Polygonal chain

A polygon or polyline is in mathematics, the union of the links of a sequence of points. Polylines are used in many areas of mathematics, such as geometry, numerical analysis, topology, analysis and theory of functions. In addition, they also come in some applications such as in computer graphics or geodesy used.

• 2.1 Definition
• 2.2 rectifiability
• 2.3 related to the area property

Polygons in geometry

Definition

Are points in the Euclidean plane or Euclidean space, then that means the union of the routes

Reference to polygons

The geometric figure whose boundary is formed by a closed polygon, ie polygon, the points are called vertices of the polygon and the routes hot sides of the polygon. If the points lie in a plane, so you call this figure a planar polygon, otherwise a wind chiefes polygon.

Use

Polygons have a variety of applications, for example in the interpolation of data points in the numerical solution of ordinary differential equations with the Euler's Polygonzugverfahren as well as in modeling in computer graphics and computer -aided design. For the purposes of polygons in surveying see polygon (geodesy ).

Polygons in the Analysis

Definition

Let now generally a real vector space and are given elements of the vector space, then that means the union

The routes

Polyline or polygon from to. Is a topological vector space, then these routes are continuous images of the unit interval and thus compact, which then also applies to the finite associations formed from them. Each polyline is always also the example of a continuum.

Rectifiability

Traverses play an essential role in measuring the length of curves in n-dimensional space.

A length alone is declared to be rectifiable curves. For detecting the rectifiability is considered for a given curve all polygons, the corners of which passes through the curve, which are so arranged to constitute the sides of the polygon formed by the corners at the same time of tendons. Such a polygon is also called Sehnenzug or tendons polygon and is said to be inscribed. To determine the rectifiability of the lengths of all inscribed polygons tendons are examined. Is understood as the length of a polygon the sum of the lengths of its routes.

If exists for these wavelengths within an upper bound, then a rectifiable curve, and only then. In this case, the length is defined as the supremum of all lengths tendons inscribed polygons. For the determination of rectifiable curves of the following criteria apply:

Connection with the area property

Polygons also did not play a role in determining when there is a space in the area and when. Here, the following theorem holds: