summaryrefslogtreecommitdiff
path: root/src/mongo/db/geo/shapes.cpp
blob: fa6018877bbe84aa998d03220db631f296e0de4b (plain)
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
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
/**
*    Copyright (C) 2008-2012 10gen Inc.
*
*    This program is free software: you can redistribute it and/or  modify
*    it under the terms of the GNU Affero General Public License, version 3,
*    as published by the Free Software Foundation.
*
*    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 Affero General Public License for more details.
*
*    You should have received a copy of the GNU Affero General Public License
*    along with this program.  If not, see <http://www.gnu.org/licenses/>.
*
*    As a special exception, the copyright holders give permission to link the
*    code of portions of this program with the OpenSSL library under certain
*    conditions as described in each individual source file and distribute
*    linked combinations including the program with the OpenSSL library. You
*    must comply with the GNU Affero General Public License in all respects for
*    all of the code used other than as permitted herein. If you modify file(s)
*    with this exception, you may extend this exception to your version of the
*    file(s), but you are not obligated to do so. If you do not wish to do so,
*    delete this exception statement from your version. If you delete this
*    exception statement from all source files in the program, then also delete
*    it in the license file.
*/

#include "mongo/db/jsobj.h"
#include "mongo/db/geo/shapes.h"
#include "mongo/util/mongoutils/str.h"

using std::abs;

// So we can get at the str namespace.
using namespace mongoutils;

namespace mongo {

////////////// Point

Point::Point() : x(0), y(0) {}

Point::Point(double x, double y) : x(x), y(y) {}

Point::Point(const BSONElement& e) {
    BSONObjIterator i(e.Obj());
    x = i.next().number();
    y = i.next().number();
}

Point::Point(const BSONObj& o) {
    BSONObjIterator i(o);
    x = i.next().number();
    y = i.next().number();
}

string Point::toString() const {
    StringBuilder buf;
    buf << "(" << x << "," << y << ")";
    return buf.str();
}

////////////// Circle

Circle::Circle() {}
Circle::Circle(double radius, Point center) : radius(radius), center(center) {}


////////////// Box

Box::Box() {}

Box::Box(double x, double y, double size) : _min(x, y), _max(x + size, y + size) {}

Box::Box(const Point& ptA, const Point& ptB) {
    init(ptA, ptB);
}

void Box::init(const Point& ptA, const Point& ptB) {
    _min.x = min(ptA.x, ptB.x);
    _min.y = min(ptA.y, ptB.y);
    _max.x = max(ptA.x, ptB.x);
    _max.y = max(ptA.y, ptB.y);
}

void Box::init(const Box& other) {
    init(other._min, other._max);
}

BSONArray Box::toBSON() const {
    return BSON_ARRAY(BSON_ARRAY(_min.x << _min.y) << BSON_ARRAY(_max.x << _max.y));
}

string Box::toString() const {
    StringBuilder buf;
    buf << _min.toString() << " -->> " << _max.toString();
    return buf.str();
}

bool Box::between(double min, double max, double val, double fudge) const {
    return val + fudge >= min && val <= max + fudge;
}

bool Box::onBoundary(double bound, double val, double fudge) const {
    return (val >= bound - fudge && val <= bound + fudge);
}

bool Box::mid(double amin, double amax, double bmin, double bmax, bool min, double* res) const {
    verify(amin <= amax);
    verify(bmin <= bmax);

    if (amin < bmin) {
        if (amax < bmin)
            return false;
        *res = min ? bmin : amax;
        return true;
    }
    if (amin > bmax)
        return false;
    *res = min ? amin : bmax;
    return true;
}

bool Box::intersects(const Box& other) const {
    bool intersectX = between(_min.x, _max.x, other._min.x)  // contain part of other range
        || between(_min.x, _max.x, other._max.x)             // contain part of other range
        || between(other._min.x, other._max.x, _min.x);      // other range contains us

    bool intersectY = between(_min.y, _max.y, other._min.y) ||
        between(_min.y, _max.y, other._max.y) || between(other._min.y, other._max.y, _min.y);

    return intersectX && intersectY;
}

double Box::legacyIntersectFraction(const Box& other) const {
    Point boundMin(0, 0);
    Point boundMax(0, 0);

    if (!mid(_min.x, _max.x, other._min.x, other._max.x, true, &boundMin.x) ||
        !mid(_min.x, _max.x, other._min.x, other._max.x, false, &boundMax.x) ||
        !mid(_min.y, _max.y, other._min.y, other._max.y, true, &boundMin.y) ||
        !mid(_min.y, _max.y, other._min.y, other._max.y, false, &boundMax.y))
        return 0;

    Box intersection(boundMin, boundMax);
    return intersection.area() / area();
}

double Box::area() const {
    return (_max.x - _min.x) * (_max.y - _min.y);
}

double Box::maxDim() const {
    return max(_max.x - _min.x, _max.y - _min.y);
}

Point Box::center() const {
    return Point((_min.x + _max.x) / 2, (_min.y + _max.y) / 2);
}

void Box::truncate(double min, double max) {
    if (_min.x < min)
        _min.x = min;
    if (_min.y < min)
        _min.y = min;
    if (_max.x > max)
        _max.x = max;
    if (_max.y > max)
        _max.y = max;
}

void Box::fudge(double error) {
    _min.x -= error;
    _min.y -= error;
    _max.x += error;
    _max.y += error;
}

void Box::expandToInclude(const Point& pt) {
    _min.x = min(_min.x, pt.x);
    _min.y = min(_min.y, pt.y);
    _max.x = max(_max.x, pt.x);
    _max.y = max(_max.y, pt.y);
}

bool Box::onBoundary(Point p, double fudge) const {
    return onBoundary(_min.x, p.x, fudge) || onBoundary(_max.x, p.x, fudge) ||
        onBoundary(_min.y, p.y, fudge) || onBoundary(_max.y, p.y, fudge);
}

bool Box::inside(Point p, double fudge) const {
    bool res = inside(p.x, p.y, fudge);
    return res;
}

bool Box::inside(double x, double y, double fudge) const {
    return between(_min.x, _max.x, x, fudge) && between(_min.y, _max.y, y, fudge);
}

bool Box::contains(const Box& other, double fudge) const {
    return inside(other._min, fudge) && inside(other._max, fudge);
}

////////////// Polygon

Polygon::Polygon() {}

Polygon::Polygon(const vector<Point>& points) {
    init(points);
}

void Polygon::init(const vector<Point>& points) {
    _points.clear();
    _bounds.reset();
    _centroid.reset();

    _points.insert(_points.begin(), points.begin(), points.end());
}

void Polygon::init(const Polygon& other) {
    init(other._points);
}

int Polygon::size(void) const {
    return _points.size();
}

bool Polygon::contains(const Point& p) const {
    return contains(p, 0) > 0;
}

/*
 * Return values:
 * -1 if no intersection
 * 0 if maybe an intersection (using fudge)
 * 1 if there is an intersection
 *
 * A ray casting intersection method is used.
 */
int Polygon::contains(const Point& p, double fudge) const {
    Box fudgeBox(Point(p.x - fudge, p.y - fudge), Point(p.x + fudge, p.y + fudge));

    int counter = 0;
    Point p1 = _points[0];
    for (int i = 1; i <= size(); i++) {
        // XXX: why is there a mod here?
        Point p2 = _points[i % size()];

        // We need to check whether or not this segment intersects our error box
        if (fudge > 0 &&
            // Points not too far below box
            fudgeBox._min.y <= std::max(p1.y, p2.y) &&
            // Points not too far above box
            fudgeBox._max.y >= std::min(p1.y, p2.y) &&
            // Points not too far to left of box
            fudgeBox._min.x <= std::max(p1.x, p2.x) &&
            // Points not too far to right of box
            fudgeBox._max.x >= std::min(p1.x, p2.x)) {
            // If our box contains one or more of these points, we need to do an exact
            // check.
            if (fudgeBox.inside(p1)) {
                return 0;
            }
            if (fudgeBox.inside(p2)) {
                return 0;
            }

            // Do intersection check for vertical sides
            if (p1.y != p2.y) {
                double invSlope = (p2.x - p1.x) / (p2.y - p1.y);

                double xintersT = (fudgeBox._max.y - p1.y) * invSlope + p1.x;
                if (fudgeBox._min.x <= xintersT && fudgeBox._max.x >= xintersT) {
                    return 0;
                }

                double xintersB = (fudgeBox._min.y - p1.y) * invSlope + p1.x;
                if (fudgeBox._min.x <= xintersB && fudgeBox._max.x >= xintersB) {
                    return 0;
                }
            }

            // Do intersection check for horizontal sides
            if (p1.x != p2.x) {
                double slope = (p2.y - p1.y) / (p2.x - p1.x);

                double yintersR = (p1.x - fudgeBox._max.x) * slope + p1.y;
                if (fudgeBox._min.y <= yintersR && fudgeBox._max.y >= yintersR) {
                    return 0;
                }

                double yintersL = (p1.x - fudgeBox._min.x) * slope + p1.y;
                if (fudgeBox._min.y <= yintersL && fudgeBox._max.y >= yintersL) {
                    return 0;
                }
            }
        } else if (fudge == 0) {
            // If this is an exact vertex, we won't intersect, so check this
            if (p.y == p1.y && p.x == p1.x)
                return 1;
            else if (p.y == p2.y && p.x == p2.x)
                return 1;

            // If this is a horizontal line we won't intersect, so check this
            if (p1.y == p2.y && p.y == p1.y) {
                // Check that the x-coord lies in the line
                if (p.x >= std::min(p1.x, p2.x) && p.x <= std::max(p1.x, p2.x))
                    return 1;
            }
        }

        // Normal intersection test.
        // TODO: Invert these for clearer logic?
        if (p.y > std::min(p1.y, p2.y)) {
            if (p.y <= std::max(p1.y, p2.y)) {
                if (p.x <= std::max(p1.x, p2.x)) {
                    if (p1.y != p2.y) {
                        double xinters = (p.y - p1.y) * (p2.x - p1.x) / (p2.y - p1.y) + p1.x;
                        // Special case of point on vertical line
                        if (p1.x == p2.x && p.x == p1.x) {
                            // Need special case for the vertical edges, for example:
                            // 1) \e   pe/----->
                            // vs.
                            // 2) \ep---e/----->
                            //
                            // if we count exact as intersection, then 1 is in but 2 is out
                            // if we count exact as no-int then 1 is out but 2 is in.

                            return 1;
                        } else if (p1.x == p2.x || p.x <= xinters) {
                            counter++;
                        }
                    }
                }
            }
        }

        p1 = p2;
    }

    if (counter % 2 == 0) {
        return -1;
    } else {
        return 1;
    }
}

const Point& Polygon::centroid() const {
    if (_centroid) {
        return *_centroid;
    }

    _centroid.reset(new Point());

    double signedArea = 0.0;
    double area = 0.0;  // Partial signed area

    /// For all vertices except last
    int i = 0;
    for (i = 0; i < size() - 1; ++i) {
        area = _points[i].x * _points[i + 1].y - _points[i + 1].x * _points[i].y;
        signedArea += area;
        _centroid->x += (_points[i].x + _points[i + 1].x) * area;
        _centroid->y += (_points[i].y + _points[i + 1].y) * area;
    }

    // Do last vertex
    area = _points[i].x * _points[0].y - _points[0].x * _points[i].y;
    _centroid->x += (_points[i].x + _points[0].x) * area;
    _centroid->y += (_points[i].y + _points[0].y) * area;
    signedArea += area;
    signedArea *= 0.5;
    _centroid->x /= (6 * signedArea);
    _centroid->y /= (6 * signedArea);

    return *_centroid;
}

const Box& Polygon::bounds() const {
    if (_bounds) {
        return *_bounds;
    }

    _bounds.reset(new Box(_points[0], _points[0]));

    for (int i = 1; i < size(); i++) {
        _bounds->expandToInclude(_points[i]);
    }

    return *_bounds;
}

R2Annulus::R2Annulus() : _inner(0.0), _outer(0.0) {}

R2Annulus::R2Annulus(const Point& center, double inner, double outer)
    : _center(center), _inner(inner), _outer(outer) {}

const Point& R2Annulus::center() const {
    return _center;
}

double R2Annulus::getInner() const {
    return _inner;
}

double R2Annulus::getOuter() const {
    return _outer;
}

bool R2Annulus::contains(const Point& point) const {
    // See if we're inside the inner radius
    if (distanceCompare(point, _center, _inner) < 0) {
        return false;
    }

    // See if we're outside the outer radius
    if (distanceCompare(point, _center, _outer) > 0) {
        return false;
    }

    return true;
}

Box R2Annulus::getR2Bounds() const {
    return Box(
        _center.x - _outer, _center.y - _outer, 2 * _outer);  // Box(_min.x, _min.y, edgeLength)
}

bool R2Annulus::fastContains(const Box& other) const {
    return circleContainsBox(Circle(_outer, _center), other) &&
        !circleInteriorIntersectsWithBox(Circle(_inner, _center), other);
}

bool R2Annulus::fastDisjoint(const Box& other) const {
    return !circleIntersectsWithBox(Circle(_outer, _center), other) ||
        circleInteriorContainsBox(Circle(_inner, _center), other);
}

string R2Annulus::toString() const {
    return str::stream() << "center: " << _center.toString() << " inner: " << _inner
                         << " outer: " << _outer;
}

/////// Other methods

double S2Distance::distanceRad(const S2Point& pointA, const S2Point& pointB) {
    S1Angle angle(pointA, pointB);
    return angle.radians();
}

double S2Distance::minDistanceRad(const S2Point& point, const S2Polyline& line) {
    int tmp;
    S1Angle angle(point, line.Project(point, &tmp));
    return angle.radians();
}

double S2Distance::minDistanceRad(const S2Point& point, const S2Polygon& polygon) {
    S1Angle angle(point, polygon.Project(point));
    return angle.radians();
}

double S2Distance::minDistanceRad(const S2Point& point, const S2Cap& cap) {
    S1Angle angleToCenter(point, cap.axis());
    return (angleToCenter - cap.angle()).radians();
}

/**
 * Distance method that compares x or y coords when other direction is zero,
 * avoids numerical error when distances are very close to radius but axis-aligned.
 *
 * An example of the problem is:
 * (52.0 - 51.9999) - 0.0001 = 3.31965e-15 and 52.0 - 51.9999 > 0.0001 in double arithmetic
 * but:
 * 51.9999 + 0.0001 <= 52.0
 *
 * This avoids some (but not all!) suprising results in $center queries where points are
 * (radius + center.x, center.y) or vice-versa.
 */
bool distanceWithin(const Point& p1, const Point& p2, double radius) {
    return distanceCompare(p1, p2, radius) <= 0.0;
}

// Compare the distance between p1 and p2 with the radius.
// Float-number comparison might be inaccurate.
//
// > 0: distance is greater than radius
// = 0: distance equals radius
// < 0: distance is less than radius
double distanceCompare(const Point& p1, const Point& p2, double radius) {
    double a = p2.x - p1.x;
    double b = p2.y - p1.y;

    if (a == 0) {
        //
        // Note:  For some, unknown reason, when a 32-bit g++ optimizes this call, the sum is
        // calculated imprecisely.  We need to force the compiler to always evaluate it
        // correctly, hence the weirdness.
        //
        // On some 32-bit linux machines, removing the volatile keyword or calculating the sum
        // inline will make certain geo tests fail.  Of course this check will force volatile
        // for all 32-bit systems, not just affected systems.
        if (sizeof(void*) <= 4) {
            volatile double sum = p2.y > p1.y ? p1.y + radius : p2.y + radius;
            return p2.y > p1.y ? p2.y - sum : p1.y - sum;
        } else {
            // Original math, correct for most systems
            return p2.y > p1.y ? p2.y - (p1.y + radius) : p1.y - (p2.y + radius);
        }
    }

    if (b == 0) {
        if (sizeof(void*) <= 4) {
            volatile double sum = p2.x > p1.x ? p1.x + radius : p2.x + radius;
            return p2.x > p1.x ? p2.x - sum : p1.x - sum;
        } else {
            return p2.x > p1.x ? p2.x - (p1.x + radius) : p1.x - (p2.x + radius);
        }
    }

    return sqrt((a * a) + (b * b)) - radius;
}

// note: multiply by earth radius for distance
double spheredist_rad(const Point& p1, const Point& p2) {
    // this uses the n-vector formula: http://en.wikipedia.org/wiki/N-vector
    // If you try to match the code to the formula, note that I inline the cross-product.

    double sinx1(sin(p1.x)), cosx1(cos(p1.x));
    double siny1(sin(p1.y)), cosy1(cos(p1.y));
    double sinx2(sin(p2.x)), cosx2(cos(p2.x));
    double siny2(sin(p2.y)), cosy2(cos(p2.y));

    double cross_prod =
        (cosy1 * cosx1 * cosy2 * cosx2) + (cosy1 * sinx1 * cosy2 * sinx2) + (siny1 * siny2);

    if (cross_prod >= 1 || cross_prod <= -1) {
        // fun with floats
        verify(fabs(cross_prod) - 1 < 1e-6);
        return cross_prod > 0 ? 0 : M_PI;
    }

    return acos(cross_prod);
}

// @param p1 A point on the sphere where x and y are degrees.
// @param p2 A point on the sphere where x and y are degrees.
// @return The distance between the two points in RADIANS.  Multiply by radius to get arc
// length.
double spheredist_deg(const Point& p1, const Point& p2) {
    return spheredist_rad(Point(deg2rad(p1.x), deg2rad(p1.y)), Point(deg2rad(p2.x), deg2rad(p2.y)));
}

// Technically lat/long bounds, not really tied to earth radius.
bool isValidLngLat(double lng, double lat) {
    return abs(lng) <= 180 && abs(lat) <= 90;
}

double distance(const Point& p1, const Point& p2) {
    double a = p1.x - p2.x;
    double b = p1.y - p2.y;

    // Avoid numerical error if possible...
    if (a == 0)
        return abs(b);
    if (b == 0)
        return abs(a);

    return sqrt((a * a) + (b * b));
}

static inline Vector2_d toVector2(const Point& p) {
    return Vector2_d(p.x, p.y);
}

// Given a segment (A, B) and a segment (C, D), check whether they intersect.
bool linesIntersect(const Point& pA, const Point& pB, const Point& pC, const Point& pD) {
    Vector2_d a = toVector2(pA);
    Vector2_d b = toVector2(pB);
    Vector2_d c = toVector2(pC);
    Vector2_d d = toVector2(pD);

    // The normal of line AB
    Vector2_d normalAB = (b - a).Ortho();

    // Dot products of AC and the normal of AB
    // = 0 : C is on the line AB
    // > 0 : C is on one side
    // < 0 : C is on the other side
    double dotProdNormalAB_AC = normalAB.DotProd(c - a);
    double dotProdNormalAB_AD = normalAB.DotProd(d - a);

    // C and D can not on the same side of line AB
    if (dotProdNormalAB_AC * dotProdNormalAB_AD > 0)
        return false;

    // AB and CD are on the same line
    if (dotProdNormalAB_AC == 0 && dotProdNormalAB_AD == 0) {
        // Test if C or D is on segment AB.
        return (c - a).DotProd(c - b) <= 0 || (d - a).DotProd(d - b) <= 0;
    }

    // Check if A and B are on different sides of line CD.
    Vector2_d normalCD = (d - c).Ortho();
    double dotProdNormalCD_CA = normalCD.DotProd(a - c);
    double dotProdNormalCD_CB = normalCD.DotProd(b - c);
    return dotProdNormalCD_CA * dotProdNormalCD_CB <= 0;  // Perhaps A or B is on line CD
}

static bool circleContainsBoxInternal(const Circle& circle,
                                      const Box& box,
                                      bool includeCircleBoundary) {
    // NOTE: a circle of zero radius is a point, and there are NO points contained inside a
    // zero-radius circle, not even the point itself.

    const Point& a = box._min;
    const Point& b = box._max;
    double compareLL = distanceCompare(circle.center, a, circle.radius);  // Lower left
    double compareUR = distanceCompare(circle.center, b, circle.radius);  // Upper right
    // Upper Left
    double compareUL = distanceCompare(circle.center, Point(a.x, b.y), circle.radius);
    // Lower right
    double compareLR = distanceCompare(circle.center, Point(b.x, a.y), circle.radius);
    if (includeCircleBoundary) {
        return compareLL <= 0 && compareUR <= 0 && compareUL <= 0 && compareLR <= 0;
    } else {
        return compareLL < 0 && compareUR < 0 && compareUL < 0 && compareLR < 0;
    }
}

bool circleContainsBox(const Circle& circle, const Box& box) {
    return circleContainsBoxInternal(circle, box, true);
}

bool circleInteriorContainsBox(const Circle& circle, const Box& box) {
    return circleContainsBoxInternal(circle, box, false);
}

// Check the intersection by measuring the distance between circle center and box center.
static bool circleIntersectsWithBoxInternal(const Circle& circle,
                                            const Box& box,
                                            bool includeCircleBoundary) {
    // NOTE: a circle of zero radius is a point, and there are NO points to intersect inside a
    // zero-radius circle, not even the point itself.
    if (circle.radius == 0.0 && !includeCircleBoundary)
        return false;

    /* Collapses the four quadrants down into one.
     *   ________
     * r|___B___ \  <- a quarter round corner here. Let's name it "D".
     *  |       | |
     * h|       | |
     *  |   A   |C|
     *  |_______|_|
     *      w    r
     */

    Point boxCenter = box.center();
    double dx = abs(circle.center.x - boxCenter.x);
    double dy = abs(circle.center.y - boxCenter.y);
    double w = (box._max.x - box._min.x) / 2;
    double h = (box._max.y - box._min.y) / 2;
    const double& r = circle.radius;

    // Check if circle.center is in A, B or C.
    // The circle center could be above the box (B) or right to the box (C), but close enough.
    if (includeCircleBoundary) {
        if ((dx <= w + r && dy <= h) || (dx <= w && dy <= h + r))
            return true;
    } else {
        if ((dx < w + r && dy < h) || (dx < w && dy < h + r))
            return true;
    }

    // Now check if circle.center is in the round corner "D".
    double compareResult = distanceCompare(Point(dx, dy), Point(w, h), r);
    return compareResult < 0 || (compareResult == 0 && includeCircleBoundary);
}

bool circleIntersectsWithBox(const Circle& circle, const Box& box) {
    return circleIntersectsWithBoxInternal(circle, box, true);
}

bool circleInteriorIntersectsWithBox(const Circle& circle, const Box& box) {
    return circleIntersectsWithBoxInternal(circle, box, false);
}

bool lineIntersectsWithBox(const Point& a, const Point& b, const Box& box) {
    Point upperLeft(box._min.x, box._max.y);
    Point lowerRight(box._max.x, box._min.y);

    return linesIntersect(a, b, upperLeft, box._min) ||
        linesIntersect(a, b, box._min, lowerRight) || linesIntersect(a, b, lowerRight, box._max) ||
        linesIntersect(a, b, box._max, upperLeft);
}

// Doc: The last point specified is always implicitly connected to the first.
// [[ 0 , 0 ], [ 3 , 6 ], [ 6 , 0 ]]
bool edgesIntersectsWithBox(const vector<Point>& vertices, const Box& box) {
    for (size_t i = 0; i < vertices.size() - 1; i++) {
        if (lineIntersectsWithBox(vertices[i], vertices[i + 1], box))
            return true;
    }
    // The last point and first point.
    return lineIntersectsWithBox(vertices[vertices.size() - 1], vertices[0], box);
}

bool polygonContainsBox(const Polygon& polygon, const Box& box) {
    // All vertices of box have to be inside the polygon.
    if (!polygon.contains(box._min) || !polygon.contains(box._max) ||
        !polygon.contains(Point(box._min.x, box._max.y)) ||
        !polygon.contains(Point(box._max.x, box._min.y)))
        return false;

    // No intersection between the polygon edges and the box.
    return !edgesIntersectsWithBox(polygon.points(), box);
}

bool polygonIntersectsWithBox(const Polygon& polygon, const Box& box) {
    // 1. Polygon contains the box.
    // Check the relaxed condition that whether the polygon include any vertex of the box.
    if (polygon.contains(box._min) || polygon.contains(box._max) ||
        polygon.contains(Point(box._min.x, box._max.y)) ||
        polygon.contains(Point(box._max.x, box._min.y)))
        return true;

    // 2. Box contains polygon.
    // Check the relaxed condition that whether the box include any vertex of the polygon.
    for (vector<Point>::const_iterator it = polygon.points().begin(); it != polygon.points().end();
         it++) {
        if (box.inside(*it))
            return true;
    }

    // 3. Otherwise they intersect on a portion of both shapes.
    // Edges intersects
    return edgesIntersectsWithBox(polygon.points(), box);
}

bool ShapeProjection::supportsProject(const PointWithCRS& point, const CRS crs) {
    // Can always trivially project or project from SPHERE->FLAT
    if (point.crs == crs || point.crs == SPHERE)
        return true;

    invariant(point.crs == FLAT);
    // If crs is FLAT, we might be able to upgrade the point to SPHERE if it's a valid SPHERE
    // point (lng/lat in bounds).  In this case, we can use FLAT data with SPHERE predicates.
    return isValidLngLat(point.oldPoint.x, point.oldPoint.y);
}

bool ShapeProjection::supportsProject(const PolygonWithCRS& polygon, const CRS crs) {
    return polygon.crs == crs || (polygon.crs == STRICT_SPHERE && crs == SPHERE);
}

void ShapeProjection::projectInto(PointWithCRS* point, CRS crs) {
    dassert(supportsProject(*point, crs));

    if (point->crs == crs)
        return;

    if (FLAT == point->crs) {
        // Prohibit projection to STRICT_SPHERE CRS
        invariant(SPHERE == crs);

        // Note that it's (lat, lng) for S2 but (lng, lat) for MongoDB.
        S2LatLng latLng = S2LatLng::FromDegrees(point->oldPoint.y, point->oldPoint.x).Normalized();
        dassert(latLng.is_valid());
        point->point = latLng.ToPoint();
        point->cell = S2Cell(point->point);
        point->crs = SPHERE;
        return;
    }

    // Prohibit projection to STRICT_SPHERE CRS
    invariant(SPHERE == point->crs && FLAT == crs);
    // Just remove the additional spherical information
    point->point = S2Point();
    point->cell = S2Cell();
    point->crs = FLAT;
}

void ShapeProjection::projectInto(PolygonWithCRS* polygon, CRS crs) {
    if (polygon->crs == crs)
        return;

    // Only project from STRICT_SPHERE to SPHERE
    invariant(STRICT_SPHERE == polygon->crs && SPHERE == crs);
    polygon->crs = SPHERE;
}

}  // namespace mongo