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166:: Any two convex polygons with no points in common have a separator line. If the polygons are closed and at least one of them is compact, then there are even two parallel separator lines (with a gap between them).
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Every line segment between two points in the interior, or between two points on the boundary but not on the same edge, is strictly interior to the polygon (except at its endpoints if they are on the edges).
150:: For every collection of at least three convex polygons: if all intersections of all but one polygon are nonempty, then the intersection of all the polygons is nonempty.
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of its vertices. Thus it is fully defined by the set of its vertices, and one only needs the corners of the polygon to recover the entire polygon shape.
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property: Of all triangles contained in a convex polygon, there exists a triangle with a maximal area whose vertices are all polygon vertices.
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For each edge, the interior points and the boundary points not contained in the edge are on the same side of the line that the edge defines.
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between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a
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are less than or equal to 180 degrees, while in a strictly convex polygon all interior angles are strictly less than 180 degrees.
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The angle at each vertex contains all other vertices in its interior (except the given vertex and the two adjacent vertices).
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Chandran, Sharat; Mount, David M. (1992). "A parallel algorithm for enclosed and enclosing triangles".
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Every polygon inscribed in a circle (such that all vertices of the polygon touch the circle), if not
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For each edge, the interior points are all on the same side of the line that the edge defines.
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between two points inside or on the boundary of the polygon remains inside or on the boundary.
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is a convex polygon such that no line contains two of its edges. In a convex polygon, all
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The polygon is entirely contained in a closed half-plane defined by each of its edges.
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The following properties of a simple polygon are all equivalent to strict convexity:
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419:{\displaystyle 0.5{\text{ × Area}}(R)\leq {\text{Area}}(C)\leq 2{\text{ × Area}}(r)}
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453:. So its width is the diameter of a circle with the same perimeter as the polygon.
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The angle at each vertex contains all other vertices in its edges and interior.
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The following properties of a simple polygon are all equivalent to convexity:
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461:, is convex. However, not every convex polygon can be inscribed in a circle.
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that does not contain any edge intersects the polygon in at most two points.
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Definition and properties of convex polygons with interactive animation.
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Lassak, M. (1993). "Approximation of convex bodies by rectangles".
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International
Journal of Computational Geometry & Applications
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Implicit curve § Smooth approximation of convex polygons
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The intersection of two convex polygons is a convex polygon.
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of a convex polygon is equal to its perimeter divided by
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can be inscribed in a triangle of area at most equal to
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Pages displaying short descriptions of redirect targets
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Every internal angle is strictly less than 180 degrees.
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355:and the positive homothety ratio is at most 2 and
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125:Additional properties of convex polygons include:
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653:"What's the average width of a convex polygon?"
63:). Equivalently, a polygon is convex if every
692:http://www.rustycode.com/tutorials/convex.html
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561:
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16:Polygon that is the boundary of a convex set
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737:
508: – Simple polygon which is not convex
251:in the plane, we can inscribe a rectangle
697:Schorn, Peter; Fisher, Frederick (1994),
178:property: every convex polygon with area
699:"I.2 Testing the convexity of a polygon"
18:
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221:. Equality holds (exclusively) for a
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23:An example of a convex polygon: a
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520: – Points on a common circle
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231:property: For every convex body
229:Inscribed/inscribing rectangles
701:, in Heckbert, Paul S. (ed.),
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164:Hyperplane separation theorem
94:is less than or equal to 180
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711:(Academic Press), pp.
502: – Type of plane curve
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156:: A convex polygon is the
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599:"Triangle Circumscribing"
576:10.1142/S0218195992000123
132:A convex polygon may be
335:is circumscribed about
72:strictly convex polygon
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51:. This means that the
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486:Every non-degenerate
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101:Every point on every
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984:Nonagon/Enneagon (9)
914:Tangential trapezoid
490:is strictly convex.
446:{\displaystyle \pi }
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154:Krein–Milman theorem
1096:Megagon (1,000,000)
864:Isosceles trapezoid
657:Math Stack Exchange
618:Geometriae Dedicata
597:Weisstein, Eric W.
176:Inscribing triangle
117:The polygon is the
1066:Icositetragon (24)
675:Weisstein, Eric W.
630:10.1007/BF01263495
603:Wolfram Math World
532:Tangential polygon
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416:
345:
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214:{\displaystyle 2A}
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170:Inscribed triangle
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1286:Types of polygons
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1267:
1109:
1108:
1086:Myriagon (10,000)
1071:Triacontagon (30)
1035:Heptadecagon (17)
1025:Pentadecagon (15)
1020:Tetradecagon (14)
959:Quadrilateral (4)
829:Antiparallelogram
459:self-intersecting
405:
404: × Area
385:
368:
367: × Area
348:{\displaystyle C}
328:{\displaystyle r}
308:{\displaystyle R}
284:{\displaystyle C}
264:{\displaystyle r}
244:{\displaystyle C}
191:{\displaystyle A}
142:fan triangulation
61:self-intersecting
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1081:Chiliagon (1000)
1061:Icositrigon (23)
1040:Octadecagon (18)
1030:Hexadecagon (16)
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704:Graphics Gems IV
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678:"Convex polygon"
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465:Strict convexity
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1015:Tridecagon (13)
1005:Hendecagon (11)
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512:Convex polytope
506:Concave polygon
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148:Helly's theorem
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76:interior angles
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1010:Dodecagon (12)
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667:External links
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570:(2): 191–214.
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518:Cyclic polygon
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879:Parallelogram
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874:Orthodiagonal
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854:Ex-tangential
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1054:>20 sides
989:Decagon (10)
974:Heptagon (7)
964:Pentagon (5)
954:Triangle (3)
849:Equidiagonal
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500:Convex curve
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291:such that a
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134:triangulated
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103:line segment
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53:line segment
43:that is the
36:
30:
1250:Star-shaped
1225:Rectilinear
1195:Equilateral
1190:Equiangular
1154:Hendecagram
998:11–20 sides
979:Octagon (8)
969:Hexagon (6)
944:Monogon (1)
786:Equilateral
651:Belk, Jim.
624:: 111–117.
158:convex hull
138:linear time
119:convex hull
1275:Categories
1255:Tangential
1159:Dodecagram
937:1–10 sides
928:By number
909:Tangential
889:Right kite
539:References
431:mean width
293:homothetic
140:through a
82:Properties
49:convex set
1235:Reinhardt
1144:Enneagram
1134:Heptagram
1124:Pentagram
1091:65537-gon
949:Digon (2)
919:Trapezoid
884:Rectangle
834:Bicentric
796:Isosceles
773:Triangles
683:MathWorld
638:119508642
441:π
397:≤
380:≤
27:pentagon.
1210:Isotoxal
1205:Isogonal
1149:Decagram
1139:Octagram
1129:Hexagram
930:of sides
859:Harmonic
760:Polygons
494:See also
488:triangle
45:boundary
33:geometry
1230:Regular
1175:Concave
1168:Classes
1076:257-gon
899:Rhombus
839:Crossed
584:1168956
96:degrees
41:polygon
25:regular
1240:Simple
1185:Cyclic
1180:Convex
904:Square
844:Cyclic
806:Obtuse
801:Kepler
719:
636:
582:
90:Every
1215:Magic
811:Right
791:Ideal
781:Acute
634:S2CID
295:copy
59:(not
47:of a
39:is a
1245:Skew
869:Kite
764:List
717:ISBN
713:7–15
429:The
384:Area
65:line
35:, a
626:doi
572:doi
363:0.5
315:of
271:in
136:in
31:In
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186:A
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