# Traverse (mathematics)

In mathematics, a **polygon** or **line** is the union of the connecting lines of a sequence of points . Polygonal lines are used in many branches of mathematics , for example in geometry , numerics , topology , analysis and function theory . In addition, they are also used in some application areas such as computer graphics or geodesy .

## Polygon courses in the geometry

### definition

If points are in the Euclidean plane or in Euclidean space , then it is called the union of the lines

Line or polygon from to . If they fall and collapse, one speaks of a **closed polygon course** , otherwise of an **open polygon course** .

### Relation to polygons

The geometric figure , the edge of which is formed by a closed polygon course, is called a polygon , the points are called *corner points of* the polygon and the lines are called the *sides of* the polygon. If the points lie in one plane , this figure is called a *flat polygon* , otherwise a *skewed polygon* .

### use

Polygon courses have a wide range of possible uses, for example in the interpolation of data points, in the numerical solution of ordinary differential equations with Euler's polygon method , and in modeling in computer graphics and computer-aided design . For the use of polygons in surveying, see polygon (geodesy) .

## Polygon courses in analysis

### definition

Now, let in general be a real vector space and given elements of the vector space, then the union is called

the routes

Line or polygon from to . Is a topological vector space , these routes are continuous images of the unit interval and compact, which then also for the formed from them finite associations applies. Every route is always an example of a continuum .

### Rectifiability

Polygonal lines play an essential role for the length measurement of curves in dimensional space.

A *length* is only declared for rectifiable curves. To prove the rectifiability, one considers for a given curve all polygons from to , through the corners of which the curve runs in this order, which are made in such a way that the sides of the polygon formed by the corners also represent chords from . Such a polygon is also called *Sehnenzug* or *tendon polygon* designated and is said to *be **inscribed* . To determine the rectifiability of between and the *lengths of all inscribed chord polygons are* examined. The *length of a polygon is* the *sum of the lengths of its lines* .
* *

If there is an upper bound for all these lengths within , then there is a rectifiable curve, and only then. In this case the **length is defined** as the *supremum of** all lengths of inscribed chord polygons* (everything for the curve segment to ). The following criterion applies to determining the rectifiability of curves :

- A curve with the continuous parameterization can be rectified precisely when the
*coordinate functions are*of limited variation .

### Connection with the property of the area

The polygons also play a role in determining when there is an area in space and when it is not. The following sentence applies here :

- An open subset of a topological vector space (and in particular of the -dimensional space) is connected if and only if two points of can be connected by a polygon lying entirely in .

## See also

## literature

- Rudolf Bereis: Descriptive Geometry I (= mathematical textbooks and monographs . Volume 11 ). Akademie-Verlag, Berlin 1964.
- Charles O. Christenson, William L. Voxman: Aspects of Topology (= Monographs and Textbooks in Pure and Applied Mathematics . Volume 39 ). Marcel Dekker, New York / Basel 1977, ISBN 0-8247-6331-9 .
- Jürgen Elstrodt : Measure and integration theory (= basic knowledge of mathematics (Springer textbook) ). 6th, corrected edition. Berlin / Heidelberg 2009, ISBN 978-3-540-89727-9 .
- György Hajós : Introduction to Geometry . BG Teubner Verlag, Leipzig (Hungarian: Bevezetés A Geometriába . Translated by G. Eisenreich [Leipzig, also editing]).
- Michael Henle: A Combinatorial Introduction to Topology (= A Series of Books in Mathematical Sciences ). WH Freeman and Company, San Francisco 1979, ISBN 0-7167-0083-2 .
- Harro Heuser : Textbook of Analysis. Part 2 (= mathematical guidelines ). 5th revised edition. Teubner Verlag, Wiesbaden 1990, ISBN 3-519-42222-0 .
- Konrad Knopp : Theory of functions I. Basics of the general theory of analytical functions (= Göschen Collection . Volume 668 ). Walter de Gruyter Verlag, Berlin 1965.
- Willi Rinow : Textbook of Topology . German Science Publishers, Berlin 1975.
- Hans von Mangoldt , Konrad Knopp : Introduction to higher mathematics . 13th edition. Volume 2: differential calculus, infinite series, elements of differential geometry and function theory. S. Hirzel Verlag, Stuttgart 1968.
- Hans von Mangoldt , Konrad Knopp : Introduction to higher mathematics . 13th edition. Volume 3: Integral calculus and its applications, function theory, differential equations. S. Hirzel Verlag, Stuttgart 1967.

## References and comments

- ^ Willi Rinow : Textbook of Topology . Deutscher Verlag der Wissenschaften, Berlin 1975, p. 22-23 .
- ↑ Harro Heuser : Textbook of Analysis. Part 2 (= mathematical guidelines ). 5th revised edition. Teubner Verlag, Wiesbaden 1990, ISBN 3-519-42222-0 , pp. 349 ff .
- ↑ Hans von Mangoldt , Konrad Knopp : Introduction to higher mathematics . 13th edition. Volume 2: differential calculus, infinite series, elements of differential geometry and function theory. S. Hirzel Verlag, Stuttgart 1968, p. 296 ff .
- ^ Charles O. Christenson, William L. Voxman: Aspects of Topology (= Monographs and Textbooks in Pure and Applied Mathematics . Volume 39 ). Marcel Dekker, New York / Basel 1977, ISBN 0-8247-6331-9 , pp. 63-64 .
- ↑ Rudolf Bereis: Descriptive Geometry I (= Math textbooks and monographs . Band 11 ). Akademie-Verlag , Berlin 1964, p. 117 ff .
- ^ György Hajós : Introduction to Geometry . BG Teubner Verlag, Leipzig, p. 32 ff . (Hungarian: Bevezetés A Geometriába . Translated by G. Eisenreich [Leipzig, also editor]).
- ↑ As a rule, the borderline case that only consists of a single line or even just a single point is excluded. Polygon courses usually consist of at least two lines.
- ↑ Hans von Mangoldt , Konrad Knopp : Introduction to higher mathematics . 13th edition. Volume 3: Integral calculus and its applications, function theory, differential equations. S. Hirzel Verlag, Stuttgart 1967, p. 306-307 .
- ↑ Hans von Mangoldt , Konrad Knopp : Introduction to higher mathematics . 13th edition. Volume 2: differential calculus, infinite series, elements of differential geometry and function theory. S. Hirzel Verlag, Stuttgart 1968, p. 415 ff .
- ↑ Hans von Mangoldt , Konrad Knopp : Introduction to higher mathematics . 13th edition. Volume 3: Integral calculus and its applications, function theory, differential equations. S. Hirzel Verlag, Stuttgart 1967, p. 224 ff .
- ↑ Jürgen Elstrodt : Measure and integration theory (= basic knowledge of mathematics (Springer textbook) ). 6th, corrected edition. Berlin / Heidelberg 2009, ISBN 978-3-540-89727-9 , pp. 78, 308 ff .
- ↑ Konrad Knopp : Function Theory I. Fundamentals of the General Theory of Analytical Functions (= Göschen Collection . Volume 668 ). Walter de Gruyter Verlag, Berlin 1965, p. 22-23 .
- ^ Willi Rinow : Textbook of Topology . Deutscher Verlag der Wissenschaften, Berlin 1975, p. 150 .