Diameter

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Circle with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta. Circle-withsegments.svg
Circle with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta.

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 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 .

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 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, yA } .

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.

Symbol

Sign [?] in a technical drawing Technical Drawing Hole 01.svg
Sign ⌀ in a technical drawing
Sign
.mw-parser-output .monospaced{font-family:monospace,monospace}
U+2205 [?]
.mw-parser-output span.smallcaps{font-variant:small-caps}.mw-parser-output span.smallcaps-smaller{font-size:85%}
EMPTY SET from an AutoCAD drawing in dim.shx font with an angle 16deg. This font does not contain
U+2300 [?]
DIAMETER SIGN. Sign diameter.png
Sign U+2205EMPTY SET from an AutoCAD drawing in dim.shx font with an angle 16°. This font does not contain U+2300DIAMETER SIGN.

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

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.

Encodings

The symbol has a Unicode code point at U+2300DIAMETER 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⌘ 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]

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 ⌥ 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.

See also

Related Research Articles

Circle Simple curve of Euclidean geometry

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Convex set In geometry, set that intersects every line into a single line segment

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.

Ellipse Plane curve: conic section

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.

Perpendicular Relationship between two lines that meet at a right angle (90 degrees)

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Parallelogram quadrilateral with two pairs of parallel sides

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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.

Midpoint middle point of a line segment

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.

Hyperbolic orthogonality

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.

Focus (geometry) fixed reference point; special points with reference to which any of a variety of curves is constructed

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.

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Rotating calipers Measure method in Computational Geometry

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Conic section Curve obtained by intersecting a cone and a plane

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Line segment Part of a line that is bounded by two distinct end points; line with two endpoints

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The null sign U+2205EMPTY 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.

Rytzs construction method of finding axes and veritces of an ellipse

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.

References

  1. 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)
  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.