24:
1237:
983:
591:
243:
970:
713:
1209:, but these operations do not always produce a simple polygon as their result. They can be defined in a way that always produces a two-dimensional region, but this requires careful definitions of the intersection and difference operations in order to avoid creating one-dimensional features or isolated points.
1114:
Constructing a triangulation of a simple polygon can also be performed in linear time, although the algorithm is complicated. A modification of the same algorithm can also be used to test whether a closed polygonal chain forms a simple polygon (that is, whether it avoids self-intersections) in linear
822:
on a triangulation of the polygon: it is always possible to color the vertices with three colors, so that each side or diagonal in the triangulation has two endpoints of different colors. Each point of the polygon is visible to a vertex of each color, for instance one of the three vertices of the
977:
emanating from the point and count its intersections with the polygon. If it crosses only interior points of edges, an odd number of times, the point lies inside the polygon; if even, the point lies outside. Rays through polygon vertices or containing its edges need special
1180:
of an interior point of a simple polygon, the points that are directly visible from the given point by line segments interior to the polygon, can be constructed in linear time. The same is true for the subset that is visible from at least one point of a given line
920:
intersects the interior of the polygon in a connected set. Equivalently, it is a polygon whose boundary can be partitioned into two monotone polygonal chains, subsequences of edges whose vertices, when projected perpendicularly onto
1333:, formed by the polygon sides. The computational complexity of reconstructing a polygon that has a given graph as its visibility graph, with a specified Hamiltonian cycle as its cycle of sides, remains an open problem.
1244:
Every finite set of points in the plane that does not lie on a single line can be connected to form the vertices of a simple polygon (allowing 180° angles); for instance, one such polygon is the solution to the
1168:
in the plane that connects two points interior to a polygon, without crossing to the exterior, may be found in linear time by an algorithm that uses triangulation as a subroutine. The same is true for the
509:, the turning angle from one directed side to the next. The external angle is positive at a convex vertex or negative at a concave vertex. For every simple polygon, the sum of the external angles is
381:, with infinite area. Although the formal definition of a simple polygon is typically as a system of line segments, it is also possible (and common in informal usage) to define a simple polygon as a
1149:
points, although not necessarily using the optimal number of points for a given polygon. Although it is possible to transform any two triangulations of the same polygon into each other by
1229:
provides a method to explicitly construct a map from a disk to any simple polygon using specified vertex angles and pre-images of the polygon vertices on the boundary of the disk. These
703:, a vertex whose two neighbors are the endpoints of a line segment that is otherwise entirely exterior to the polygon. The polygons that have exactly two ears and one mouth are called
1147:
855:
776:
186:
1085:; alternatively, it is possible to process a given polygon into a data structure, in linear time, so that subsequent point in polygon tests can be performed in logarithmic time.
507:
1320:
585:
472:
530:
98:
452:
428:
668:
642:
148:
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Other computational problems studied for simple polygons include constructions of the longest diagonal or the longest line segment interior to a polygon, of the
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1039:
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323:(180°), while others disallow this, instead requiring collinear segments of a closed polygonal chain to be merged into a single longer side. Two vertices are
346:'s original proof of this theorem took the special case of simple polygons (stated without proof) as its starting point. The region inside the polygon (its
778:
of the vertices with the property that every point in the polygon is visible from one of the selected vertices. This means that, for each point
392:
of a simple polygon is any line segment that has two polygon vertices as its endpoints, and that otherwise is entirely interior to the polygon.
319:
can be used to avoid this ambiguity. The number of edges always equals the number of vertices. Some sources allow two line segments to form a
2591:
Cabello, Sergio; Cibulka, Josef; KynÄl, Jan; Saumell, Maria; Valtr, Pavel (2017). "Peeling potatoes near-optimally in near-linear time".
2361:
Aichholzer, Oswin; Mulzer, Wolfgang; Pilz, Alexander (2015). "Flip distance between triangulations of a simple polygon is NP-complete".
2051:
2415:; De Carufel, Jean-Lou; Korman, Matias; Oh, Eunjin (2016). "A linear-time algorithm for the geodesic center of a simple polygon".
695:, vertices whose two neighbors are the endpoints of a diagonal. A related theorem states that every simple polygon that is not a
1728:
1802:
Margalit, Avraham; Knott, Gary D. (1989). "An algorithm for computing the union, intersection or difference of two polygons".
2165:
1451:
1329:
of a simple polygon connects its vertices by edges representing the sides and diagonals of the polygon. It always contains a
406:
of a simple polygon, at one of its vertices, is the angle spanned by the interior of the polygon at that vertex. A vertex is
2974:
Ghosh, Subir Kumar; Goswami, Partha P. (2013). "Unsolved problems in visibility graphs of points, segments, and polygons".
2766:
2647:
2554:
2417:
2363:
2276:
1960:
598:
Every simple polygon can be partitioned into non-overlapping triangles by a subset of its diagonals. When the polygon has
676:. The shape of a triangulated simple polygon can be uniquely determined by the internal angles of the polygon and by the
266:. Two line segments meet at every endpoint, and there are no other points of intersection between the line segments. No
3057:
2831:
2337:
2224:
2002:
1938:
1837:
1752:
1747:
1673:
44:
1528:
1476:
2113:
362:
2173:
1933:. Pure and Applied Undergraduate Texts. Vol. 63 (2nd ed.). American Mathematical Society. p. 421.
1240:
The black polygon is the shortest loop connecting every red dot, a solution to the traveling salesperson problem.
1226:
1104:
215:
204:
2684:
Cheng, Siu-Wing; Mencel, Liam; Vigneron, Antoine (2016). "A faster algorithm for computing straight skeletons".
2480:(1987). "Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons".
2686:
2085:
1889:
1593:
1153:
that replace one diagonal at a time, determining whether one can do so using only a limited number of flips is
378:
2121:
Algorithms â ESA 2008, 16th Annual
European Symposium, Karlsruhe, Germany, September 15â17, 2008. Proceedings
1246:
818:
to a selected vertex, passing only through interior points of the polygon. One way to prove this is to use
2937:; Snoeyink, Jack (1993). "An efficient algorithm for finding the CSG representation of a simple polygon".
27:
Two simple polygons (green and blue) and a self-intersecting polygon (red, in the lower right, not simple)
3180:
3160:
1342:
1257:
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826:
747:
227:
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3155:
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that constructs the polygon (as a closed set including the interior) from unions and intersections of
986:
A simple polygon (interior shaded blue) and its convex hull (surrounding both blue and yellow regions)
716:
This 42-vertex polygonal art gallery is entirely visible from cameras placed at the 4 marked vertices.
23:
2593:
1173:, a point in the polygon that minimizes the maximum length of its geodesic paths to all other points.
1348:
873:, the polygons that have a point (interior or on their boundary) from which every point is visible.
3215:
1115:
time. This also leads to a linear time algorithm for solving the art gallery problem using at most
486:
3140:
2219:
1236:
1218:
1165:
823:
triangle containing that point in the chosen triangulation. One of the colors is used by at most
705:
208:
1287:
307:
are more formal, but may be ambiguous in contexts that also involve the edges and vertices of a
3165:
3050:
2317:
1716:
1566:
991:
555:
192:
48:
1928:
3566:
3506:
3145:
2976:
2878:
2829:
Quintas, L. V.; Supnick, Fred (1965). "On some properties of shortest
Hamiltonian circuits".
2313:
1433:
1366:
1264:, with each side of the polygon appearing once as a half-plane in the formula. Converting an
1189:(the largest convex polygon within the given simple polygon), and of various one-dimensional
672:
457:
121:
3450:
3220:
3150:
3092:
2960:
2912:
2860:
2815:
2789:
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2624:
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2503:
2448:
2394:
2347:
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2031:
1983:
1910:
1866:
1781:
1702:
1614:
1551:
1497:
512:
339:
308:
80:
1201:. Researchers have also studied producing other polygons from simple polygons using their
437:
413:
8:
3556:
3531:
3501:
3496:
3455:
3170:
2080:
1720:
1425:
870:
721:
647:
621:
481:
151:
127:
63:
994:, several important computational tasks involve inputs in the form of a simple polygon.
3561:
3102:
2985:
2930:
2848:
2695:
2638:
2602:
2465:
2426:
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2019:
1997:
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1024:
1004:
944:
924:
903:
883:
801:
781:
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601:
535:
103:
2215:
532:(one full turn, 360°). Thus the sum of the internal angles, for a simple polygon with
3541:
3135:
3043:
3018:
2534:
2333:
2098:
2065:
2046:
1934:
1902:
1815:
1606:
1542:
1523:
1489:
1447:
1360:
1330:
1198:
1097:
342:
can be used to prove that such a polygon divides the plane into two regions. Indeed,
291:
2808:
Proceedings of the
International Congress of Mathematicians, Vol. III (Berlin, 1998)
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3021:
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2094:
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2011:
1969:
1924:
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1811:
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1485:
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1250:
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998:
877:
688:
594:
A triangulated polygon with 11 vertices: 11 sides and 8 diagonals form 9 triangles.
231:
223:
196:
67:
2143:
1351:, a process of reflecting pockets of a non-convex simple polygon to make it convex
982:
3536:
3516:
3511:
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3200:
3175:
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2926:
2908:
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2724:
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2620:
2573:
2499:
2444:
2390:
2343:
2295:
2249:
2128:
2123:. Lecture Notes in Computer Science. Vol. 5193. Springer. pp. 744â755.
2027:
1979:
1906:
1862:
1777:
1698:
1610:
1547:
1493:
1369:, a generalization of simple polygons allowing the edges to touch in limited ways
385:
in the plane, the union of these line segments with the interior of the polygon.
263:
255:
2806:; Driscoll, Tobin A. (1998). "SchwarzâChristoffel mapping in the computer era".
2521:(1981). "A linear algorithm for computing the visibility polygon from a point".
3526:
3491:
3486:
3117:
3097:
2803:
1640:
1519:
1515:
974:
866:
819:
696:
476:
402:
343:
320:
74:
59:
2780:
2761:
2440:
2386:
1652:
1591:(1979). "A linear algorithm for finding the convex hull of a simple polygon".
1443:
3581:
3521:
3372:
3265:
3185:
3127:
2477:
1832:
1636:
1471:
1222:
1206:
681:
354:
267:
219:
2999:
1750:(1992). "The Jordan-Schönflies theorem and the classification of surfaces".
242:
3551:
3421:
3377:
3341:
3331:
3326:
2939:
2874:
2757:
2482:
2473:
2083:; Supowit, Kenneth J. (1981). "Testing a simple polygon for monotonicity".
1884:
1628:
1511:
1202:
1186:
590:
335:
259:
251:
55:
51:
1357:, a simple polygon that can be folded and glued to form a given polyhedron
3460:
3367:
3346:
3336:
2903:
1438:. Texts and Monographs in Computer Science. Springer-Verlag. p. 18.
1194:
1154:
1108:
1082:
677:
351:
2881:(2022). "Area-optimal simple polygonalizations: the CG challenge 2019".
869:
is a simple polygon. Another important class of simple polygons are the
3465:
3321:
3311:
3195:
2952:
2852:
2661:
2642:
2616:
2568:
2549:
2518:
2495:
2290:
2271:
2245:
2194:
2023:
1974:
1955:
1858:
1773:
1694:
1588:
1261:
1150:
969:
382:
712:
3440:
3430:
3407:
3397:
3387:
3316:
3225:
3190:
3026:
1627:
1107:
can also be found in linear time, faster than algorithms for finding
358:
2894:
2844:
2709:
2237:
2186:
2015:
1726:. From Insight to Proof: Festschrift in Honour of Andrzej Trybulec.
1686:
1400:
3445:
3435:
3392:
3351:
3280:
3270:
3260:
3079:
2431:
2412:
1161:
214:
Other constructions in geometry related to simple polygons include
32:
3035:
2990:
2700:
2607:
2377:
327:
if they are the two endpoints of one of the sides of the polygon.
3402:
3382:
3295:
3290:
3285:
3275:
3250:
3205:
3066:
2149:. In Toth, Csaba D.; O'Rourke, Joseph; Goodman, Jacob E. (eds.).
1284:-sided polygon into this representation can be performed in time
40:
1835:; Zuckerman, H. S. (1967). "Lattice points and polygonal area".
1474:(1995). "Negative results on characterizing visibility graphs".
1345:, on continuous motion of a simple polygon into a convex polygon
3210:
2925:
691:, every simple polygon that is not a triangle has at least two
370:
1887:(1990). "Computing the longest diagonal of a simple polygon".
3255:
2725:"Computing mitered offset curves based on straight skeletons"
973:
To test whether a point is inside the polygon, construct any
2873:
2643:"Finding the medial axis of a simple polygon in linear time"
2464:
1249:. Connecting points to form a polygon in this way is called
744:
vertices, it is always possible to find a subset of at most
270:
of the line segments has the same properties. The qualifier
1387:
Milnor, John W. (1950). "On the total curvature of knots".
1089:
366:
200:
2877:; Fekete, SĂĄndor P.; Keldenich, Phillip; Krupke, Dominik;
2590:
2153:(3rd ed.). Chapman and Hall/CRC. pp. 1005â1023.
2114:"How reliable are practical point-in-polygon strategies?"
2762:"Minkowski sums of monotone and general simple polygons"
1256:
Every simple polygon can be represented by a formula in
3016:
1221:, any simply connected open subset of the plane can be
798:
in the polygon, there exists a line segment connecting
2550:"A polynomial solution for the potato-peeling problem"
2410:
1111:
of points that have not been connected into a polygon.
369:. The polygon itself is topologically equivalent to a
1290:
1270:
1121:
1067:
1047:
1027:
1007:
947:
927:
906:
900:, is a polygon for which every line perpendicular to
886:
829:
804:
784:
750:
730:
650:
624:
604:
558:
538:
515:
489:
460:
440:
416:
281:
The line segments that form a polygon are called its
160:
130:
106:
83:
2360:
2316:(2000). "Art gallery and illumination problems". In
1510:
684:
formed by pairs of triangles that share a diagonal.
2683:
1645:
Computational
Geometry: Algorithms and Applications
1001:testing involves determining, for a simple polygon
2796:
1827:
1825:
1420:
1418:
1416:
1414:
1412:
1410:
1314:
1276:
1141:
1073:
1053:
1033:
1013:
953:
933:
912:
892:
849:
810:
790:
770:
736:
662:
636:
610:
579:
544:
524:
501:
466:
446:
422:
191:Simple polygons are commonly seen as the input to
180:
142:
112:
92:
2802:
2208:
2073:
1797:
1795:
1793:
1791:
1529:Computational Geometry: Theory & Applications
1524:"On compatible triangulations of simple polygons"
1477:Computational Geometry: Theory & Applications
1465:
1463:
1363:, an analogous concept on the surface of a sphere
3579:
2723:Palfrader, Peter; Held, Martin (February 2015).
2637:
2214:
2079:
1923:
1878:
1876:
1092:of the interior of a polygon. These include the
2822:
2324:. Amsterdam: North-Holland. pp. 973â1027.
2272:"Triangulating a simple polygon in linear time"
2151:Handbook of Discrete and Computational Geometry
1822:
1582:
1580:
1424:
1407:
670:diagonals. The resulting partition is called a
18:Shape bounded by non-intersecting line segments
2828:
2716:
1831:
1788:
1671:Meisters, G. H. (1975). "Polygons have ears".
1469:
1460:
58:. These polygons include as special cases the
3051:
2967:
2722:
2047:"A short proof of ChvĂĄtal's watchman theorem"
1956:"Cross-ratios and angles determine a polygon"
1930:A Discrete Transition to Advanced Mathematics
1873:
1801:
1100:for polygons with integer vertex coordinates.
2973:
2516:
2510:
1882:
1721:"Jordan's proof of the Jordan curve theorem"
1586:
1577:
1136:
1122:
1088:Simple formulae are known for computing the
844:
830:
765:
751:
175:
161:
2810:. Documenta Mathematica. pp. 533â542.
2119:. In Halperin, Dan; Mehlhorn, Kurt (eds.).
1947:
3058:
3044:
2749:
1666:
1664:
1662:
1564:
47:itself and has no holes. That is, it is a
2989:
2902:
2779:
2755:
2740:
2699:
2660:
2606:
2567:
2541:
2430:
2406:
2404:
2376:
2289:
2064:
2052:Journal of Combinatorial Theory, Series B
1996:
1990:
1973:
1746:
1740:
1558:
1541:
1504:
964:
2883:ACM Journal of Experimental Algorithmics
2547:
2460:
2458:
2266:
2260:
2141:
2135:
1953:
1670:
1235:
1212:
981:
968:
711:
589:
480:at the same vertex is defined to be its
241:
22:
2641:; Snoeyink, Jack; Wang, Cao An (1999).
2312:
2306:
2111:
2105:
1659:
1435:Computational Geometry: An Introduction
1193:approximating its shape, including the
289:. An endpoint of a segment is called a
3580:
2729:Computer-Aided Design and Applications
2401:
2163:
2157:
1729:Studies in Logic, Grammar and Rhetoric
1647:(3rd ed.). Springer. p. 58.
1567:"Are precise definitions a good idea?"
1386:
857:of the vertices, proving the theorem.
434:if the internal angle is greater than
3039:
3017:
2867:
2767:Discrete & Computational Geometry
2648:Discrete & Computational Geometry
2584:
2555:Discrete & Computational Geometry
2455:
2418:Discrete & Computational Geometry
2364:Discrete & Computational Geometry
2277:Discrete & Computational Geometry
1961:Discrete & Computational Geometry
1917:
1715:
1621:
1380:
330:Simple polygons are sometimes called
2919:
2044:
2038:
2000:(1991). "Anthropomorphic polygons".
1709:
1233:are typically computed numerically.
274:is sometimes omitted, with the word
3065:
2677:
2631:
2354:
2222:(February 1993). "Pick's theorem".
1142:{\displaystyle \lfloor n/3\rfloor }
850:{\displaystyle \lfloor n/3\rfloor }
771:{\displaystyle \lfloor n/3\rfloor }
410:if its internal angle is less than
181:{\displaystyle \lfloor n/3\rfloor }
13:
2322:Handbook of Computational Geometry
880:, with respect to a straight line
278:assumed to mean a simple polygon.
154:its interior is visible from some
14:
3599:
3010:
2832:The American Mathematical Monthly
2548:Chang, J. S.; Yap, C.-K. (1986).
2225:The American Mathematical Monthly
2003:The American Mathematical Monthly
1838:The American Mathematical Monthly
1753:The American Mathematical Monthly
1674:The American Mathematical Monthly
2330:10.1016/B978-044482537-7/50023-1
1573:. American Mathematical Society.
1205:, unions and intersections, and
860:
2174:The College Mathematics Journal
1105:convex hull of a simple polygon
262:, meeting end-to-end to form a
205:convex hull of a simple polygon
2687:ACM Transactions on Algorithms
2086:Information Processing Letters
1890:Information Processing Letters
1851:10.1080/00029890.1967.12000095
1766:10.1080/00029890.1992.11995820
1736:(23). University of BiaĆystok.
1594:Information Processing Letters
1309:
1294:
571:
559:
373:, and the region outside (the
237:
1:
2166:"The surveyor's area formula"
1373:
1247:traveling salesperson problem
430:(a straight angle, 180°) and
395:
150:of its diagonals, and by the
2742:10.1080/16864360.2014.997637
2535:10.1016/0196-6774(81)90019-5
2129:10.1007/978-3-540-87744-8_62
2099:10.1016/0020-0190(81)90091-0
2066:10.1016/0095-8956(78)90059-X
1903:10.1016/0020-0190(90)90167-V
1816:10.1016/0097-8493(89)90059-9
1607:10.1016/0020-0190(79)90069-3
1543:10.1016/0925-7721(93)90028-5
1490:10.1016/0925-7721(95)00021-Z
1096:for arbitrary polygons, and
941:, have the same order along
502:{\displaystyle \pi -\theta }
365:, with a finite but nonzero
311:; the more colloquial terms
100:. Every simple polygon with
54:consisting of finitely many
7:
2411:Ahn, Hee-Kap; Barba, Luis;
1927:; Richmond, Thomas (2023).
1565:Malkevitch, Joseph (2016).
1336:
1258:constructive solid geometry
1227:SchwarzâChristoffel mapping
724:, in a simple polygon with
454:. If the internal angle is
230:formulas for polygons, and
228:constructive solid geometry
222:involving simple polygons,
216:SchwarzâChristoffel mapping
10:
3604:
1315:{\displaystyle O(n\log n)}
3474:
3420:
3360:
3304:
3243:
3234:
3126:
3078:
2781:10.1007/s00454-005-1206-y
2594:SIAM Journal on Computing
2441:10.1007/s00454-016-9796-0
2387:10.1007/s00454-015-9709-7
2320:; Urrutia, Jorge (eds.).
1653:10.1007/978-3-540-77974-2
1444:10.1007/978-1-4612-1098-6
961:as they do in the chain.
580:{\displaystyle (n-2)\pi }
363:JordanâSchönflies theorem
246:Parts of a simple polygon
2112:Schirra, Stefan (2008).
1804:Computers & Graphics
1343:Carpenter's rule problem
706:anthropomorphic polygons
644:triangles, separated by
355:topologically equivalent
295:(plural: vertices) or a
209:Euclidean shortest paths
3000:10.1145/2543581.2543589
2142:Snoeyink, Jack (2017).
1954:Snoeyink, Jack (1999).
1219:Riemann mapping theorem
467:{\displaystyle \theta }
77:of a simple polygon is
2879:Mitchell, Joseph S. B.
1316:
1278:
1241:
1143:
1081:. It can be solved in
1075:
1055:
1035:
1015:
992:computational geometry
987:
979:
965:Computational problems
955:
935:
914:
894:
851:
812:
792:
772:
738:
717:
664:
638:
612:
595:
581:
546:
526:
503:
468:
448:
424:
260:straight line segments
250:A simple polygon is a
247:
193:computational geometry
182:
144:
114:
94:
28:
2977:ACM Computing Surveys
2523:Journal of Algorithms
2164:Braden, Bart (1986).
1389:Annals of Mathematics
1367:Weakly simple polygon
1317:
1279:
1239:
1213:Related constructions
1144:
1076:
1056:
1036:
1016:
985:
972:
956:
936:
915:
895:
852:
813:
793:
773:
739:
715:
673:polygon triangulation
665:
639:
618:sides, this produces
613:
593:
582:
547:
527:
525:{\displaystyle 2\pi }
504:
469:
449:
425:
245:
207:, triangulation, and
183:
145:
115:
95:
93:{\displaystyle 2\pi }
26:
3291:Nonagon/Enneagon (9)
3221:Tangential trapezoid
2081:Preparata, Franco P.
1426:Preparata, Franco P.
1288:
1268:
1119:
1065:
1045:
1025:
1005:
945:
925:
904:
884:
871:star-shaped polygons
827:
802:
782:
748:
728:
648:
622:
602:
556:
536:
513:
487:
458:
447:{\displaystyle \pi }
438:
423:{\displaystyle \pi }
414:
340:Jordan curve theorem
195:problems, including
158:
128:
104:
81:
64:star-shaped polygons
3403:Megagon (1,000,000)
3171:Isosceles trapezoid
2804:Trefethen, Lloyd N.
2639:Chin, Francis Y. L.
1998:Toussaint, Godfried
1430:Shamos, Michael Ian
722:art gallery theorem
663:{\displaystyle n-3}
637:{\displaystyle n-2}
334:, because they are
152:art gallery theorem
143:{\displaystyle n-3}
3373:Icositetragon (24)
3019:Weisstein, Eric W.
2953:10.1007/BF01908629
2662:10.1007/PL00009429
2617:10.1137/16M1079695
2569:10.1007/BF02187692
2517:El Gindy, Hossam;
2496:10.1007/BF01840360
2318:Sack, Jörg-RĂŒdiger
2291:10.1007/BF02574703
1975:10.1007/PL00009481
1748:Thomassen, Carsten
1587:McCallum, Duncan;
1571:AMS Feature Column
1349:ErdĆsâNagy theorem
1312:
1274:
1242:
1223:conformally mapped
1178:visibility polygon
1139:
1071:
1051:
1031:
1021:and a query point
1011:
988:
980:
951:
931:
910:
890:
847:
808:
788:
768:
734:
718:
660:
634:
608:
596:
577:
542:
522:
499:
464:
444:
420:
379:connected open set
377:) is an unbounded
248:
178:
140:
110:
90:
29:
3588:Types of polygons
3575:
3574:
3416:
3415:
3393:Myriagon (10,000)
3378:Triacontagon (30)
3342:Heptadecagon (17)
3332:Pentadecagon (15)
3327:Tetradecagon (14)
3266:Quadrilateral (4)
3136:Antiparallelogram
2984:(2): 22:1â22:29.
2935:Hershberger, John
2694:(3): 44:1â44:21.
2478:Tarjan, Robert E.
2472:; Leven, Daniel;
2470:Hershberger, John
2268:Chazelle, Bernard
2045:Fisk, S. (1978).
1925:Richmond, Bettina
1845:(10): 1195â1200.
1453:978-1-4612-1098-6
1361:Spherical polygon
1331:Hamiltonian cycle
1277:{\displaystyle n}
1217:According to the
1199:straight skeleton
1074:{\displaystyle P}
1061:lies interior to
1054:{\displaystyle q}
1034:{\displaystyle q}
1014:{\displaystyle P}
954:{\displaystyle L}
934:{\displaystyle L}
913:{\displaystyle L}
893:{\displaystyle L}
811:{\displaystyle p}
791:{\displaystyle p}
737:{\displaystyle n}
720:According to the
687:According to the
611:{\displaystyle n}
545:{\displaystyle n}
232:visibility graphs
203:computation, the
188:of its vertices.
113:{\displaystyle n}
68:monotone polygons
3595:
3388:Chiliagon (1000)
3368:Icositrigon (23)
3347:Octadecagon (18)
3337:Hexadecagon (16)
3241:
3240:
3060:
3053:
3046:
3037:
3036:
3032:
3031:
3022:"Simple polygon"
3004:
3003:
2993:
2971:
2965:
2964:
2931:Guibas, Leonidas
2923:
2917:
2916:
2906:
2875:Demaine, Erik D.
2871:
2865:
2864:
2826:
2820:
2819:
2800:
2794:
2793:
2783:
2753:
2747:
2746:
2744:
2720:
2714:
2713:
2703:
2681:
2675:
2674:
2664:
2635:
2629:
2628:
2610:
2601:(5): 1574â1602.
2588:
2582:
2581:
2571:
2545:
2539:
2538:
2514:
2508:
2507:
2466:Guibas, Leonidas
2462:
2453:
2452:
2434:
2408:
2399:
2398:
2380:
2358:
2352:
2351:
2310:
2304:
2303:
2293:
2264:
2258:
2257:
2216:GrĂŒnbaum, Branko
2212:
2206:
2205:
2203:
2197:. Archived from
2170:
2161:
2155:
2154:
2148:
2144:"Point Location"
2139:
2133:
2132:
2118:
2109:
2103:
2102:
2077:
2071:
2070:
2068:
2042:
2036:
2035:
1994:
1988:
1987:
1977:
1951:
1945:
1944:
1921:
1915:
1914:
1883:Aggarwal, Alok;
1880:
1871:
1870:
1829:
1820:
1819:
1799:
1786:
1785:
1744:
1738:
1737:
1725:
1717:Hales, Thomas C.
1713:
1707:
1706:
1668:
1657:
1656:
1625:
1619:
1618:
1584:
1575:
1574:
1562:
1556:
1555:
1545:
1508:
1502:
1501:
1470:Everett, Hazel;
1467:
1458:
1457:
1422:
1405:
1404:
1384:
1355:Net (polyhedron)
1327:visibility graph
1321:
1319:
1318:
1313:
1283:
1281:
1280:
1275:
1251:polygonalization
1148:
1146:
1145:
1140:
1132:
1094:shoelace formula
1080:
1078:
1077:
1072:
1060:
1058:
1057:
1052:
1040:
1038:
1037:
1032:
1020:
1018:
1017:
1012:
999:Point in polygon
960:
958:
957:
952:
940:
938:
937:
932:
919:
917:
916:
911:
899:
897:
896:
891:
878:monotone polygon
856:
854:
853:
848:
840:
817:
815:
814:
809:
797:
795:
794:
789:
777:
775:
774:
769:
761:
743:
741:
740:
735:
689:two ears theorem
669:
667:
666:
661:
643:
641:
640:
635:
617:
615:
614:
609:
586:
584:
583:
578:
551:
549:
548:
543:
531:
529:
528:
523:
508:
506:
505:
500:
473:
471:
470:
465:
453:
451:
450:
445:
429:
427:
426:
421:
224:polygonalization
197:point in polygon
187:
185:
184:
179:
171:
149:
147:
146:
141:
119:
117:
116:
111:
99:
97:
96:
91:
49:piecewise-linear
3603:
3602:
3598:
3597:
3596:
3594:
3593:
3592:
3578:
3577:
3576:
3571:
3470:
3424:
3412:
3356:
3322:Tridecagon (13)
3312:Hendecagon (11)
3300:
3236:
3230:
3201:Right trapezoid
3122:
3074:
3064:
3013:
3008:
3007:
2972:
2968:
2924:
2920:
2895:10.1145/3504000
2872:
2868:
2845:10.2307/2313333
2827:
2823:
2801:
2797:
2754:
2750:
2721:
2717:
2710:10.1145/2898961
2682:
2678:
2636:
2632:
2589:
2585:
2546:
2542:
2515:
2511:
2463:
2456:
2413:Bose, Prosenjit
2409:
2402:
2359:
2355:
2340:
2311:
2307:
2265:
2261:
2238:10.2307/2323771
2220:Shephard, G. C.
2213:
2209:
2201:
2187:10.2307/2686282
2168:
2162:
2158:
2146:
2140:
2136:
2116:
2110:
2106:
2078:
2074:
2043:
2039:
2016:10.2307/2324033
1995:
1991:
1952:
1948:
1941:
1922:
1918:
1881:
1874:
1830:
1823:
1800:
1789:
1745:
1741:
1723:
1714:
1710:
1687:10.2307/2319703
1669:
1660:
1641:Schwarzkopf, O.
1633:van Kreveld, M.
1626:
1622:
1585:
1578:
1563:
1559:
1520:Souvaine, Diane
1516:Seidel, Raimund
1509:
1505:
1468:
1461:
1454:
1423:
1408:
1401:10.2307/1969467
1385:
1381:
1376:
1339:
1289:
1286:
1285:
1269:
1266:
1265:
1215:
1171:geodesic center
1128:
1120:
1117:
1116:
1066:
1063:
1062:
1046:
1043:
1042:
1026:
1023:
1022:
1006:
1003:
1002:
967:
946:
943:
942:
926:
923:
922:
905:
902:
901:
885:
882:
881:
863:
836:
828:
825:
824:
803:
800:
799:
783:
780:
779:
757:
749:
746:
745:
729:
726:
725:
649:
646:
645:
623:
620:
619:
603:
600:
599:
557:
554:
553:
537:
534:
533:
514:
511:
510:
488:
485:
484:
459:
456:
455:
439:
436:
435:
415:
412:
411:
398:
332:Jordan polygons
264:polygonal chain
256:Euclidean plane
240:
226:of point sets,
218:, used to find
167:
159:
156:
155:
129:
126:
125:
105:
102:
101:
82:
79:
78:
75:external angles
60:convex polygons
19:
12:
11:
5:
3601:
3591:
3590:
3573:
3572:
3570:
3569:
3564:
3559:
3554:
3549:
3544:
3539:
3534:
3529:
3527:Pseudotriangle
3524:
3519:
3514:
3509:
3504:
3499:
3494:
3489:
3484:
3478:
3476:
3472:
3471:
3469:
3468:
3463:
3458:
3453:
3448:
3443:
3438:
3433:
3427:
3425:
3418:
3417:
3414:
3413:
3411:
3410:
3405:
3400:
3395:
3390:
3385:
3380:
3375:
3370:
3364:
3362:
3358:
3357:
3355:
3354:
3349:
3344:
3339:
3334:
3329:
3324:
3319:
3317:Dodecagon (12)
3314:
3308:
3306:
3302:
3301:
3299:
3298:
3293:
3288:
3283:
3278:
3273:
3268:
3263:
3258:
3253:
3247:
3245:
3238:
3232:
3231:
3229:
3228:
3223:
3218:
3213:
3208:
3203:
3198:
3193:
3188:
3183:
3178:
3173:
3168:
3163:
3158:
3153:
3148:
3143:
3138:
3132:
3130:
3128:Quadrilaterals
3124:
3123:
3121:
3120:
3115:
3110:
3105:
3100:
3095:
3090:
3084:
3082:
3076:
3075:
3063:
3062:
3055:
3048:
3040:
3034:
3033:
3012:
3011:External links
3009:
3006:
3005:
2966:
2918:
2866:
2839:(9): 977â980.
2821:
2795:
2774:(2): 223â240.
2748:
2735:(4): 414â424.
2715:
2676:
2655:(3): 405â420.
2630:
2583:
2562:(2): 155â182.
2540:
2529:(2): 186â197.
2509:
2490:(2): 209â233.
2454:
2425:(4): 836â859.
2400:
2371:(2): 368â389.
2353:
2338:
2314:Urrutia, Jorge
2305:
2284:(5): 485â524.
2259:
2232:(2): 150â161.
2207:
2204:on 2012-11-07.
2181:(4): 326â337.
2156:
2134:
2104:
2093:(4): 161â164.
2072:
2037:
1989:
1968:(4): 619â631.
1946:
1939:
1916:
1872:
1821:
1810:(2): 167â183.
1787:
1760:(2): 116â130.
1739:
1708:
1681:(6): 648â651.
1658:
1637:Overmars, Mark
1620:
1601:(5): 201â206.
1576:
1557:
1503:
1472:Corneil, Derek
1459:
1452:
1406:
1391:. 2nd Series.
1378:
1377:
1375:
1372:
1371:
1370:
1364:
1358:
1352:
1346:
1338:
1335:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1273:
1214:
1211:
1207:Minkowski sums
1183:
1182:
1174:
1158:
1138:
1135:
1131:
1127:
1124:
1112:
1101:
1098:Pick's theorem
1086:
1070:
1050:
1030:
1010:
966:
963:
950:
930:
909:
889:
867:convex polygon
862:
859:
846:
843:
839:
835:
832:
820:graph coloring
807:
787:
767:
764:
760:
756:
753:
733:
697:convex polygon
682:quadrilaterals
659:
656:
653:
633:
630:
627:
607:
576:
573:
570:
567:
564:
561:
541:
521:
518:
498:
495:
492:
477:external angle
463:
443:
419:
403:internal angle
397:
394:
344:Camille Jordan
321:straight angle
258:consisting of
239:
236:
220:conformal maps
177:
174:
170:
166:
163:
139:
136:
133:
109:
89:
86:
43:that does not
37:simple polygon
17:
9:
6:
4:
3:
2:
3600:
3589:
3586:
3585:
3583:
3568:
3567:Weakly simple
3565:
3563:
3560:
3558:
3555:
3553:
3550:
3548:
3545:
3543:
3540:
3538:
3535:
3533:
3530:
3528:
3525:
3523:
3520:
3518:
3515:
3513:
3510:
3508:
3507:Infinite skew
3505:
3503:
3500:
3498:
3495:
3493:
3490:
3488:
3485:
3483:
3480:
3479:
3477:
3473:
3467:
3464:
3462:
3459:
3457:
3454:
3452:
3449:
3447:
3444:
3442:
3439:
3437:
3434:
3432:
3429:
3428:
3426:
3423:
3422:Star polygons
3419:
3409:
3408:Apeirogon (â)
3406:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3384:
3381:
3379:
3376:
3374:
3371:
3369:
3366:
3365:
3363:
3359:
3353:
3352:Icosagon (20)
3350:
3348:
3345:
3343:
3340:
3338:
3335:
3333:
3330:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3310:
3309:
3307:
3303:
3297:
3294:
3292:
3289:
3287:
3284:
3282:
3279:
3277:
3274:
3272:
3269:
3267:
3264:
3262:
3259:
3257:
3254:
3252:
3249:
3248:
3246:
3242:
3239:
3233:
3227:
3224:
3222:
3219:
3217:
3214:
3212:
3209:
3207:
3204:
3202:
3199:
3197:
3194:
3192:
3189:
3187:
3186:Parallelogram
3184:
3182:
3181:Orthodiagonal
3179:
3177:
3174:
3172:
3169:
3167:
3164:
3162:
3161:Ex-tangential
3159:
3157:
3154:
3152:
3149:
3147:
3144:
3142:
3139:
3137:
3134:
3133:
3131:
3129:
3125:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3089:
3086:
3085:
3083:
3081:
3077:
3072:
3068:
3061:
3056:
3054:
3049:
3047:
3042:
3041:
3038:
3029:
3028:
3023:
3020:
3015:
3014:
3001:
2997:
2992:
2987:
2983:
2979:
2978:
2970:
2962:
2958:
2954:
2950:
2946:
2942:
2941:
2936:
2932:
2928:
2927:Dobkin, David
2922:
2914:
2910:
2905:
2904:1721.1/146480
2900:
2896:
2892:
2889:: A2.4:1â12.
2888:
2884:
2880:
2876:
2870:
2862:
2858:
2854:
2850:
2846:
2842:
2838:
2834:
2833:
2825:
2817:
2813:
2809:
2805:
2799:
2791:
2787:
2782:
2777:
2773:
2769:
2768:
2763:
2759:
2758:Sharir, Micha
2756:Oks, Eduard;
2752:
2743:
2738:
2734:
2730:
2726:
2719:
2711:
2707:
2702:
2697:
2693:
2689:
2688:
2680:
2672:
2668:
2663:
2658:
2654:
2650:
2649:
2644:
2640:
2634:
2626:
2622:
2618:
2614:
2609:
2604:
2600:
2596:
2595:
2587:
2579:
2575:
2570:
2565:
2561:
2557:
2556:
2551:
2544:
2536:
2532:
2528:
2524:
2520:
2513:
2505:
2501:
2497:
2493:
2489:
2485:
2484:
2479:
2475:
2474:Sharir, Micha
2471:
2467:
2461:
2459:
2450:
2446:
2442:
2438:
2433:
2428:
2424:
2420:
2419:
2414:
2407:
2405:
2396:
2392:
2388:
2384:
2379:
2374:
2370:
2366:
2365:
2357:
2349:
2345:
2341:
2339:0-444-82537-1
2335:
2331:
2327:
2323:
2319:
2315:
2309:
2301:
2297:
2292:
2287:
2283:
2279:
2278:
2273:
2269:
2263:
2255:
2251:
2247:
2243:
2239:
2235:
2231:
2227:
2226:
2221:
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2100:
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2017:
2013:
2009:
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1985:
1981:
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1885:Suri, Subhash
1879:
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1512:Aronov, Boris
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1234:
1232:
1228:
1225:onto a disk.
1224:
1220:
1210:
1208:
1204:
1203:offset curves
1200:
1196:
1192:
1188:
1179:
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1166:shortest path
1163:
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976:
971:
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861:Special cases
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336:Jordan curves
333:
328:
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322:
318:
314:
310:
306:
302:
298:
294:
293:
288:
284:
279:
277:
273:
269:
268:proper subset
265:
261:
257:
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244:
235:
234:of polygons.
233:
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206:
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120:sides can be
107:
87:
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76:
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61:
57:
56:line segments
53:
50:
46:
42:
38:
34:
25:
21:
16:
3546:
3361:>20 sides
3296:Decagon (10)
3281:Heptagon (7)
3271:Pentagon (5)
3261:Triangle (3)
3156:Equidiagonal
3025:
2981:
2975:
2969:
2944:
2940:Algorithmica
2938:
2921:
2886:
2882:
2869:
2836:
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2512:
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2483:Algorithmica
2481:
2422:
2416:
2368:
2362:
2356:
2321:
2308:
2281:
2275:
2262:
2229:
2223:
2210:
2199:the original
2178:
2172:
2159:
2150:
2137:
2120:
2107:
2090:
2084:
2075:
2056:
2050:
2040:
2010:(1): 31â35.
2007:
2001:
1992:
1965:
1959:
1949:
1929:
1919:
1897:(1): 13â18.
1894:
1888:
1842:
1836:
1807:
1803:
1757:
1751:
1742:
1733:
1727:
1711:
1678:
1672:
1644:
1623:
1598:
1592:
1570:
1560:
1536:(1): 27â35.
1533:
1527:
1506:
1484:(2): 51â63.
1481:
1475:
1434:
1392:
1388:
1382:
1324:
1255:
1243:
1231:pre-vertices
1230:
1216:
1190:
1187:convex skull
1184:
1170:
1109:convex hulls
989:
875:
864:
719:
704:
700:
692:
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678:cross-ratios
671:
597:
475:
431:
407:
401:
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389:
387:
374:
347:
331:
329:
324:
316:
312:
304:
300:
296:
290:
286:
282:
280:
275:
271:
252:closed curve
249:
213:
190:
122:triangulated
72:
52:Jordan curve
36:
30:
20:
15:
3557:Star-shaped
3532:Rectilinear
3502:Equilateral
3497:Equiangular
3461:Hendecagram
3305:11â20 sides
3286:Octagon (8)
3276:Hexagon (6)
3251:Monogon (1)
3093:Equilateral
2947:(1): 1â23.
2519:Avis, David
1833:Niven, Ivan
1629:de Berg, M.
1589:Avis, David
1395:: 248â257.
1262:half-planes
1195:medial axis
1155:NP-complete
1083:linear time
352:bounded set
238:Definitions
73:The sum of
3562:Tangential
3466:Dodecagram
3244:1â10 sides
3235:By number
3216:Tangential
3196:Right kite
2432:1501.00561
2059:(3): 374.
1374:References
1164:path, the
1041:, whether
482:supplement
396:Properties
383:closed set
350:) forms a
3542:Reinhardt
3451:Enneagram
3441:Heptagram
3431:Pentagram
3398:65537-gon
3256:Digon (2)
3226:Trapezoid
3191:Rectangle
3141:Bicentric
3103:Isosceles
3080:Triangles
3027:MathWorld
2991:1012.5187
2701:1405.4691
2608:1406.1368
2378:1209.0579
1304:
1191:skeletons
1137:⌋
1123:⌊
845:⌋
831:⌊
766:⌋
752:⌊
655:−
629:−
575:π
566:−
552:sides is
520:π
497:θ
494:−
491:π
462:θ
442:π
418:π
359:open disk
325:neighbors
199:testing,
176:⌋
162:⌊
135:−
88:π
45:intersect
3582:Category
3517:Isotoxal
3512:Isogonal
3456:Decagram
3446:Octagram
3436:Hexagram
3237:of sides
3166:Harmonic
3067:Polygons
2760:(2006).
2270:(1991).
1719:(2007).
1643:(2008).
1522:(1993).
1432:(1985).
1337:See also
1181:segment.
1162:geodesic
390:diagonal
375:exterior
348:interior
305:vertices
33:geometry
3537:Regular
3482:Concave
3475:Classes
3383:257-gon
3206:Rhombus
3146:Crossed
2961:1230699
2913:4390039
2861:0188872
2853:2313333
2816:1648186
2790:2195052
2671:1672988
2625:3708542
2578:0834056
2504:0895445
2449:3561791
2395:3372115
2348:1746693
2300:1115104
2254:1212401
2246:2323771
2195:2686282
2032:1083611
2024:2324033
1984:1721028
1911:1069001
1867:0225216
1859:2315660
1782:1144352
1774:2324180
1703:0367792
1695:2319703
1615:0552534
1552:1222755
1498:1353288
680:of the
432:concave
361:by the
317:corners
276:polygon
254:in the
41:polygon
3547:Simple
3492:Cyclic
3487:Convex
3211:Square
3151:Cyclic
3113:Obtuse
3108:Kepler
2959:
2911:
2859:
2851:
2814:
2788:
2669:
2623:
2576:
2502:
2447:
2393:
2346:
2336:
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2252:
2244:
2193:
2030:
2022:
1982:
1937:
1909:
1865:
1857:
1780:
1772:
1701:
1693:
1613:
1550:
1496:
1450:
865:Every
699:has a
474:, the
408:convex
371:circle
357:to an
338:; the
297:corner
292:vertex
272:simple
66:, and
3522:Magic
3118:Right
3098:Ideal
3088:Acute
2986:arXiv
2849:JSTOR
2696:arXiv
2603:arXiv
2427:arXiv
2373:arXiv
2242:JSTOR
2202:(PDF)
2191:JSTOR
2169:(PDF)
2147:(PDF)
2117:(PDF)
2020:JSTOR
1855:JSTOR
1770:JSTOR
1724:(PDF)
1691:JSTOR
1151:flips
978:care.
701:mouth
313:sides
309:graph
301:Edges
287:sides
283:edges
39:is a
3552:Skew
3176:Kite
3071:List
2334:ISBN
1935:ISBN
1448:ISBN
1325:The
1197:and
1176:The
1103:The
1090:area
693:ears
400:The
367:area
315:and
303:and
201:area
35:, a
2996:doi
2949:doi
2899:hdl
2891:doi
2841:doi
2776:doi
2737:doi
2706:doi
2657:doi
2613:doi
2564:doi
2531:doi
2492:doi
2437:doi
2383:doi
2326:doi
2286:doi
2234:doi
2230:100
2183:doi
2125:doi
2095:doi
2061:doi
2012:doi
1970:doi
1899:doi
1847:doi
1812:doi
1762:doi
1683:doi
1649:doi
1603:doi
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1440:doi
1397:doi
1301:log
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285:or
124:by
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