Traverse (mathematics)

An open traverse

A closed traverse

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 ${\displaystyle P_{1},P_{2},\dotsc ,P_{m}}$

${\displaystyle [P_{1}P_{2}]\cup [P_{2}P_{3}]\cup \cdots \cup [P_{m-1}P_{m}]}$

Line or polygon from to . If they fall and collapse, one speaks of a closed polygon course , otherwise of an open polygon course . ${\displaystyle P_{1}}$ ${\displaystyle P_{m}}$ ${\displaystyle P_{1}}$ ${\displaystyle P_{m}}$

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 . ${\displaystyle P_{1},\ldots ,P_{m-1}}$ ${\displaystyle [P_{1}P_{2}],\ldots ,[P_{m-1}P_{m}]}$

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 ${\displaystyle V}$ ${\displaystyle x_{1},x_{2},\dotsc ,x_{m}\in V}$

${\displaystyle {\mathcal {P}}=\bigcup _{i=1}^{m-1}[x_{i}x_{i+1}]}$

the routes

${\displaystyle [x_{i}x_{i+1}]=\{(1-\lambda )x_{i}+\lambda x_{i+1}\mid \lambda \in [0,1]\}}$

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 . ${\displaystyle x_{1}}$ ${\displaystyle x_{m}}$ ${\displaystyle V}$

Rectifiability

Polygonal lines play an essential role for the length measurement of curves in dimensional space. ${\displaystyle n}$

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 . ${\displaystyle {\mathcal {K}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle x_{1}}$ ${\displaystyle x_{\text{ende}}}$ ${\displaystyle x_{1},x_{2},\dotsc ,x_{m}=x_{\text{ende}}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle x_{1}}$ ${\displaystyle x_{\text{ende}}}$

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 : ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle L({\mathcal {K}})}$ ${\displaystyle x_{1}}$ ${\displaystyle x_{\text{ende}}}$

A curve with the continuous parameterization can be rectified precisely when the coordinate functions are of limited variation . ${\displaystyle {\mathbb {R} }^{n}}$ ${\displaystyle \gamma =({\gamma }_{1},\dots ,{\gamma }_{n})\colon I\to {\mathbb {R} }^{n}}$   ${\displaystyle (I=[a,b]\subset \mathbb {R} )}$ ${\displaystyle {\gamma }_{1},\dots ,{\gamma }_{n}}$

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 . ${\displaystyle G}$ ${\displaystyle n}$ ${\displaystyle G}$ ${\displaystyle G}$

See also

• polyhedron
• simplex
• Way (math)

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

1. ^ Willi Rinow : Textbook of Topology . Deutscher Verlag der Wissenschaften, Berlin 1975, p. 22-23 .
2. ↑ Harro Heuser : Textbook of Analysis. Part 2 (=  mathematical guidelines ). 5th revised edition. Teubner Verlag, Wiesbaden 1990, ISBN 3-519-42222-0 , pp. 349 ff .
3. ↑ 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 .
4. ^ 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 .
5. ↑ Rudolf Bereis: Descriptive Geometry I (=  Math textbooks and monographs . Band 11 ). Akademie-Verlag , Berlin 1964, p. 117 ff .
6. ^ György Hajós : Introduction to Geometry . BG Teubner Verlag, Leipzig, p. 32 ff . (Hungarian: Bevezetés A Geometriába . Translated by G. Eisenreich [Leipzig, also editor]).
7. ↑ 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. ${\displaystyle {\mathcal {P}}}$
8. ↑ 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 .
9. ↑ 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 .
10. ↑ 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 .
11. ↑ 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 .
12. ↑ 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 .
13. ^ Willi Rinow : Textbook of Topology . Deutscher Verlag der Wissenschaften, Berlin 1975, p. 150 .