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# Diameter ( Elementary Geometry )

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Rank: 100%     In , a diameter of a is any straight 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 . The word "diameter" is derived from διάμετρος ( diametros), "diameter of a circle", from δια- ( dia-), "across, through" μέτρον ( metron), "measure". Online Etymology Dictionary 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 r.

$d = 2r \quad \Rightarrow \quad r = \frac\left\{d\right\}\left\{2\right\}.$

For a in the plane, the diameter is defined to be the largest distance that can be formed between two opposite tangent to its boundary, and the width is defined to be the smallest such distance. Both quantities can be calculated efficiently using . For a curve of constant width such as the , the width and diameter are the same because all such pairs of parallel tangent lines have the same distance.

For an , the standard terminology is different. A diameter of an ellipse is any chord passing through the midpoint of the ellipse.http://www.cut-the-knot.org/Curriculum/Geometry/ConjugateDiameters.shtml Cut-the-Knot 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 .

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 or a set of scattered points. The diameter of a of a is the of the set of all distances between pairs of points in the subset. So, if A is the subset, the diameter is
{ d( x, y) | x, yA } .
If the d is viewed here as having R (the set of all ), this implies that the diameter of the (the case ) equals −∞ (). Some authors prefer to treat the empty set as a special case, assigning it a diameter equal to 0, Re: diameter of an empty set 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 , the diameter of the object or set is the same as the diameter of its . 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 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.

Diameter symbol
The or variable for diameter, ⌀, is to ø, the Latin small letter o with stroke. In it is defined as . On an Apple , the diameter symbol can be entered via the character palette (this is opened by pressing in most applications), where it can be found in the Technical Symbols category.

The character will sometimes not display correctly, however, since many do not include it. In many situations the letter ø is an acceptable substitute, which in Unicode is . It can be obtained in UNIX-like operating systems using a by pressing, in sequence, and on a Macintosh by pressing (the letter o, not the number 0).

In the diameter symbol can be acquired by typing 2300 and then pressing Alt X.

In the diameter symbol can be obtained with the command \diameter from the wasysym package.

The diameter symbol ⌀ is distinct from the symbol ∅, from an () uppercase phi Φ, and from the Nordic vowel Ø..

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