SherbyNet - /page/wikibook/entry.mu

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Disc_(disambiguation)" class="mw-redirect mw-disambig" title="Disc (disambiguation)">Disc (disambiguation)</a>.</div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>open unit disk</b> (or <b>disc</b>) around <i>P</i> (where <i>P</i> is a given point in the <a href="Plane_(mathematics)" title="Plane (mathematics)">plane</a>), is the set of points whose distance from <i>P</i> is less than 1:

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<annotation encoding="application/x-tex">{\displaystyle D_{1}(P)=\{Q:\vert P-Q\vert <1\}.\,}</annotation>

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</math></span><img src="./8f7cbe54a7a4b2a0ae79471f591ab893b9727187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.745ex; height:2.843ex;" alt="{\displaystyle D_{1}(P)=\{Q:\vert P-Q\vert <1\}.\,}" loading="lazy"></span></dd></dl>

<p>The <b>closed unit disk</b> around <i>P</i> is the set of points whose distance from <i>P</i> is less than or equal to one:

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<annotation encoding="application/x-tex">{\displaystyle {\bar {D}}_{1}(P)=\{Q:|P-Q|\leq 1\}.\,}</annotation>

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</math></span><img src="./285469e34acb6006195d4bf0fa4df27e1f725cdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.745ex; height:3.009ex;" alt="{\displaystyle {\bar {D}}_{1}(P)=\{Q:|P-Q|\leq 1\}.\,}" loading="lazy"></span></dd></dl>

<p>Unit disks are special cases of <a href="Disk_(mathematics)" title="Disk (mathematics)">disks</a> and <a href="Unit_ball" class="mw-redirect" title="Unit ball">unit balls</a>; as such, they contain the interior of the <a href="Unit_circle" title="Unit circle">unit circle</a> and, in the case of the closed unit disk, the unit circle itself.

</p><p>Without further specifications, the term <i>unit disk</i> is used for the open unit disk about the <a href="Origin_(mathematics)" title="Origin (mathematics)">origin</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{1}(0)}">

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<annotation encoding="application/x-tex">{\displaystyle D_{1}(0)}</annotation>

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</math></span><img src="./44b8b170ccd7ad61c55cf138411488d92790ad7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.95ex; height:2.843ex;" alt="{\displaystyle D_{1}(0)}" loading="lazy"></span>, with respect to the <a href="Euclidean_distance" title="Euclidean distance">standard Euclidean metric</a>. It is the interior of a <a href="Circle" title="Circle">circle</a> of radius 1, centered at the origin. This set can be identified with the set of all <a href="Complex_number" title="Complex number">complex numbers</a> of <a href="Absolute_value" title="Absolute value">absolute value</a> less than one. When viewed as a subset of the complex plane (<b>C</b>), the unit disk is often denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {D} }</annotation>

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</math></span><img src="./b932b553742ca27776057f1262527014ebbb46a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {D} }" loading="lazy"></span>.

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<div class="mw-heading mw-heading2"><h2 id="The_open_unit_disk,_the_plane,_and_the_upper_half-plane">The open unit disk, the plane, and the upper half-plane</h2></div>

<p>The function

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<annotation encoding="application/x-tex">{\displaystyle f(z)={\frac {z}{1-|z|^{2}}}}</annotation>

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</math></span><img src="./ce405ba74d309fcb0d68e00970dafe4f83b8aa9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.55ex; height:5.843ex;" alt="{\displaystyle f(z)={\frac {z}{1-|z|^{2}}}}" loading="lazy"></span></dd></dl>

<p>is an example of a real <a href="Analytic_function" title="Analytic function">analytic</a> and <a href="Bijective" class="mw-redirect" title="Bijective">bijective</a> function from the open unit disk to the plane; its inverse function is also analytic. Considered as a real 2-dimensional <a href="Analytic_manifold" title="Analytic manifold">analytic manifold</a>, the open unit disk is therefore isomorphic to the whole plane. In particular, the open unit disk is <a href="Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to the whole plane.

</p><p>There is however no <a href="Conformal_map" title="Conformal map">conformal</a> bijective map between the open unit disk and the plane. Considered as a <a href="Riemann_surface" title="Riemann surface">Riemann surface</a>, the open unit disk is therefore different from the <a href="Complex_plane" title="Complex plane">complex plane</a>.

</p><p>There are conformal bijective maps between the open unit disk and the open <a href="Upper_half-plane" title="Upper half-plane">upper half-plane</a>. 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.

</p><p>Much more generally, the <a href="Riemann_mapping_theorem" title="Riemann mapping theorem">Riemann mapping theorem</a> states that every <a href="Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a> <a href="Open_set" title="Open set">open subset</a> of the complex plane that is different from the complex plane itself admits a conformal and bijective map to the open unit disk.

</p><p>One bijective conformal map from the open unit disk to the open upper half-plane is the <a href="M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformation</a>

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<annotation encoding="application/x-tex">{\displaystyle g(z)=i{\frac {1+z}{1-z}}}</annotation>

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</math></span><img src="./e37b24ad5b904b97f8357aafa65e88731168f3fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.841ex; height:5.343ex;" alt="{\displaystyle g(z)=i{\frac {1+z}{1-z}}}" loading="lazy"></span>   which is the inverse of the <a href="Cayley_transform#Complex_homography" title="Cayley transform">Cayley transform</a>.</dd></dl>

<p>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 <a href="Stereographic_projection" title="Stereographic projection">stereographic projections</a>: 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.

</p><p>The unit disk and the upper half-plane are not interchangeable as domains for <a href="Hardy_spaces" class="mw-redirect" title="Hardy spaces">Hardy spaces</a>. Contributing to this difference is the fact that the unit circle has finite (one-dimensional) <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> while the real line does not.

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<div class="mw-heading mw-heading2"><h2 id="Hyperbolic_plane">Hyperbolic plane</h2></div>

<p>The open unit disk forms the set of points for the <a href="Poincar%C3%A9_disk_model" title="Poincaré disk model">Poincaré disk model</a> of the hyperbolic plane. <a href="Circular_arc" title="Circular arc">Circular arcs</a> perpendicular to the unit circle form the "lines" in this model. The unit circle is the <a href="Cayley_absolute" class="mw-redirect" title="Cayley absolute">Cayley absolute</a> that determines a <a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> on the disk through use of <a href="Cross-ratio" title="Cross-ratio">cross-ratio</a> in the style of the <a href="Cayley%E2%80%93Klein_metric" title="Cayley–Klein metric">Cayley–Klein metric</a>. In the language of differential geometry, the circular arcs perpendicular to the unit circle are <a href="Geodesic" title="Geodesic">geodesics</a> that show the shortest distance between points in the model. The model includes <a href="Motion_(geometry)" title="Motion (geometry)">motions</a> which are expressed by the special unitary group <a href="SU(1%2C1)" class="mw-redirect" title="SU(1,1)">SU(1,1)</a>. The disk model can be transformed to the <a href="Poincar%C3%A9_half-plane_model" title="Poincaré half-plane model">Poincaré half-plane model</a> by the mapping <i>g</i> given above.

</p><p>Both the Poincaré disk and the Poincaré half-plane are <i>conformal</i> models of the hyperbolic plane, which is to say that angles between intersecting curves are preserved by motions of their isometry groups.

</p><p>Another model of hyperbolic space is also built on the open unit disk: the <a href="Beltrami%E2%80%93Klein_model" title="Beltrami–Klein model">Beltrami–Klein model</a>. It is <i>not conformal</i>, but has the property that the geodesics are straight lines.

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<div class="mw-heading mw-heading2"><h2 id="Unit_disks_with_respect_to_other_metrics">Unit disks with respect to other metrics</h2></div>

<p>One also considers unit disks with respect to other <a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metrics</a>. For instance, with the <a href="Taxicab_geometry" title="Taxicab geometry">taxicab metric</a> and the <a href="Chebyshev_distance" title="Chebyshev distance">Chebyshev metric</a> disks look like squares (even though the underlying <a href="Topological_space" title="Topological space">topologies</a> are the same as the Euclidean one).

</p><p>The area of the Euclidean unit disk is <a href="Pi" title="Pi">π</a> and its <a href="Perimeter" title="Perimeter">perimeter</a> is 2π. In contrast, the perimeter (relative to the taxicab metric) of the unit disk in the taxicab geometry is 8. In 1932, <a href="Stanis%C5%82aw_Go%C5%82%C4%85b" title="Stanisław Gołąb">Stanisław Gołąb</a> proved that in metrics arising from a <a href="Norm_(mathematics)" title="Norm (mathematics)">norm</a>, 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 <a href="Hexagon" title="Hexagon">hexagon</a> or a <a href="Parallelogram" title="Parallelogram">parallelogram</a>, respectively.

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<ul><li><a href="Unit_disk_graph" title="Unit disk graph">Unit disk graph</a></li>

<li><a href="Unit_sphere" title="Unit sphere">Unit sphere</a></li>

<li><a href="De_Branges's_theorem" title="De Branges's theorem">De Branges's theorem</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li>S. Golab, "Quelques problèmes métriques de la géometrie de Minkowski", Trav. de l'Acad. Mines Cracovie 6 (1932), 179.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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</style><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/UnitDisk.html">"Unit disk"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060907065432/http://www.math.poly.edu/%7Ealvarez/pdfs/disc.pdf">On the Perimeter and Area of the Unit Disc</a>, by J.C. Álvarez Pavia and A.C. Thompson</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-15" href="https://en.wikipedia.org/wiki/?title=Unit_disk&oldid=1285664502">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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