Английская Википедия:Annulus (mathematics)

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An annulus
An annulus
Файл:Mamikon annulus area visualisation.svg
Illustration of Mamikon's visual calculus method showing that the areas of two annuli with the same chord length are the same regardless of inner and outer radii.[1]

In mathematics, an annulus (Шаблон:Plural form: annuli or annuluses) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word anulus or annulus meaning 'little ring'. The adjectival form is annular (as in annular eclipse).

The open annulus is topologically equivalent to both the open cylinder Шаблон:Math and the punctured plane.

Area

The area of an annulus is the difference in the areas of the larger circle of radius Шаблон:Math and the smaller one of radius Шаблон:Math:

<math>A = \pi R^2 - \pi r^2 = \pi\left(R^2 - r^2\right).</math>
Файл:Annuli with same area around unit regular polygons.svg
As a corollary of the chord formula, the area bounded by the circumcircle and incircle of every unit convex regular polygon is Шаблон:Pi/4

The area of an annulus is determined by the length of the longest line segment within the annulus, which is the chord tangent to the inner circle, Шаблон:Math in the accompanying diagram. That can be shown using the Pythagorean theorem since this line is tangent to the smaller circle and perpendicular to its radius at that point, so Шаблон:Math and Шаблон:Math are sides of a right-angled triangle with hypotenuse Шаблон:Math, and the area of the annulus is given by

<math>A = \pi\left(R^2 - r^2\right) = \pi d^2.</math>

The area can also be obtained via calculus by dividing the annulus up into an infinite number of annuli of infinitesimal width Шаблон:Math and area Шаблон:Math and then integrating from Шаблон:Math to Шаблон:Math:

<math>A = \int_r^R\!\! 2\pi\rho\, d\rho = \pi\left(R^2 - r^2\right).</math>

The area of an annulus sector of angle Шаблон:Math, with Шаблон:Math measured in radians, is given by

<math> A = \frac{\theta}{2} \left(R^2 - r^2\right). </math>

Complex structure

In complex analysis an annulus Шаблон:Math in the complex plane is an open region defined as

<math> r < |z - a| < R. </math>

If Шаблон:Math is Шаблон:Math, the region is known as the punctured disk (a disk with a point hole in the center) of radius Шаблон:Math around the point Шаблон:Math.

As a subset of the complex plane, an annulus can be considered as a Riemann surface. The complex structure of an annulus depends only on the ratio Шаблон:Math. Each annulus Шаблон:Math can be holomorphically mapped to a standard one centered at the origin and with outer radius 1 by the map

<math>z \mapsto \frac{z - a}{R}.</math>

The inner radius is then Шаблон:Math.

The Hadamard three-circle theorem is a statement about the maximum value a holomorphic function may take inside an annulus.

The Joukowsky transform conformally maps an annulus onto an ellipse with a slit cut between foci.

See also

References

Шаблон:Reflist

External links

Шаблон:Wiktionary

Шаблон:Compact topological surfaces