1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
|
// Bezier triangulation routines written by Orest Shardt, 2008.
private real fuzz=sqrtEpsilon;
real duplicateFuzz=1e-3; // Work around font errors.
private real[][] intersections(pair a, pair b, path p)
{
pair delta=fuzz*unit(b-a);
return intersections(a-delta--b+delta,p,fuzz);
}
int countIntersections(path[] p, pair start, pair end)
{
int intersects=0;
for(path q : p)
intersects += intersections(start,end,q).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;
pair direction=I*dir(inners[curveIndex],0,1); // Use outgoing direction
if(direction == 0) // Try a random direction
direction=expi(2pi*unitrand());
pair start=point(inners[curveIndex],0);
// find first intersection of line segment with outer curve
real[][] ints=intersections(start,start+d*direction,outer);
assert(ints.length != 0);
real endtime=ints[0][1]; // endtime is time on outer
pair end = point(outer,endtime);
// find first intersection of end--start with any inner curve
real starttime=0; // starttime is time on inners[curveIndex]
real earliestTime=1;
for(int j=0; j < inners.length; ++j) {
real[][] ints=intersections(end,start,inners[j]);
if(ints.length > 0 && ints[0][0] < earliestTime) {
earliestTime=ints[0][0]; // time on end--start
starttime=ints[0][1]; // time on inner curve
curveIndex=j;
}
}
start=point(inners[curveIndex],starttime);
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);
}
}
bool checkSegment(path g, pair p, pair q)
{
pair mid=0.5*(p+q);
return intersections(p,q,g).length == 2 &&
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) {
warning("subdivisions","too many subdivisions",position=true);
} else {
p=subdivide(p);
i=-1;
}
}
}
}
}
if(length(p) <= 4)
patch.append(p);
}
return patch;
}
|
