsilon;
real duplicateFuzz=1e-3; // Work around font errors.
int countIntersections(path[] p, pair start, pair end)
{
int intersects=0;
for(path q : p)
intersects += intersections(q,start--end,fuzz).length;
return intersects;
}
path[][] containmentTree(path[] paths)
{
path[][] result;
for(int i=0; i < paths.length; ++i) {
bool classified=false;
// check if current curve contains or is contained in a group of curves
for(int j=0; !classified && j < result.length; ++j)
{
int test = inside(paths[i],result[j][0],zerowinding);
if(test == 1) // current curve contains group's toplevel curve
{
// replace toplevel curve with current curve
result[j].insert(0,paths[i]);
classified = true;
}
else if(test == -1) // current curve contained in group's toplevel curve
{
result[j].push(paths[i]);
classified = true;
}
}
// create a new group if this curve does not belong to another group
if(!classified)
result.push(new path[] {paths[i]});
}
// sort group so that later paths in the array are contained in previous paths
bool comparepaths(path i, path j) {return inside(i,j,zerowinding)==1;}
for(int i=0; i < result.length; ++i)
result[i] = sort(result[i],comparepaths);
return result;
}
bool isDuplicate(pair a, pair b, real relSize)
{
return abs(a-b) <= duplicateFuzz*relSize;
}
path removeDuplicates(path p)
{
real relSize = abs(max(p)-min(p));
bool cyclic=cyclic(p);
for(int i=0; i < length(p); ++i) {
if(isDuplicate(point(p,i),point(p,i+1),relSize)) {
p=subpath(p,0,i)&subpath(p,i+1,length(p));
--i;
}
}
return cyclic ? p&cycle : p;
}
path section(path p, real t1, real t2, bool loop=false)
{
if(t2 < t1 || loop && t1 == t2)
t2 += length(p);
return subpath(p,t1,t2);
}
path uncycle(path p, real t)
{
return subpath(p,t,t+length(p));
}
// returns outer paths
void connect(path[] paths, path[] result, path[] patch)
{
path[][] tree=containmentTree(paths);
for(path[] group : tree) {
path outer = group[0];
group.delete(0);
path[][] innerTree = containmentTree(group);
path[] remainingCurves;
path[] inners;
for(path[] innerGroup:innerTree)
{
inners.push(innerGroup[0]);
if(innerGroup.length>1)
remainingCurves.append(innerGroup[1:]);
}
connect(remainingCurves,result,patch);
real d=2*abs(max(outer)-min(outer));
while(inners.length > 0) {
int curveIndex = 0;
real starttime = 0; // starttime is time on inners[curveIndex]
pair direction=I*dir(inners[curveIndex],starttime);
pair start=point(inners[curveIndex],starttime);
// find first intersection of line segment with outer curve
path line = start--start+d*direction;
real[][] ints=intersections(line,outer,fuzz);
real endtime=ints.length > 0 ? ints[0][1] : 0; // endtime is time on outer
pair end = point(outer,endtime);
line = start--end;
path rline = reverse(line);
// find first intersection of rline segment with any inner curve
real earliestTime=1;
for(int j=0; j < inners.length; ++j) {
real[][] ints=intersections(rline,inners[j],fuzz);
if(ints.length > 0 && ints[0][0] < earliestTime) {
earliestTime=ints[0][0]; // time on rline
starttime=ints[0][1]; // time on inner curve
curveIndex=j;
}
}
start=point(inners[curveIndex],starttime);
line = start--end;
real timeoffset=2;
bool found=false;
path portion;
path[] allCurves = {outer};
allCurves.append(inners);
while(!found && timeoffset > fuzz) {
timeoffset /= 2;
if(countIntersections(allCurves,start,
point(outer,endtime+timeoffset)) == 2)
{
portion = subpath(outer,endtime,endtime+timeoffset)--start--cycle;
found=true;
// check if an inner curve is inside the portion
for(int k = 0; found && k < inners.length; ++k)
{
if(k!=curveIndex &&
inside(portion,point(inners[k],0),zerowinding))
found = false;
}
}
}
if(!found)timeoffset=-2;
while(!found && timeoffset < -fuzz) {
timeoffset /= 2;
if(countIntersections(allCurves,start,
point(outer,endtime+timeoffset))==2)
{
portion = subpath(outer,endtime+timeoffset,endtime)--start--cycle;
found = true;
// check if an inner curve is inside the portion
for(int k = 0; found && k < inners.length; ++k)
{
if(k!=curveIndex &&
inside(portion,point(inners[k],0),zerowinding))
found = false;
}
}
}
assert(found);
endtime=min(endtime,endtime+timeoffset);
timeoffset=abs(timeoffset);
// depends on the curves having opposite orientations
path remainder=section(outer,endtime+timeoffset,endtime)
--uncycle(inners[curveIndex],
starttime)--cycle;
inners.delete(curveIndex);
outer = remainder;
patch.append(portion);
}
result.append(outer);
}
}
int countIntersections(path g, pair p, pair q)
{
int ints=0;
int l=length(g);
for(int i=1; i <= l; ++i)
ints += intersections(subpath(g,i-1,i),p--q,fuzz).length;
return ints;
}
bool checkSegment(path g, pair p, pair q)
{
pair mid=0.5*(p+q);
return countIntersections(g,p,q) == 4 && inside(g,mid,zerowinding) &&
intersections(g,mid).length == 0;
}
path subdivide(path p)
{
path q;
int l=length(p);
for(int i=0; i < l; ++i)
q=q&subpath(p,i,i+0.5)&subpath(p,i+0.5,i+1);
return cyclic(p) ? q&cycle : q;
}
path[] bezulate(path[] p)
{
if(p.length == 1 && length(p[0]) <= 4) return p;
path[] patch;
path[] result;
connect(p,result,patch);
for(int i=0; i < result.length; ++i) {
path p=result[i];
int refinements=0;
if(size(p) <= 1) return p;
if(!cyclic(p))
abort("path must be cyclic and nonselfintersecting.");
p=removeDuplicates(p);
if(length(p) > 4) {
static real SIZE_STEPS=10;
static real factor=1.05/SIZE_STEPS;
for(int k=1; k <= SIZE_STEPS; ++k) {
real L=factor*k*abs(max(p)-min(p));
for(int i=0; length(p) > 4 && i < length(p); ++i) {
bool found=false;
pair start=point(p,i);
//look for quadrilaterals and triangles with one line, 4 | 3 curves
for(int desiredSides=4; !found && desiredSides >= 3;
--desiredSides) {
if(desiredSides == 3 && length(p) <= 3)
break;
pair end;
int endi=i+desiredSides-1;
end=point(p,endi);
found=checkSegment(p,start,end) && abs(end-start) < L;
if(found) {
path p1=subpath(p,endi,i+length(p))--cycle;
patch.append(subpath(p,i,endi)--cycle);
p=removeDuplicates(p1);
i=-1; // increment will make i be 0
}
}
if(!found && k == SIZE_STEPS && length(p) > 4 && i == length(p)-1) {
// avoid infinite recursion
++refinements;
if(refinements > mantissaBits) {
write("warning: too many subdivisions");
} else {
p=subdivide(p);
i=-1;
}
}
}
}
}
if(length(p) <= 4)
patch.append(p);
}
return patch;
}
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