Convex polygon

Convex Polygon A simple polygon is a finite region of the plane bounded by a non-self-sealing closed broken line. Taking into account how a simple polygon is with respect to a line that contains two of its points, two classes of polygons result: convex polygons and concave polygons.

Summary

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• 1 Definition
  • 1 Examples
• 2 elements
  • 1 Topological concepts
• 3 Features
• 4 Sources
• 5 References
• 6 See also

Definition

A polygon is convex when a line that contains any of its sides determines two half planes, such that in one of them the whole polygon remains; in case no such property occurs the polygon “is not convex”. ^[one]

Examples

• Triangles
• Square, rectangle, rhombus, rhomboid, trapezoid and trapezoids.
• Pentagons, hexagons, octagons, enegons, decagons, hexagons, dodecagons, etc.

Elements

• Side of the polygon is any segment of the broken line
• Vertex any end of one side, this vertex on two sides called adjacent
• Diagonal: Any segment that joins two non-consecutive vertices.
• From the vertex of an entering angle, a decomposition into convex figures with a smaller number of sides is possible, if possible into triangles.

Topological concepts

• Inner point of a convex polygon. If we draw any line through point I that cuts two sides of the polygon at M and N respectively, if I is between M and N it is an interior pointof the polygon.
• Interior: It is the set of all interior points.
• Polygonal region: It is the union of the polygon and its interior.
• Border is the broken line that delimits the polygon.
• Exterior point: It is a point that is not in the polygonal region. The set of all exterior points is the exteriorof the polygon.
• The interior, exterior and polygon establish a plane partition, so that the intersection of two of them is empty and the union of the three sets is equal to the entire plane.
• Accumulation point of a polygon Π is a point P _athat has an environment V (P _a  ; δ) so that it contains at least one point Π other than P _a .
• Isolated point of a polygon K is a point P _ithat is in K and that has an environment V (P _i  ; δ), which does not contain any point of K, outside it.

characteristics

1. In a convex polygon all its diagonals are on the same polygon; otherwise they are inside except for the ends that are on the border.
2. Any two different points on the polygon determine a segment that is on the polygon.
3. If a line is drawn through an interior point, it is cut at least two of its sides.
4. Three-sided polygons ( triangles) are the only polygons that cannot be concave, since none of their three angles can exceed 180 degrees or π radians.