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:

 $D_{1}(P)=\{Q:|P-Q|<1\}.$

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

 $\overline{D}_{1}(P)=\{Q:|P-Q|\leq 1\}$

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