# Diameter

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In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere.

## Contents

In more modern usage, the length ${\displaystyle d}$ of a diameter is also called the diameter. In this sense one speaks of the diameter rather than a diameter (which refers to the line segment itself), because all diameters of a circle or sphere have the same length, this being twice the radius ${\displaystyle r.}$

${\displaystyle d=2r\qquad {\text{or equivalently}}\qquad r={\frac {d}{2}}.}$

For a convex shape in the plane, the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and the width is often defined to be the smallest such distance. Both quantities can be calculated efficiently using rotating calipers. [1] For a curve of constant width such as the Reuleaux triangle, the width and diameter are the same because all such pairs of parallel tangent lines have the same distance.

For an ellipse, the standard terminology is different. A diameter of an ellipse is any chord passing through the centre of the ellipse. [2] For example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one diameter is parallel to the conjugate diameter. The longest diameter is called the major axis.

The word "diameter" is derived from Ancient Greek : διάμετρος (diametros), "diameter of a circle", from διά (dia), "across, through" and μέτρον (metron), "measure". [3] It is often abbreviated ${\displaystyle {\text{DIA}},{\text{dia}},d,}$ or ${\displaystyle \varnothing .}$

## Generalizations

The definitions given above are only valid for circles, spheres and convex shapes. However, they are special cases of a more general definition that is valid for any kind of ${\displaystyle n}$-dimensional (convex or non-convex) object, such as a hypercube or a set of scattered points. The diameter or metric diameter of a subset of a metric space is the least upper bound of the set of all distances between pairs of points in the subset. Explicitly, if ${\displaystyle S}$ is the subset and if ${\displaystyle \rho }$ is the metric, the diameter is

${\displaystyle \operatorname {diam} (S)=\sup _{x,y\in S}\rho (x,y).}$

If the metric ${\displaystyle \rho }$ is viewed here as having codomain ${\displaystyle \mathbb {R} }$ (the set of all real numbers), this implies that the diameter of the empty set (the case ${\displaystyle S=\varnothing }$) equals ${\displaystyle -\infty }$ (negative infinity). Some authors prefer to treat the empty set as a special case, assigning it a diameter of ${\displaystyle 0,}$ [4] which corresponds to taking the codomain of ${\displaystyle d}$ to be the set of nonnegative reals.

For any solid object or set of scattered points in ${\displaystyle n}$-dimensional Euclidean space, the diameter of the object or set is the same as the diameter of its convex hull. In medical parlance concerning a lesion or in geology concerning a rock, the diameter of an object is the least upper bound of the set of all distances between pairs of points in the object.

In differential geometry, the diameter is an important global Riemannian invariant.

In planar geometry, a diameter of a conic section is typically defined as any chord which passes through the conic's centre; such diameters are not necessarily of uniform length, except in the case of the circle, which has eccentricity ${\displaystyle e=0.}$

## Symbol

The symbol or variable for diameter, , is sometimes used in technical drawings or specifications as a prefix or suffix for a number (e.g. "⌀ 55 mm"), indicating that it represents diameter. For example, photographic filter thread sizes are often denoted in this way.

In German, the diameter symbol (German Durchmesserzeichen ) is also used as an average symbol (Durchschnittszeichen).

### Similar symbols

The Latin small letter o with stroke ø is similar in size and design to this. The diameter symbol ⌀ is distinct from the empty set symbol , from an (italic) uppercase phi Φ, and from the Nordic vowel Ø (Latin capital letter O with stroke). [5] See also slashed zero.

### Encodings

The symbol has a Unicode code point at U+2300DIAMETER SIGN, in the Miscellaneous Technical set. On an Apple Macintosh, the diameter symbol can be entered via the character palette (this is opened by pressing ⌥ Opt⌘ CmdT in most applications), where it can be found in the Technical Symbols category. In Unix/Linux/ChromeOS systems, it is generated using Ctrl+⇧ Shift+U 2300space. It can be obtained in Unix-like operating systems using a Compose key by pressing, in sequence, Compose di. [6] In Windows, it can be entered in most programs with Alt code 8960.

The character will sometimes not display correctly, however, since many fonts do not include it. In many situations, the Nordic letter ø at Unicode U+00F8øLATIN SMALL LETTER O WITH STROKE (&oslash;) is an acceptable substitute. It can be entered on a Macintosh by pressing ⌥ OptO (the letter o, not the number 0). In Unix/Linux/ChromeOS systems, it is generated using Ctrl+⇧ Shift+U F8space or Composeo/. AutoCAD uses U+2205EMPTY SET available as a shortcut string %%c.

In Microsoft Word, the diameter symbol can be acquired by typing 2300 and then pressing Alt +X.

In LaTeX, the diameter symbol can be obtained with the command \diameter from the "wasysym" package. [7]

The diameter of a circle is exactly twice its radius. However, this is true only for a circle, and only in the Euclidean metric. The page on Jung's theorem discusses some more general inequalities relating the diameter to the radius.

## Related Research Articles

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

A sphere is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the centre of the sphere, and r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle. The condition of perpendicularity may be represented graphically using the perpendicular symbol, ⟂. It can be defined between two lines, between a line and a plane, and between two planes.

In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.

In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turns. The winding number depends on the orientation of the curve, and it is negative if the curve travels around the point clockwise.

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold. It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

Ø is a letter used in the Danish, Norwegian, Faroese, and Southern Sámi languages. It is mostly used as a representation of mid front rounded vowels, such as (listen) and (listen), except for Southern Sámi where it is used as an diphthong.

In mathematics, the upper half-plane, is the set of points (x, y) in the Cartesian plane with y > 0.

In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.

In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked together closely in Leiden in the 1920s on the spacetime structure of the universe.

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

A cylinder has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.

In geometry, the relation of hyperbolic orthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity to define simultaneous events. Two events will be simultaneous when they are on a line hyperbolically orthogonal to a particular time line. This dependence on a certain time line is determined by velocity, and is the basis for the relativity of simultaneity.

Given the problem of the aerodynamic design of the nose cone section of any vehicle or body meant to travel through a compressible fluid medium, an important problem is the determination of the nose cone geometrical shape for optimum performance. For many applications, such a task requires the definition of a solid of revolution shape that experiences minimal resistance to rapid motion through such a fluid medium.

In elementary plane geometry, the power of a point is a real number that reflects the relative distance of a given point from a given circle. It was introduced by Jakob Steiner in 1826.

In general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.

In geometry, a Steiner chain is a set of n circles, all of which are tangent to two given non-intersecting circles, where n is finite and each circle in the chain is tangent to the previous and next circles in the chain. In the usual closed Steiner chains, the first and last circles are also tangent to each other; by contrast, in open Steiner chains, they need not be. The given circles α and β do not intersect, but otherwise are unconstrained; the smaller circle may lie completely inside or outside of the larger circle. In these cases, the centers of Steiner-chain circles lie on an ellipse or a hyperbola, respectively.

In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.

The Rytz’s axis construction is a basic method of descriptive geometry to find the axes, the semi-major axis and semi-minor axis and the vertices of an ellipse, starting from two conjugated half-diameters. If the center and the semi axis of an ellipse are determined the ellipse can be drawn using an ellipsograph or by hand.

## References

1. Toussaint, Godfried T. (1983). "Solving geometric problems with the rotating calipers". Proc. MELECON '83, Athens. CiteSeerX  .{{cite web}}: Missing or empty |url= (help)
2. Bogomolny, Alexander. "Conjugate Diameters in Ellipse". www.cut-the-knot.org.
3. "diameter - Origin and meaning of diameter by Online Etymology Dictionary". www.etymonline.com.
4. "Re: diameter of an empty set". at.yorku.ca.
5. Korpela, Jukka K. (2006), Unicode Explained, O'Reilly Media, Inc., pp. 23–24, ISBN   978-0-596-10121-3 .
6. Monniaux, David. "UTF-8 (Unicode) compose sequence" . Retrieved 2018-07-13.
7. "wasysym – LaTeX support for the wasy fonts". Comprehensive TeX Archive Network . Retrieved 2022-03-11.