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{{Other uses2|Disc}}
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[[Image:Unit disk open.svg|thumb|right|An open Euclidean unit disk]]
In [[mathematics]], the '''open unit disk''' (or '''disc''') around ''P'' (where ''P'' is a given point in the [[plane (mathematics)|plane]]), is the set of points whose distance from ''P'' is less than 1:
 
:<math>D_1(P) = \{ Q : \vert P-Q\vert<1\}.\,</math>
 
The '''closed unit disk''' 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 disks are special cases of [[disk (mathematics)|disks]] and [[unit ball]]s.
 
Without further specifications, the term ''unit disk'' is used for the open unit disk about the [[origin (mathematics)|origin]], <math>D_1(0)</math>, with respect to the [[Euclidean distance|standard Euclidean metric]]. It is the interior of a [[circle]] of radius 1, centered at the origin. This set can be identified with the set of all [[complex number]]s of [[absolute value]] less than one. When viewed as a subset of the complex plane ('''C'''), the unit disk is often denoted <math>\mathbb{D}</math>.
 
== The open unit disk, the plane, and the upper half-plane ==
The function
 
:<math>f(z)=\frac{z}{1-|z|^2}</math>
 
is an example of a real [[analytic function|analytic]] and [[bijective]] function from the open unit disk to the plane; its inverse function is also analytic. Considered as a real 2-dimensional [[analytic manifold]], the open unit disk is therefore isomorphic to the whole plane. In particular, the open unit disk is [[homeomorphic]] to the whole plane.
 
There is however no [[conformal map|conformal]] bijective map between the open unit disk and the plane. Considered as a [[Riemann surface]], the open unit disk is therefore different from the [[complex plane]].  
 
There are conformal bijective maps between the open unit disk and the open [[upper half-plane]]. So considered as a Riemann surface, the open unit disk is isomorphic ("biholomorphic", or "conformally equivalent") 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 set|open subset]] of the complex plane that is different from the complex plane itself admits a conformal and bijective map to the open unit disk.
 
One bijective conformal map from the open unit disk to the open upper half-plane is the [[Möbius transformation]]
 
:<math>g(z)=i\frac{1+z}{1-z}</math> &nbsp; which is the inverse of the [[Cayley transform#Conformal map|Cayley transform]].
 
Geometrically, one can imagine the real axis being bent and shrunk so that the upper half-plane becomes the disk's interior and the real axis forms the disk's circumference, save for one point at the top, the "point at infinity". A bijective conformal map from the open unit disk to the open upper half-plane can also be constructed as the composition of two [[stereographic projection]]s: first the unit disk is stereographically projected upward onto the unit upper half-sphere, taking the "south-pole" of the unit sphere as the projection center, and then this half-sphere is projected sideways onto a vertical half-plane touching the sphere, taking the point on the half-sphere opposite to the touching point as projection center.
 
The unit disk and the upper half-plane are not interchangeable as domains for [[Hardy spaces]]. Contributing to this difference is the fact that the unit circle has finite (one-dimensional) [[Lebesgue measure]] while the real line does not.
 
==Hyperbolic space==
The open unit disk is commonly used as a model for the [[Hyperbolic space|hyperbolic plane]], by introducing a new metric on it, the [[Poincaré metric]]. Using the above mentioned conformal map between the open unit disk and the upper half-plane, this model can be turned into the [[Poincaré half-plane model]] of the hyperbolic plane. Both the Poincaré disk and the Poincaré half-plane are ''conformal'' models of hyperbolic space, i.e. angles measured in the model coincide with angles in hyperbolic space, and consequently the shapes (but not the sizes) of small figures are preserved.
 
Another model of hyperbolic space is also built on the open unit disk: the [[Klein model]]. It is not conformal, but has the property that straight lines in the model correspond to straight lines in hyperbolic space.
 
== Unit disks with respect to other metrics ==
[[Image:Unit disc.svg|thumb|right|From top to bottom: open unit disk in the [[Euclidean metric]], [[taxicab metric]], and [[Chebyshev metric]].]]
One also considers unit disks with respect to other [[Metric (mathematics)|metric]]s. For instance, with the [[taxicab geometry|taxicab metric]] and the [[Chebyshev distance|Chebyshev metric]] disks look like squares (even though the underlying [[topological space|topologies]] are the same as the Euclidean one).
 
The area of the Euclidean unit disk is [[&pi;]] and its [[perimeter]] is 2π. In contrast, the perimeter (relative to the taxicab metric) of the unit disk in the taxicab geometry is 8. In 1932, [[Stanisław Gołąb]] proved that in metrics arising from a [[norm (mathematics)|norm]], the perimeter of the unit disk can take any value in between 6 and 8, and that these extremal values are obtained if and only if the unit disk is a regular [[hexagon]] or a [[parallelogram]], respectively.
 
==See also==
* [[Unit disk graph]]
* [[Bieberbach conjecture]]
 
==References==
* S. Golab, "Quelques problèmes métriques de la géometrie de Minkowski", Trav. de l'Acad. Mines Cracovie 6 (1932), 179.
 
==External links==
* {{mathworld | urlname = UnitDisk | title = Unit disk}}
* [http://www.math.poly.edu/~alvarez/pdfs/disc.pdf On the Perimeter and Area of the Unit Disc], by J.C. Álvarez Pavia and A.C. Thompson
 
[[Category:Circles]]
[[Category:One]]

Latest revision as of 14:56, 7 July 2014

Today there are a lot of women who go for professional treatment to get this corrected. The primary purpose of the melanocyte cell is to make melanin. Stir it well to mix the beeswax thoroughly with the oil, then allow it to cool to about 90 degrees. These proteins are collectively known as Functional Keratin, and along with another unusual ingredient called Phytessence Wakame, will leave your skin virtually wrinkle free. There are various home remedies for dry skin available for all.

Kindly consult the product website or a vendor for latest prices. All these individuals are looking for for the way through which they are going to make their skin glance very good. Hydroquinone-induced ochronosis is often difficult to treat, but it may respond to topical steroids and chemical peeling. The skin will get better faster if you let it fall all by itself, when you bath, otherwise you risk the potential of scarring the skin or further darken it. This has so many skin care benefits, one of which is fading acne scars.

This can effectively whiten your skin and can be naturally performed by making a paste, which combines brown sugar and oatmeal with milk. Although the contention that parabens in a single product is limited to low levels, an individual uses multiple products as part of personal hygiene and aesthetic enhancements more than once in a single day. This fruit can ease inflammation and lighten acne scars with consistent use. For more details, read a complete review of Photomatix HDR. The Obagi Nu-Derm system is clinically proven to ensure younger, healthier and radiant looking skin.

Vitamins will also prevent the pimple from developing to a more serious swelling such as boils. While it is true that physical beauty is just skin deep, it is still no reason why people, particularly women, should disregard basic skin care. Its antiseptic and anti-inflammatory effects will help you achieve a flawless skin. Obviously - If you are allergic to bananas, do not use banana peels to whiten your teeth. Darker skin produces more melanin, which makes the skin appear darker.

If you put junk in you, chances are your skin will reflect junk. Finally, Emu Oil also fights against bacteria and inflammation. For those with darker skin colors, these discolorations can appear as patches or areas of ashen-gray skin. Essential ingredients used for the preparation of Golden Glow consist of jatamansi, ocimum sanctum, turmeric and neem. In the time of applying Yogurt on the skin for about ten minutes to a minimum of thirty minutes, the skin is going to be expected to turn into lighter and more vibrant and this really is recognized by many to include a great deal of lactic acid.

If you have any inquiries about in which and how to use natural ways to lighten skin - click through the following post,, you can contact us at our web-page.

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