unit disc

The open unit disc around $P$ (where $P$ is a given point on the plane), is the set of points whose distance from $P$ is less than one:

A closed unit disc is the set of points whose distance from $P$ is less than or equal to one:

$${\overline{D}}_1(P)=\{Q:|P-Q|\le 1\}$$

Unit discs are a special case of unit ball. Without further specifications, the term ”unit disc” is used for the open unit disc about the origin, $D_1(0)$.

The specific set of points in the unit disc, and hence, its visual appearance, depends on the metric being used.

With the standard metric, unit discs look like circles of radius one, but changing the metric changes also the corresponding set of points and therefore the shape of the discs.

For instance, with the taxicab metric discs look like squares, while on the Chebyshev metric a disc is shaped like a rhombus (even though the underlying topologies are the same as the Euclidean one).

Title	unit disc
Canonical name	UnitDisc
Date of creation	2013-03-22 15:47:01
Last modified on	2013-03-22 15:47:01
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	7
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 51-00
Classification	msc 54-00
Synonym	unit ball
Related topic	UnitDisk
Related topic	OpenBall
Related topic	Ball
Defines	closed unit disk
Defines	open unit disk