Unit disc

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Image:Unit disc.svg The open unit disc around P (where P is a given point in the plane), is the set of points whose distance from P is less than one:

<math>D_1(P) = \{ Q : \vert P-Q\vert<1\}</math>.

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

<math>\bar D_1(P)=\{Q:|P-Q| \leq 1\}</math>.

Unit discs are a special case of unit balls.

Without further specifications, the term unit disc is used for the open unit disc about the origin, <math>D_1(0)</math>, with respect to the standard Euclidean metric. It looks like the interior of an ordinary circle of radius 1, centered at the origin. This set can be identified with the set of all complex numbers of absolute value less than one.

1 Unit discs with respect to other metrics
2 The open unit disk, the plane, and the upper half plane
3 Hyperbolic space
4 See also
5 References
6 External links

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Unit discs with respect to other metrics

One also considers unit discs with respect to other metrics. For instance, with the taxicab metric and the Chebyshev metric discs look like squares (even though the underlying topologies are the same as the Euclidean one).

The area of the Euclidean unit disk is π and its perimeter is 2π. In contrast, the perimeter (relative to the taxicab metric) of the unit disc in the taxicab geometry is 8. In 1932, Stanisław Gołąb proved that in metrics arising from a norm, the perimeter of the unit disc can take any value in between 6 and 8, and that these extremal values are obtained if and only if the unit disc is a regular hexagon respectively a parallelogram.

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The open unit disk, the plane, and the upper half plane

The function

is an example of a real analytic and bijective function from the unit disk to the plane; its inverse function is also analytic. So considered as a real 2-dimensional analytic manifold, the open unit disc is isomorphic to the whole plane. In particular, the open unit disk is homeomorphic to the whole plane.

There is however no conformal bijective map between the unit disc and the plane. So considered as a Riemann surface, the unit disc is different from the complex plane.

There is a conformal map between the open unit disc and the open upper half plane, for instance the Möbius transformation

So as Riemann surface, the open unit disc is isomorphic to the upper half plane, and the two are often used interchangeably.

Much more generally, the Riemann mapping theorem states that every simply connected open subset of the complex plane that is different from the complex plane itself admits a conformal and bijective map to the open unit disc.

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