Polygon

( Polygṓnion of ancient Greek πολυγώνιον, polygon '; due to πολύς polys, much ' and γωνία Gonia, angle ') polygon or polygon is a term used in geometry and in particular the planimetry. A polygon is a geometric figure obtained by reacting at least three distinct points in the plane of the drawing ( the corners) by stretching ( the edges ) to one another, so that an area enclosed by the resultant trace (polygon ). These resulting surface is often called polygon. Triangles, squares and hexagons are from everyday life known examples of polygons.

• 3.1 Comparison of the values
• 3.2 calculations
• 3.3 naming familiar polygons

Mathematical definition and designations

A polygon is a figure, which is defined by a tuple of different points.

And are referred to as sides of the polygon.

Sometimes additional conditions for the definition of a polygon are provided, but which mathem. are not necessary:

• The polygon has at least three pairwise different from each other vertices. This includes " gon " from.

Mathematical relationships

Interior angle

In a non- battered, flat -Eck is the sum of the interior angles

In addition, where all interior angles equal, they have the value

Number of diagonals

For not troubled polygons following consideration applies for calculating the number of diagonals:

So a non- proposed -gon has exactly diagonal.

Surface

Gaussian trapezoidal formula

If the vertices of a planar simple (see below) polygon are given by Cartesian coordinates, one can calculate the area of the polygon using the Gaussian trapezoidal formula:

Where the indices that are greater than ever modulo must be considered, that is, is meant by:

As determinants:

Signed sum of triangle areas

In addition to the Gaussian trapezoidal formula, the area of ​​a polygon can be calculated by a signed sum of the areas of triangles formed by the edges of the polygon as a base and a fixed point (eg the origin ) as a point. The areas of the triangles with a fixed point remote from the base ( as an edge of the polygon ) are provided with a negative sign.

Approximation

In computer science are important approximations of complex polygons, the convex hull and the surrounding minimal rectangle. In algorithms is often tested only on the basis of the approximation to a possible non- empty intersection with another geometric object (or the excluded ) and only then the whole polygon is loaded into the memory, and calculates a more precise cut.

Regular Polygons

Polygons can be equilateral or equiangular. A polygon having equal sides, and the same interior angle, then it is referred to as a regular or regular polygon. Regular polygons are isogonal, ie its vertices are equidistant, ie under the same central angle in a circle.

A regular -gon always has a radius of radius and an inscribed circle with radius. The length of each side is labeled with the number of pages. This yields the following formulas for regular, non- battered polygons arise:

Comparison of the values

Calculations

Calculation of the characteristics of constructible polygons ( = radius of the circumcircle )

From the side length of an N -gon, the side length of a 2N -gon with the same radius as follows can be calculated:

With

Naming familiar polygons

• Triangle ( Trigon )
• Quadrangle ( Tetragon ) kite quadrilateral
• Parallelogram
• Square
• Diamond
• Rectangle
• Trapeze

Special types of polygons

• Cutting (touching) the edges not only at the vertices is called the polygon as a roll over.
• If the intersection of two edges is either empty or a vertex, and each vertex belongs to at most two edges ( that is, there is no self-intersection in front ), defined as the polygon as easy.
• Not proposed to convex polygons ( all interior angles are less than 180 °) or nichtkonvex (at least one internal angle greater than 180 °) to be.
• A distinction is in-plane (planar ) and lying in the space ( non-planar ) polygons.
• Planar turned-over regular polygons are also called star polygons because of their appearance.
• Wherein all of the edges orthogonal polygons meet at a right angle (ie, the internal angle of each edge is either 90 ° or 270 °).

Famous polygons

• The " Pentagon " ( seat of the U.S. Department of Defense );
• The Pentagon in Kronach: Fortress Rosenberg shows a pentagon as the floor plan;
• France is also called a hexagon because of its geographical shape;
• The Carolingian octagon in plan view of the Aachen Cathedral;
• Castel del Monte in Apulia as octagon in plan.

Polygons in computer graphics

In 3D computer graphics in addition to other methods of geometric modeling arbitrary (also curved ) surfaces are modeled as a polygon mesh. Triangle meshes are particularly well suited for quick display of surfaces, however, can not be so well interpolated by subdivision surfaces. For storing polygonal networks, there are a number of known data structures.

Not Mathematical Meaning

Some polygons have in addition to the geometrical meaning as a form of architecture (Example Pentagon ) or in the symbolism ( eg pentagram ).