7dca7f0ebb8180c01f2c240fd7b80a3fe7643782
1 /*
2 * convex-cover.cpp
3 *
4 * Copyright 2006 Nathan Hurst <njh@mail.csse.monash.edu.au>
5 * Copyright 2006 Michael G. Sloan <mgsloan@gmail.com>
6 *
7 * This library is free software; you can redistribute it and/or
8 * modify it either under the terms of the GNU Lesser General Public
9 * License version 2.1 as published by the Free Software Foundation
10 * (the "LGPL") or, at your option, under the terms of the Mozilla
11 * Public License Version 1.1 (the "MPL"). If you do not alter this
12 * notice, a recipient may use your version of this file under either
13 * the MPL or the LGPL.
14 *
15 * You should have received a copy of the LGPL along with this library
16 * in the file COPYING-LGPL-2.1; if not, write to the Free Software
17 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
18 * You should have received a copy of the MPL along with this library
19 * in the file COPYING-MPL-1.1
20 *
21 * The contents of this file are subject to the Mozilla Public License
22 * Version 1.1 (the "License"); you may not use this file except in
23 * compliance with the License. You may obtain a copy of the License at
24 * http://www.mozilla.org/MPL/
25 *
26 * This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
27 * OF ANY KIND, either express or implied. See the LGPL or the MPL for
28 * the specific language governing rights and limitations.
29 *
30 */
32 #include <2geom/convex-cover.h>
33 #include <algorithm>
34 #include <map>
35 /** Todo:
36 + modify graham scan to work top to bottom, rather than around angles
37 + intersection
38 + minimum distance between convex hulls
39 + maximum distance between convex hulls
40 + hausdorf metric?
41 + check all degenerate cases carefully
42 + check all algorithms meet all invariants
43 + generalise rotating caliper algorithm (iterator/circulator?)
44 */
46 using std::vector;
47 using std::map;
48 using std::pair;
50 namespace Geom{
52 /*** SignedTriangleArea
53 * returns the area of the triangle defined by p0, p1, p2. A clockwise triangle has positive area.
54 */
55 double
56 SignedTriangleArea(Point p0, Point p1, Point p2) {
57 return cross((p1 - p0), (p2 - p0));
58 }
60 class angle_cmp{
61 public:
62 Point o;
63 angle_cmp(Point o) : o(o) {}
65 bool
66 operator()(Point a, Point b) {
67 Point da = a - o;
68 Point db = b - o;
70 #if 1
71 double aa = da[0];
72 double ab = db[0];
73 if((da[1] == 0) && (db[1] == 0))
74 return da[0] < db[0];
75 if(da[1] == 0)
76 return true; // infinite tangent
77 if(db[1] == 0)
78 return false; // infinite tangent
79 aa = da[0] / da[1];
80 ab = db[0] / db[1];
81 if(aa > ab)
82 return true;
83 #else
84 //assert((ata > atb) == (aa < ab));
85 double aa = atan2(da);
86 double ab = atan2(db);
87 if(aa < ab)
88 return true;
89 #endif
90 if(aa == ab)
91 return L2sq(da) < L2sq(db);
92 return false;
93 }
94 };
96 void
97 ConvexHull::find_pivot() {
98 // Find pivot P;
99 unsigned pivot = 0;
100 for(unsigned i = 1; i < boundary.size(); i++)
101 if(boundary[i] <= boundary[pivot])
102 pivot = i;
104 std::swap(boundary[0], boundary[pivot]);
105 }
107 void
108 ConvexHull::angle_sort() {
109 // sort points by angle (resolve ties in favor of point farther from P);
110 // we leave the first one in place as our pivot
111 std::sort(boundary.begin()+1, boundary.end(), angle_cmp(boundary[0]));
112 }
114 void
115 ConvexHull::graham_scan() {
116 unsigned stac = 2;
117 for(unsigned int i = 2; i < boundary.size(); i++) {
118 double o = SignedTriangleArea(boundary[stac-2],
119 boundary[stac-1],
120 boundary[i]);
121 if(o == 0) { // colinear - dangerous...
122 stac--;
123 } else if(o < 0) { // anticlockwise
124 } else { // remove concavity
125 while(o >= 0 && stac > 2) {
126 stac--;
127 o = SignedTriangleArea(boundary[stac-2],
128 boundary[stac-1],
129 boundary[i]);
130 }
131 }
132 boundary[stac++] = boundary[i];
133 }
134 boundary.resize(stac);
135 }
137 void
138 ConvexHull::graham() {
139 find_pivot();
140 angle_sort();
141 graham_scan();
142 }
144 //Mathematically incorrect mod, but more useful.
145 int mod(int i, int l) {
146 return i >= 0 ?
147 i % l : (i % l) + l;
148 }
149 //OPT: usages can often be replaced by conditions
151 /*** ConvexHull::left
152 * Tests if a point is left (outside) of a particular segment, n. */
153 bool
154 ConvexHull::is_left(Point p, int n) {
155 return SignedTriangleArea((*this)[n], (*this)[n+1], p) > 0;
156 }
158 /*** ConvexHull::find_positive
159 * May return any number n where the segment n -> n + 1 (possibly looped around) in the hull such
160 * that the point is on the wrong side to be within the hull. Returns -1 if it is within the hull.*/
161 int
162 ConvexHull::find_left(Point p) {
163 int l = boundary.size(); //Who knows if C++ is smart enough to optimize this?
164 for(int i = 0; i < l; i++) {
165 if(is_left(p, i)) return i;
166 }
167 return -1;
168 }
169 //OPT: do a spread iteration - quasi-random with no repeats and full coverage.
171 /*** ConvexHull::contains_point
172 * In order to test whether a point is inside a convex hull we can travel once around the outside making
173 * sure that each triangle made from an edge and the point has positive area. */
174 bool
175 ConvexHull::contains_point(Point p) {
176 return find_left(p) == -1;
177 }
179 /*** ConvexHull::add_point
180 * to add a point we need to find whether the new point extends the boundary, and if so, what it
181 * obscures. Tarjan? Jarvis?*/
182 void
183 ConvexHull::merge(Point p) {
184 std::vector<Point> out;
186 int l = boundary.size();
188 if(l < 2) {
189 boundary.push_back(p);
190 return;
191 }
193 bool pushed = false;
195 bool pre = is_left(p, -1);
196 for(int i = 0; i < l; i++) {
197 bool cur = is_left(p, i);
198 if(pre) {
199 if(cur) {
200 if(!pushed) {
201 out.push_back(p);
202 pushed = true;
203 }
204 continue;
205 }
206 else if(!pushed) {
207 out.push_back(p);
208 pushed = true;
209 }
210 }
211 out.push_back(boundary[i]);
212 pre = cur;
213 }
215 boundary = out;
216 }
217 //OPT: quickly find an obscured point and find the bounds by extending from there. then push all points not within the bounds in order.
218 //OPT: use binary searches to find the actual starts/ends, use known rights as boundaries. may require cooperation of find_left algo.
220 /*** ConvexHull::is_clockwise
221 * We require that successive pairs of edges always turn right.
222 * proposed algorithm: walk successive edges and require triangle area is positive.
223 */
224 bool
225 ConvexHull::is_clockwise() const {
226 if(is_degenerate())
227 return true;
228 Point first = boundary[0];
229 Point second = boundary[1];
230 for(std::vector<Point>::const_iterator it(boundary.begin()+2), e(boundary.end());
231 it != e;) {
232 if(SignedTriangleArea(first, second, *it) > 0)
233 return false;
234 first = second;
235 second = *it;
236 ++it;
237 }
238 return true;
239 }
241 /*** ConvexHull::top_point_first
242 * We require that the first point in the convex hull has the least y coord, and that off all such points on the hull, it has the least x coord.
243 * proposed algorithm: track lexicographic minimum while walking the list.
244 */
245 bool
246 ConvexHull::top_point_first() const {
247 std::vector<Point>::const_iterator pivot = boundary.begin();
248 for(std::vector<Point>::const_iterator it(boundary.begin()+1),
249 e(boundary.end());
250 it != e; it++) {
251 if((*it)[1] < (*pivot)[1])
252 pivot = it;
253 else if(((*it)[1] == (*pivot)[1]) &&
254 ((*it)[0] < (*pivot)[0]))
255 pivot = it;
256 }
257 return pivot == boundary.begin();
258 }
259 //OPT: since the Y values are orderly there should be something like a binary search to do this.
261 /*** ConvexHull::no_colinear_points
262 * We require that no three vertices are colinear.
263 proposed algorithm: We must be very careful about rounding here.
264 */
265 bool
266 ConvexHull::no_colinear_points() const {
267 // XXX: implement me!
268 }
270 bool
271 ConvexHull::meets_invariants() const {
272 return is_clockwise() && top_point_first() && no_colinear_points();
273 }
275 /*** ConvexHull::is_degenerate
276 * We allow three degenerate cases: empty, 1 point and 2 points. In many cases these should be handled explicitly.
277 */
278 bool
279 ConvexHull::is_degenerate() const {
280 return boundary.size() < 3;
281 }
284 /* Here we really need a rotating calipers implementation. This implementation is slow and incorrect.
285 This incorrectness is a problem because it throws off the algorithms. Perhaps I will come up with
286 something better tomorrow. The incorrectness is in the order of the bridges - they must be in the
287 order of traversal around. Since the a->b and b->a bridges are seperated, they don't need to be merge
288 order, just the order of the traversal of the host hull. Currently some situations make a n->0 bridge
289 first.*/
290 pair< map<int, int>, map<int, int> >
291 bridges(ConvexHull a, ConvexHull b) {
292 map<int, int> abridges;
293 map<int, int> bbridges;
295 for(unsigned ia = 0; ia < a.boundary.size(); ia++) {
296 for(unsigned ib = 0; ib < b.boundary.size(); ib++) {
297 Point d = b[ib] - a[ia];
298 Geom::Coord e = cross(d, a[ia - 1] - a[ia]), f = cross(d, a[ia + 1] - a[ia]);
299 Geom::Coord g = cross(d, b[ib - 1] - a[ia]), h = cross(d, b[ib + 1] - a[ia]);
300 if (e > 0 && f > 0 && g > 0 && h > 0) abridges[ia] = ib;
301 else if(e < 0 && f < 0 && g < 0 && h < 0) bbridges[ib] = ia;
302 }
303 }
305 return make_pair(abridges, bbridges);
306 }
308 std::vector<Point> bridge_points(ConvexHull a, ConvexHull b) {
309 vector<Point> ret;
310 pair< map<int, int>, map<int, int> > indices = bridges(a, b);
311 for(map<int, int>::iterator it = indices.first.begin(); it != indices.first.end(); it++) {
312 ret.push_back(a[it->first]);
313 ret.push_back(b[it->second]);
314 }
315 for(map<int, int>::iterator it = indices.second.begin(); it != indices.second.end(); it++) {
316 ret.push_back(b[it->first]);
317 ret.push_back(a[it->second]);
318 }
319 return ret;
320 }
322 unsigned find_bottom_right(ConvexHull const &a) {
323 unsigned it = 1;
324 while(it < a.boundary.size() &&
325 a.boundary[it][Y] > a.boundary[it-1][Y])
326 it++;
327 return it-1;
328 }
330 /*** ConvexHull sweepline_intersection(ConvexHull a, ConvexHull b);
331 * find the intersection between two convex hulls. The intersection is also a convex hull.
332 * (Proof: take any two points both in a and in b. Any point between them is in a by convexity,
333 * and in b by convexity, thus in both. Need to prove still finite bounds.)
334 * This algorithm works by sweeping a line down both convex hulls in parallel, working out the left and right edges of the new hull.
335 */
336 ConvexHull sweepline_intersection(ConvexHull const &a, ConvexHull const &b) {
337 ConvexHull ret;
339 unsigned al = 0;
340 unsigned bl = 0;
342 while(al+1 < a.boundary.size() &&
343 (a.boundary[al+1][Y] > b.boundary[bl][Y])) {
344 al++;
345 }
346 while(bl+1 < b.boundary.size() &&
347 (b.boundary[bl+1][Y] > a.boundary[al][Y])) {
348 bl++;
349 }
350 // al and bl now point to the top of the first pair of edges that overlap in y value
351 //double sweep_y = std::min(a.boundary[al][Y],
352 // b.boundary[bl][Y]);
353 return ret;
354 }
356 /*** ConvexHull intersection(ConvexHull a, ConvexHull b);
357 * find the intersection between two convex hulls. The intersection is also a convex hull.
358 * (Proof: take any two points both in a and in b. Any point between them is in a by convexity,
359 * and in b by convexity, thus in both. Need to prove still finite bounds.)
360 */
361 ConvexHull intersection(ConvexHull /*a*/, ConvexHull /*b*/) {
362 ConvexHull ret;
363 /*
364 int ai = 0, bi = 0;
365 int aj = a.boundary.size() - 1;
366 int bj = b.boundary.size() - 1;
367 */
368 /*while (true) {
369 if(a[ai]
370 }*/
371 return ret;
372 }
374 /*** ConvexHull merge(ConvexHull a, ConvexHull b);
375 * find the smallest convex hull that surrounds a and b.
376 */
377 ConvexHull merge(ConvexHull a, ConvexHull b) {
378 ConvexHull ret;
380 pair< map<int, int>, map<int, int> > bpair = bridges(a, b);
381 map<int, int> ab = bpair.first;
382 map<int, int> bb = bpair.second;
384 ab[-1] = 0;
385 bb[-1] = 0;
387 int i = -1; // XXX: i is int but refers to vector indices
389 if(a.boundary[0][1] > b.boundary[0][1]) goto start_b;
390 while(true) {
391 for(; ab.count(i) == 0; i++) {
392 ret.boundary.push_back(a[i]);
393 if(i >= (int)a.boundary.size()) return ret;
394 }
395 if(ab[i] == 0 && i != -1) break;
396 i = ab[i];
397 start_b:
399 for(; bb.count(i) == 0; i++) {
400 ret.boundary.push_back(b[i]);
401 if(i >= (int)b.boundary.size()) return ret;
402 }
403 if(bb[i] == 0 && i != -1) break;
404 i = bb[i];
405 }
406 return ret;
407 }
409 ConvexHull graham_merge(ConvexHull a, ConvexHull b) {
410 ConvexHull result;
412 // we can avoid the find pivot step because of top_point_first
413 if(b.boundary[0] <= a.boundary[0])
414 std::swap(a, b);
416 result.boundary = a.boundary;
417 result.boundary.insert(result.boundary.end(),
418 b.boundary.begin(), b.boundary.end());
420 /** if we modified graham scan to work top to bottom as proposed in lect754.pdf we could replace the
421 angle sort with a simple merge sort type algorithm. furthermore, we could do the graham scan
422 online, avoiding a bunch of memory copies. That would probably be linear. -- njh*/
423 result.angle_sort();
424 result.graham_scan();
426 return result;
427 }
428 //TODO: reinstate
429 /*ConvexCover::ConvexCover(Path const &sp) : path(&sp) {
430 cc.reserve(sp.size());
431 for(Geom::Path::const_iterator it(sp.begin()), end(sp.end()); it != end; ++it) {
432 cc.push_back(ConvexHull((*it).begin(), (*it).end()));
433 }
434 }*/
436 double ConvexHull::centroid_and_area(Geom::Point& centroid) const {
437 const unsigned n = boundary.size();
438 if (n < 2)
439 return 0;
440 if(n < 3) {
441 centroid = (boundary[0] + boundary[1])/2;
442 return 0;
443 }
444 Geom::Point centroid_tmp(0,0);
445 double atmp = 0;
446 for (unsigned i = n-1, j = 0; j < n; i = j, j++) {
447 const double ai = -cross(boundary[j], boundary[i]);
448 atmp += ai;
449 centroid_tmp += (boundary[j] + boundary[i])*ai; // first moment.
450 }
451 if (atmp != 0) {
452 centroid = centroid_tmp / (3 * atmp);
453 }
454 return atmp / 2;
455 }
457 // TODO: This can be made lg(n) using golden section/fibonacci search three starting points, say 0,
458 // n/2, n-1 construct a new point, say (n/2 + n)/2 throw away the furthest boundary point iterate
459 // until interval is a single value
460 Point const * ConvexHull::furthest(Point direction) const {
461 Point const * p = &boundary[0];
462 double d = dot(*p, direction);
463 for(unsigned i = 1; i < boundary.size(); i++) {
464 double dd = dot(boundary[i], direction);
465 if(d < dd) {
466 p = &boundary[i];
467 d = dd;
468 }
469 }
470 return p;
471 }
474 // returns (a, (b,c)), three points which define the narrowest diameter of the hull as the pair of
475 // lines going through b,c, and through a, parallel to b,c TODO: This can be made linear time by
476 // moving point tc incrementally from the previous value (it can only move in one direction). It
477 // is currently n*O(furthest)
478 double ConvexHull::narrowest_diameter(Point &a, Point &b, Point &c) {
479 Point tb = boundary.back();
480 double d = INFINITY;
481 for(unsigned i = 0; i < boundary.size(); i++) {
482 Point tc = boundary[i];
483 Point n = -rot90(tb-tc);
484 Point ta = *furthest(n);
485 double td = dot(n, ta-tb)/dot(n,n);
486 if(td < d) {
487 a = ta;
488 b = tb;
489 c = tc;
490 d = td;
491 }
492 tb = tc;
493 }
494 return d;
495 }
497 };
499 /*
500 Local Variables:
501 mode:c++
502 c-file-style:"stroustrup"
503 c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
504 indent-tabs-mode:nil
505 fill-column:99
506 End:
507 */
508 // vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:encoding=utf-8:textwidth=99 :