Diameter
In
geometry, a
diameter of a
circle is any straight
line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest
chords of the circle. The word "diameter" derives from
Greek διάμετρος (
diametros), "diagonal of a circle", from
δια- (
dia-), "across, through" +
μέτρον (
metron), "a measure").
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, because all diameters of a
circle 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 defined to be the smallest such distance. 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. See also
Tangent lines to circles.
The
diameter of a
connected graph is the distance between the two
vertices which are furthest from each other. The distance between two vertices
a and
b is the length of the shortest path connecting them (for the length of a path, see
Graph theory).
Generalisations
The three definitions given above are special cases of a more general definition. The
diameter of a
subset of a
metric space is the
least upper bound of the distances between pairs of points in the subset. So, if
A is the subset, the diameter is
sup { d(
x,
y) |
x,
y ∈
A } .
In
differential geometry, the diameter is an important global
Riemannian invariant.
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 on the
numeric keypad. On an
Apple Inc. Macintosh, the diameter symbol can be entered via the character palette (this is opened by pressing
⌥⌘T in most applications), where it can be found in the Technical Symbols category.
The character often will not display correctly, however, since most
fonts do not include it. (Your browser displays "⌀" in the current font.) 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 ⌥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 uppercase
phi Φ, and the Nordic vowel Ø.
The diameter also refers to the approximate size of the corner of a frame of any given object to the nearest flat surface it represents.
Graph theory
To find diameter of a
graph, first find the shortest
path between each pair of
vertices. The greatest length of any of these paths is the diameter of the graph.
See also
