Editing Distance (section)

==Distance between sets==

[[File:Distance between sets.svg|thumb|The distances between these three sets do not satisfy the triangle inequality:<math display="block">d(A,B)>d(A,C)+d(C,B)</math>]]
There are multiple ways of measuring the physical distance between objects that [[extension (metaphysics)|consist of more than one point]]:
* One may measure the distance between representative points such as the [[center of mass]]; this is used for astronomical distances such as the [[Lunar distance (astronomy)|Earth–Moon distance]].
* One may measure the distance between the closest points of the two objects; in this sense, the [[altitude]] of an airplane or spacecraft is its distance from the Earth.  The same sense of distance is used in Euclidean geometry to define [[distance from a point to a line]], [[distance from a point to a plane]], or, more generally, [[perpendicular distance]] between [[affine subspace]]s.
: Even more generally, this idea can be used to define the distance between two [[subset]]s of a metric space.  The distance between sets {{mvar|A}} and {{mvar|B}} is the [[infimum]] of the distances between any two of their respective points:<math display="block">d(A,B)=\inf_{x\in A, y\in B} d(x,y).</math> This does not define a metric on the set of such subsets: the distance between overlapping sets is zero, and this distance does not satisfy the triangle inequality for any metric space with two or more points (consider the triple of sets consisting of two distinct singletons and their union).
* The [[Hausdorff distance]] between two subsets of a metric space can be thought of as measuring how far they are from perfectly overlapping.  Somewhat more precisely, the Hausdorff distance between {{mvar|A}} and {{mvar|B}} is either the distance from {{mvar|A}} to the farthest point of {{mvar|B}}, or the distance from {{mvar|B}} to the farthest point of {{mvar|A}}, whichever is larger.  (Here "farthest point" must be interpreted as a supremum.)  The Hausdorff distance defines a metric on the set of [[compact space|compact subsets]] of a metric space.