Regular amount

This article covers regular sets in geometry and topology; for regular sets in measure theory see regular measure , for regular sets in computer science see regular language .

A regular set (blue) in the plane completely contains its border (dark blue) and has no real one-dimensional parts

A regular set is a subset of Euclidean space in geometry that is equal to the closure of its interior . A regular set therefore has no really low-dimensional parts and completely contains its boundary . Regular set operations such as intersection , union , difference and complement can be defined on regular sets . Regular sets are used in particular in geometric modeling and computer graphics , in a more general context they are also considered in topology .

definition

A subset of Euclidean space is called regular if ${\displaystyle S\subseteq \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle S={\overline {S^{\circ }}}}$

holds, where the inside and the end of a set denote. A regular set is characterized by the fact that it is equal to the closure of its interior. The amount of the regular amounts in is denoted by. ${\displaystyle X^{\circ }}$ ${\displaystyle {\overline {X}}}$ ${\displaystyle X\subseteq \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle {\mathcal {R}}_{n}}$

Examples

Examples of regular quantities are:

• closed intervals on the number line ${\displaystyle \mathbb {R} ^{1}}$
• geometric figures in the Euclidean plane , such as polygons , circular disks or ellipses ${\displaystyle \mathbb {R} ^{2}}$
• geometric bodies in Euclidean space such as polyhedra , cylinders or spheres ${\displaystyle \mathbb {R} ^{3}}$
• Polytopes , cones and closed half-spaces in the ${\displaystyle \mathbb {R} ^{n}}$
• the empty crowd and the whole space ${\displaystyle \mathbb {R} ^{n}}$

properties

Regular sets have the following properties:

• A regular set is completely -dimensional, so it has no parts of lower dimensions . ${\displaystyle S\in {\mathcal {R}}_{n}}$ ${\displaystyle n}$
• A regular set is closed , so it always contains its entire margin . ${\displaystyle S}$ ${\displaystyle \partial S}$
• A regular set does not have to be connected , but can also consist of several components. It can also have holes or cavities.
• A regular amount can also be unlimited .

Operations

Regularization

Example of a regularization

The regularization of a set is the operation ${\displaystyle X\subseteq \mathbb {R} ^{n}}$

${\displaystyle \operatorname {reg} \colon {\mathcal {P}}(\mathbb {R} ^{n})\to {\mathcal {R}}_{n},\quad X\mapsto {\overline {X^{\circ }}}}$,

where represents the power set . The associated regular set is accordingly assigned to a set by regularization . A regular set is characterized precisely by the fact that it is equal to its own regularization, that is, it holds. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle X\subseteq \mathbb {R} ^{n}}$ ${\displaystyle \operatorname {reg} (X)\in {\mathcal {R}}_{n}}$ ${\displaystyle S\in {\mathcal {R}}_{n}}$ ${\displaystyle \operatorname {reg} (S)=S}$

Regular set operations

Constructive solid geometry with regular set operations

With the help of the regularization operation the following regular set operations can be defined for the union , the intersection and the difference between two regular sets : ${\displaystyle \operatorname {reg} }$ ${\displaystyle \cup ^{\ast },\cap ^{\ast },\setminus ^{\ast }\colon {\mathcal {R}}_{n}\times {\mathcal {R}}_{n}\to {\mathcal {R}}_{n}}$ ${\displaystyle S,T\in {\mathcal {R}}_{n}}$

${\displaystyle S\cup ^{\ast }T=\operatorname {reg} (S\cup T)=S\cup T}$

${\displaystyle S\cap ^{\ast }T=\operatorname {reg} (S\cap T)}$

${\displaystyle S\setminus ^{\ast }T=\operatorname {reg} (S\setminus T)}$

In addition, there is the regular complement formation of a set : ${\displaystyle \mathrm {C} ^{\ast }\colon {\mathcal {R}}_{n}\to {\mathcal {R}}_{n}}$ ${\displaystyle S\in {\mathcal {R}}_{n}}$

${\displaystyle \mathrm {C} ^{\ast }S=\operatorname {reg} (\mathrm {C} \,S)}$

The regular amounts under these regular set operations completed . The tuple is also a Boolean algebra . In the three-dimensional space, the regular set operations form the basic framework for the constructive solid geometry ( Constructive Solid Geometry ). ${\displaystyle {\mathcal {R}}_{n}}$ ${\displaystyle ({\mathcal {R}}_{n},\cup ^{\ast },\cap ^{\ast },\mathrm {C} ^{\ast })}$

generalization

Regular sets can also be viewed more generally in topological spaces . A subset of a topological space is called regularly closed , if ${\displaystyle S\subseteq X}$ ${\displaystyle (X,{\mathcal {T}})}$

${\displaystyle S={\overline {S^{\circ }}}}$

applies, and regular open , if

${\displaystyle S={\overline {S}}^{\circ }}$

applies. A subset of a topological space is regularly closed if and only if its complement is regularly open. With the partial order and the corresponding regular set operations, both the regularly open and the regularly closed subsets of a topological space each form a complete Boolean algebra. A topological space, whose regular open subsets form a basis of the space, is called semi- regular . Every regular space , i.e. every topological space in which all points have surrounding bases from closed sets, is also semi-regular and thus also has a base from regularly open subsets. ${\displaystyle \subseteq }$

literature

• Hans-Joachim Bungartz, Michael Griebel, Christoph Zenger: Introduction to Computer Graphics: Basics, Geometric Modeling and Algorithms . Springer, 2013, ISBN 978-3-322-92925-9 .
• Beat Brüderlin, Andreas Meier: Computer graphics and geometric modeling . Springer, 2013, ISBN 978-3-322-80111-1 .
• James D. Foley, Andries van Dam, Steven K. Feiner, John F. Hughes: Computer Graphics: Principles and Practice . Addison-Wesley, 1996, ISBN 978-0-201-84840-3 .

Individual evidence

1. ^ ^{A b c} Hans-Joachim Bungartz, Michael Griebel, Christoph Zenger: Introduction to Computer Graphics . Springer, 2013, p. 55 .
2. ↑ ^{a b} Beat Brüderlin, Andreas Meier: Computer graphics and geometric modeling . Springer, 2013, p. 196 .
3. ↑ James D. Foley, Andries van Dam, Steven K. Feiner, John F. Hughes: Computer Graphics: Principles and Practice . Addison-Wesley, 1996, p. 535-539 .
4. ↑ KP Hart, Jun-iti Nagata, JE Vaughan: Encyclopedia of General Topology . Elsevier, 2003, p. 8 .
5. ^ Roman Sikorski: Boolean Algebras . Springer, 2013, p. 66 .
6. ^ Pavel S. Aleksandrov: Textbook of set theory . 7th edition. Harri Deutsch, 2001, ISBN 3-8171-1657-8 , pp. 122 .

Web links

• MI Voitsekhovskii: Canonical set . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Cwoo: Regular open set . In: PlanetMath . (English)