Geometry | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Four- /other-dimensional | ||||||||||

Geometers | ||||||||||

by name - Aida
- Aryabhata
- Ahmes
- Alhazen
- Apollonius
- Archimedes
- Atiyah
- Baudhayana
- Bolyai
- Brahmagupta
- Cartan
- Coxeter
- Descartes
- Euclid
- Euler
- Gauss
- Gromov
- Hilbert
- Jyeṣṭhadeva
- Kātyāyana
- Khayyám
- Klein
- Lobachevsky
- Manava
- Minkowski
- Minggatu
- Pascal
- Pythagoras
- Parameshvara
- Poincaré
- Riemann
- Sakabe
- Sijzi
- al-Tusi
- Veblen
- Virasena
- Yang Hui
- al-Yasamin
- Zhang
- List of geometers
| ||||||||||

by period
| ||||||||||

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.

In more modern usage, the length 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 **r**.

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 center of the ellipse.^{ [2] } For example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one of them is parallel to the other one. The longest diameter is called the major axis.

The word "diameter" is derived from Greek διάμετρος (*diametros*), "diameter of a circle", from διά (*dia*), "across, through" and μέτρον (*metron*), "measure".^{ [3] } It is often abbreviated **DIA**, **dia**, **d**, or **⌀**.

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 *n*-dimensional convex or non-convex object, such as a hypercube or a set of scattered points. The **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. So, if *A* is the subset, the diameter is

- sup { d(
*x*,*y*) |*x*,*y*∈*A*} .

If the distance function d is viewed here as having codomain **R** (the set of all real numbers), this implies that the diameter of the empty set (the case *A* = ∅) equals −∞ (negative infinity). Some authors prefer to treat the empty set as a special case, assigning it a diameter equal to 0,^{ [4] } which corresponds to taking the codomain of d to be the set of nonnegative reals.

For any solid object or set of scattered points in 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 supremum 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 plane 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 *e* = 0.

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*).

It is similar in size and design to ø, the Latin small letter o with stroke. 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.

The symbol has a Unicode code point at U+2300⌀DIAMETER SIGN (HTML `⌀`

), 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``⌘ Cmd``T` 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` `2``3``0``0``space` It can be obtained in UNIX-like operating systems using a Compose key by pressing, in sequence, ` Compose ``d``i`^{ [6] }

The character will sometimes not display correctly, however, since many fonts do not include it. In many situations the letter ø (the Latin small letter o with stroke) is an acceptable substitute, which in Unicode is U+00F8ø (HTML `ø`

**·**`ø`

). and on a Macintosh by pressing `⌥ Opt``O` (the letter o, not the number 0). In Unix/Linux/ChromeOS systems, it is generated using `ctrl`+`⇧ Shift`+`u` `F``8``space` or `Compose``o``/`. AutoCAD uses U+2205∅EMPTY 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.

- Angular diameter
- Caliper, micrometer, tools for measuring diameters
- Conjugate diameters
- Diameter (group theory), a concept in group theory
- Eratosthenes, who calculated the diameter of the Earth around 240 BC.
- Graph or network diameter
- Hydraulic diameter
- Inside diameter
- Jung's theorem, an inequality relating the diameter to the radius of the smallest enclosing ball
- Semidiameter
- Sauter mean diameter
- Tangent lines to circles
- The diameters of a screwthread
- Ø (disambiguation)

A **circle** is a shape consisting of all points in a plane that are 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. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is **convex** if, given any two points, it contains the whole line segment that joins them. Equivalently, a **convex set** or a **convex region** is a subset that intersect every line into a single line segment . For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex.

In mathematics, an **ellipse** is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity *e*, a number ranging from *e =* 0 to *e* = 1.

In elementary geometry, the property of being **perpendicular** (**perpendicularity**) is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects.

In Euclidean geometry, a **parallelogram** is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

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

In geometry, the **midpoint** is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.

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

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.

In geometry, **focuses** or **foci**, singular **focus**, are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an n-ellipse.

The **degree symbol** or **degree sign**, **°**, is a typographical symbol that is used, among other things, to represent degrees of arc, hours, degrees of temperature, alcohol proof, or diminished quality in musical harmony. The symbol consists of a small raised circle, historically a zero glyph.

In elementary plane geometry, the **power of a point** is a real number *h* that reflects the relative distance of a given point from a given circle. Specifically, the power of a point **P** with respect to a circle *O* of radius *r* is defined by.

In computational geometry, the method of **rotating calipers** is an algorithm design technique that can be used to solve optimization problems including finding the width or diameter of a set of points.

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 (*n*^{th}) 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** 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.

In geometry, a **line segment** is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A **closed line segment** includes both endpoints, while an **open line segment** excludes both endpoints; a **half-open line segment** includes exactly one of the endpoints. In geometry, a line segment is often denoted using a line above the symbols for the two endpoints.

The null sign U+2205∅EMPTY SET denotes the empty set in mathematics. The same letter in linguistics represents zero, the lack of an element. It is commonly used in phonology, morphology, and syntax.

In geometry, two diameters of a conic section are said to be **conjugate** if each chord parallel to one diameter is bisected by the other diameter. For example, two diameters of a circle are conjugate if and only if they are perpendicular.

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.

In geometry, a **fat object** is an object in two or more dimensions, whose lengths in the different dimensions are similar. For example, a square is fat because its length and width are identical. A 2-by-1 rectangle is thinner than a square, but it is fat relative to a 10-by-1 rectangle. Similarly, a circle is fatter than a 1-by-10 ellipse and an equilateral triangle is fatter than a very obtuse triangle.

- ↑ Toussaint, Godfried T. (1983). "Solving geometric problems with the rotating calipers". Proc. MELECON '83, Athens. CiteSeerX 10.1.1.155.5671 .Cite journal requires
`|journal=`

(help) - ↑ Bogomolny, Alexander. "Conjugate Diameters in Ellipse".
*www.cut-the-knot.org*. - ↑ "diameter - Origin and meaning of diameter by Online Etymology Dictionary".
*www.etymonline.com*. - ↑ "Re: diameter of an empty set".
*at.yorku.ca*. - ↑ Korpela, Jukka K. (2006),
*Unicode Explained*, O'Reilly Media, Inc., pp. 23–24, ISBN 978-0-596-10121-3 . - ↑ Monniaux, David. "UTF-8 (Unicode) compose sequence" . Retrieved 2018-07-13.

Look up in Wiktionary, the free dictionary. diameter |

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.