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 itself), because all diameters of a circle or sphere have the same length, this being twice the radius r. d = 2 r ⇒ r = d 2 . displaystyle d=2rquad 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 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 ⌀. Contents 1 Generalizations
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Generalizations[edit] 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.
Sign ⌀ in a technical drawing Sign ⌀ from an
Not to be confused with the Scandinavian letter "Ø", the empty set
symbol "∅" or the greek letter phi (Φ).
The symbol or variable for diameter, ⌀, is similar in size and
design to ø, the Latin small letter o with stroke. In
Look up diameter in Wiktionary, the free dictionary. Angular diameter
Caliper, micrometer, tools for measuring diameters
Conjugate diameters
Notes[edit] ^ Toussaint, Godfried T. (1983). "Solving geometric problems with the
rotating calipers". Proc. MELECON '83, Athens.
^ Cut-the-Knot
^ Online Etymology Dictionary
^ Re: diameter of an empty set
^ Korpela, Jukka K. (2006),
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