Diameter
In geometry, the 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. The word "diameter" is derived from Greek διάμετρος (diametros), "diameter of a circle", from δια- (dia-), "across, through" + μέτρον (metron), "measure".^{[1]} It is often abbreviated DIA, dia, d, or ⌀.
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 itself), because all diameters of a circle or sphere have the same length, this being twice the radius r.
- d = 2r \quad \Rightarrow \quad 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 defined to be the smallest such distance. Both quantities can be calculated efficiently using rotating calipers.^{[2]} 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.
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
- supremum { 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,^{[3]} 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 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.
In medical parlance the diameter of a lesion is the longest line segment whose endpoints are within the lesion.
Diameter symbol
The symbol or variable for diameter, ⌀, is similar in size and design to ø, the Latin small letter o with stroke. Unicode provides character number 8960 (hexadecimal 2300) for the symbol, which can be encoded in HTML webpages as ⌀ or ⌀. The character can be obtained in Microsoft Windows by holding the Alt key down while entering 8960 on the numeric keypad. 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.
The character will sometimes not display correctly, however, since many fonts do not include it. In most situations the letter ø is acceptable, which is unicode 0248 (hexadecimal 00F8). It can be obtained in UNIX-like operating systems using a Compose key by pressing, in sequence, Compose/o and on a Macintosh by pressing ⌥ Opt O (in both cases, that is the letter o, not the number 0).
In LaTeX the symbol is achieved with the command \diameter which is part of the wasysym package.
The diameter symbol ⌀ is distinct from the empty set symbol ∅, from an (italic) uppercase phi Φ, and from the Nordic vowel Ø.^{[4]}
See also
- Angular diameter
- Caliper, micrometer, tools for measuring diameters
- 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
- Sauter mean diameter
- Tangent lines to circles
Notes
- ^ Online Etymology Dictionary
- ^ Toussaint, Godfried T. (1983). "Solving geometric problems with the rotating calipers". Proc. MELECON '83, Athens.
- ^ Re: diameter of an empty set
- ^ Korpela, Jukka K. (2006), Unicode Explained, O'Reilly Media, Inc., pp. 23–24, .