Diameter: Difference between revisions
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{{Short description|Straight line segment that passes through the |
{{Short description|Straight line segment that passes through the centre of a circle}} |
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{{Other uses}} |
{{Other uses}} |
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{{use British English|date=March 2021}}<!-- so centre not center --> |
{{use British English|date=March 2021}}<!-- so centre not center --> |
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{{legend-line|blue solid 2px|diameter ''D''}} |
{{legend-line|blue solid 2px|diameter ''D''}} |
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{{legend-line|red solid 2px|radius ''R''}} |
{{legend-line|red solid 2px|radius ''R''}} |
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{{legend-line|green solid 2px| |
{{legend-line|green solid 2px|centre or origin ''O''}}]] |
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{{General geometry}} |
{{General geometry}} |
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In [[geometry]], a '''diameter''' of a [[circle]] is any straight [[line segment]] that passes through the |
In [[geometry]], a '''diameter''' of a [[circle]] is any straight [[line segment]] that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest [[Chord (geometry)|chord]] of the circle. Both definitions are also valid for the diameter of a [[sphere]]. |
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In more modern usage, the length <math>d</math> of a diameter is also called the diameter. In this sense one speaks of {{em|the}} diameter rather than {{em|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]] <math>r.</math> |
In more modern usage, the length <math>d</math> of a diameter is also called the diameter. In this sense one speaks of {{em|the}} diameter rather than {{em|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]] <math>r.</math> |
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:<math>d = 2r \qquad\text{or equivalently}\qquad r = \frac{d}{2}.</math> |
:<math>d = 2r \qquad\text{or equivalently}\qquad r = \frac{d}{2}.</math> |
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For a [[convex set|convex]] shape in the [[plane (geometry)|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 {{em|width}} is often defined to be the smallest such distance. Both quantities can be calculated efficiently using [[rotating calipers]].<ref>{{cite |
For a [[convex set|convex]] shape in the [[plane (geometry)|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 {{em|width}} is often defined to be the smallest such distance. Both quantities can be calculated efficiently using [[rotating calipers]].<ref>{{cite conference |last=Toussaint |first=Godfried T. |contribution=Solving geometric problems with the rotating calipers |title=Proc. MELECON '83 |conference=Mediterranean Electrotechnical Conference, 24-26 May 1983, Athens |year=1983 |publisher=IEEE |citeseerx=10.1.1.155.5671 |contribution-url=http://www-cgrl.cs.mcgill.ca/~godfried/publications/calipers.pdf }} (pdf pages in reversed order)</ref> 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. |
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{{Anchor|Ellipse}} |
{{Anchor|Ellipse}} |
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== Symbol == |
== Symbol == |
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{{redirect|⌀}} |
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[[Image:Technical Drawing Hole 01.svg|thumb|122px|Sign ⌀ in a technical drawing]] |
[[Image:Technical Drawing Hole 01.svg|thumb|122px|Sign ⌀ in a technical drawing]] |
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[[File:Hama UV Filter 1.jpg|thumb|A [[photographic filter]] marked as having a 58mm thread diameter]] |
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[[Image:Sign diameter.png|thumb|150px|Sign {{unichar|2205|EMPTY SET|ulink=Mathematical Operators}} from an [[AutoCAD]] drawing in dim.shx font with an angle 16°. This font does not contain {{unichar|2300|DIAMETER SIGN|ulink=Miscellaneous Technical}}.]] |
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The [[symbol]] or [[Variable (mathematics)|variable]] for diameter, {{char|⌀}}, 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.<ref>{{cite book|title=Interpretation of Geometric Dimensioning and Tolerancing|first=Daniel E.|last=Puncochar|publisher=Industrial Press Inc.|year=1997|isbn=9780831130725|page=5|url=https://books.google.com/books?id=76la9qNmt24C&pg=PA5}}</ref> Photographic [[filter thread]] sizes are often denoted in this way.<ref>{{cite book|title=Introduction to Digital Photography|first=Joseph|last=Ciaglia|publisher=Prentice Hall|year=2002|isbn=9780130321367|page=9|quote=The filter diameter (in mm) usually follows the symbol ⌀}}</ref> |
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{{distinguish|text=the Scandinavian letter "[[Ø]]", the [[empty set]] symbol {{italics correction|"}}''∅''", the [[slashed zero]], or the greek letter [[phi]] (Φ)}} |
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The symbol has a [[Unicode]] [[code point]] at {{unichar|2300|Diameter sign|html=}}, in the [[Miscellaneous Technical]] set, and should not be confused with several other Unicode characters that resemble it but have unrelated meanings.<ref>{{cite book|title=Unicode Explained|first=Jukka K.|last=Korpela|publisher=O'Reilly Media, Inc.|year=2006|isbn=9780596101213|page=171|url=https://books.google.com/books?id=lxndiWaFMvMC&pg=PA171}}</ref> It has the [[compose sequence]] {{key press|[[Compose key|Compose]]|d|i|chain=}}.<ref>{{cite web |url=http://cgit.freedesktop.org/xorg/lib/libX11/plain/nls/en_US.UTF-8/Compose.pre |title=UTF-8 (Unicode) compose sequence |last=Monniaux |first=David |access-date=2018-07-13}}</ref> |
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The [[symbol]] or [[Variable (mathematics)|variable]] for diameter, {{char|⌀}}, 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. |
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In [[German (language)|German]], the diameter symbol (German ''[[:de:Durchmesserzeichen|Durchmesserzeichen]]'') is also used as an [[average]] symbol (''Durchschnittszeichen''). |
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=== Similar symbols === |
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The [[Ø|Latin small letter o with stroke]] {{char|ø}} is [[Homoglyph|similar in size and design]] to this. The diameter symbol ⌀ is distinct from the [[empty set]] symbol {{char|∅}}, from an ([[Italic script|italic]]) uppercase [[Phi (letter)|phi]] {{char|''Φ''}}, and from the Nordic vowel {{char|Ø}} ([[Ø|Latin capital letter O with stroke]]).<ref>{{citation|title=Unicode Explained|first=Jukka K.|last=Korpela|publisher=[[O'Reilly Media, Inc.]]|year=2006|isbn=978-0-596-10121-3|pages=23–24|url=https://books.google.com/books?id=lxndiWaFMvMC&pg=PA23}}.</ref> See also [[slashed zero]]. |
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=== Encodings === |
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The symbol has a [[Unicode]] [[code point]] at {{unichar|2300|Diameter sign|html=}}, in the [[Miscellaneous Technical]] set. On an [[Apple Inc.|Apple]] [[Macintosh]], the diameter symbol can be entered via the character palette (this is opened by pressing {{key press|Opt|Cmd|T|chain=}} in most applications), where it can be found in the Technical Symbols category. In Unix/Linux/ChromeOS systems, it is generated using {{key press|Ctrl|Shift|U}} {{key press|2|3|0|0|space|chain=}}. It can be obtained in [[Unix-like]] operating systems using a [[Compose key]] by pressing, in sequence, {{key press|[[Compose key|Compose]]|d|i|chain=}}.<ref>{{cite web |url=http://cgit.freedesktop.org/xorg/lib/libX11/plain/nls/en_US.UTF-8/Compose.pre |title=UTF-8 (Unicode) compose sequence |last=Monniaux |first=David |access-date=2018-07-13}}</ref> In Windows, it can be entered in most programs with [[Alt code]] 8960. |
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The character will sometimes not display correctly, however, since many [[Typeface|font]]s do not include it. In many situations, the Nordic letter [[ø]] at Unicode {{unichar|00F8|Latin small letter o with stroke|html=}} is an [[Typographic approximation|acceptable substitute]]. It can be entered on a Macintosh by pressing {{key press|Opt|O|chain=}} (the letter [[o]], not the number [[0]]). In Unix/Linux/ChromeOS systems, it is generated using {{key press|Ctrl|Shift|U}} {{key press|F|8|space|chain=}} or {{key press|Compose|o|/|chain=}}. [[AutoCAD]] uses {{unichar|2205|EMPTY SET|ulink=Mathematical Operators}} available as a shortcut string {{Kbd|%%c}}. |
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In [[Microsoft Word]], the diameter symbol can be acquired by typing {{key press|2|3|0|0|chain=}} and then pressing {{key press|[[Alt key|Alt]]|X}}. |
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In [[LaTeX]], the diameter symbol can be obtained with the command <code>\diameter</code> from the "wasysym" package.<ref name="wasysym">{{cite web |url=https://ctan.org/pkg/wasysym |title=wasysym – LaTeX support for the wasy fonts |website=[[CTAN|Comprehensive TeX Archive Network]] |access-date=2022-03-11}}</ref> |
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== Diameter vs. radius == |
== Diameter vs. radius == |
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The diameter of a circle is exactly twice its radius. However, this is true only for a circle, and only in the [[Euclidean distance|Euclidean metric]]. |
The diameter of a circle is exactly twice its radius. However, this is true only for a circle, and only in the [[Euclidean distance|Euclidean metric]]. [[Jung's theorem]] provides more general inequalities relating the diameter to the radius. |
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== See also == |
== See also == |
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{{Div col|colwidth=30em}} |
{{Div col|colwidth=30em}} |
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* {{annotated link|Angular diameter}} |
* {{annotated link|Angular diameter}} |
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* {{annotated link|Tangent lines to circles}} |
* {{annotated link|Tangent lines to circles}} |
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* The [[Screw thread#Diameters|diameters]] of a screwthread |
* The [[Screw thread#Diameters|diameters]] of a screwthread |
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* {{annotated link|Ø (disambiguation)}} |
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{{div col end}} |
{{div col end}} |
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Revision as of 00:04, 6 May 2024
Geometry |
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Geometers |
In geometry, a diameter of a circle is any straight line segment that passes through the centre 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
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 or
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 -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 is the subset and if is the metric, the diameter is
If the metric is viewed here as having codomain (the set of all real numbers), this implies that the diameter of the empty set (the case ) equals (negative infinity). Some authors prefer to treat the empty set as a special case, assigning it a diameter of [4] which corresponds to taking the codomain of to be the set of nonnegative reals.
For any solid object or set of scattered points in -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
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.[5] Photographic filter thread sizes are often denoted in this way.[6]
The symbol has a Unicode code point at U+2300 ⌀ DIAMETER SIGN, in the Miscellaneous Technical set, and should not be confused with several other Unicode characters that resemble it but have unrelated meanings.[7] It has the compose sequence Composedi.[8]
Diameter vs. radius
The diameter of a circle is exactly twice its radius. However, this is true only for a circle, and only in the Euclidean metric. Jung's theorem provides more general inequalities relating the diameter to the radius.
See also
- Angular diameter – How large a sphere or circle appears
- Caliper, micrometer, tools for measuring diameters
- Conjugate diameters – Perpendicular diameters of a circle or hyperbolic-orthogonal diameters of a hyperbola
- Diameter (group theory) – measure of a finite group's complexity , a concept in group theory
- Eratosthenes, who calculated the diameter of the Earth around 240 BC.
- Graph or network diameter – Length of shortest path between two nodes of a graph
- Hydraulic diameter – Measure of a channel flow efficiency
- Inside diameter* Semidiameter – Term in geometry; half of a shape's diameter
- Sauter mean diameter – average of particle size
- Tangent lines to circles – Line which touches a circle at exactly one point
- The diameters of a screwthread
References
- ^ Toussaint, Godfried T. (1983). "Solving geometric problems with the rotating calipers" (PDF). Proc. MELECON '83. Mediterranean Electrotechnical Conference, 24-26 May 1983, Athens. IEEE. CiteSeerX 10.1.1.155.5671. (pdf pages in reversed order)
- ^ 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.
- ^ Puncochar, Daniel E. (1997). Interpretation of Geometric Dimensioning and Tolerancing. Industrial Press Inc. p. 5. ISBN 9780831130725.
- ^ Ciaglia, Joseph (2002). Introduction to Digital Photography. Prentice Hall. p. 9. ISBN 9780130321367.
The filter diameter (in mm) usually follows the symbol ⌀
- ^ Korpela, Jukka K. (2006). Unicode Explained. O'Reilly Media, Inc. p. 171. ISBN 9780596101213.
- ^ Monniaux, David. "UTF-8 (Unicode) compose sequence". Retrieved 2018-07-13.