Editing Distance (section)

==Distances in physics and geometry<!--This article violates WP:NOBACKREF all over the place, but I think it sounds better that way, and in some places genuinely requires it.  Please think twice before changing it-->==

The distance between physical locations can be defined in different ways in different contexts.

===Straight-line or Euclidean distance===
{{main|Euclidean distance}}

The distance between two points in physical [[space]] is the [[length]] of a [[line segment|straight line]] between them, which is the shortest possible path.  This is the usual meaning of distance in [[classical physics]], including [[Newtonian mechanics]]. 

Straight-line distance is formalized mathematically as the [[Euclidean distance]] in [[two-dimensional Euclidean space|two-]] and [[three-dimensional space]].  In [[Euclidean geometry]], the distance between two points {{mvar|A}} and {{mvar|B}} is often denoted <math>|AB|</math>.  In [[Cartesian coordinate system|coordinate geometry]], Euclidean distance is computed using the [[Pythagorean theorem]]. The distance between points {{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>)}} and {{math|(''x''<sub>2</sub>, ''y''<sub>2</sub>)}} in the plane is given by:<ref name=":0">{{Cite web|last=Weisstein|first=Eric W.|title=Distance|url=https://mathworld.wolfram.com/Distance.html|access-date=2020-09-01|website=mathworld.wolfram.com|language=en}}</ref><ref>{{Cite web|title=Distance Between 2 Points|url=https://www.mathsisfun.com/algebra/distance-2-points.html|access-date=2020-09-01|website=www.mathsisfun.com}}</ref>
<math display="block">d=\sqrt{(\Delta x)^2+(\Delta y)^2}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.</math>
Similarly, given points (''x''<sub>1</sub>, ''y''<sub>1</sub>, ''z''<sub>1</sub>) and (''x''<sub>2</sub>, ''y''<sub>2</sub>, ''z''<sub>2</sub>) in three-dimensional space, the distance between them is:<ref name=":0" />
<math display="block">d=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}.</math>
This idea generalizes to higher-dimensional [[Euclidean space]]s.

==== Measurement ====
{{main|Distance measurement}}
There are many ways of [[measuring]] straight-line distances. For example, it can be done directly using a [[ruler]], or indirectly with a [[radar]] (for long distances) or [[interferometry]] (for very short distances).  The [[cosmic distance ladder]] is a set of ways of measuring extremely long distances.

===Shortest-path distance on a curved surface===
[[File:Greatcircle Jetstream routes.svg|thumb|400px|Airline routes between [[Los Angeles]] and [[Tokyo]] approximately follow a [[great circle]] going west (top) but use the [[jet stream]] (bottom) when heading eastwards. The shortest route appears as a curve rather than a straight line because the [[map projection]] does not scale all distances equally compared to the real spherical surface of the Earth.]]
{{main|Geographic distance|geodesic}}

The straight-line distance between two points on the surface of the Earth is not very useful for most purposes, since we cannot tunnel straight through the [[Earth's mantle]].  Instead, one typically measures the shortest path along the [[surface of the Earth]], [[as the crow flies]].  This is approximated mathematically by the [[great-circle distance]] on a sphere.

More generally, the shortest path between two points along a [[surface (mathematics)|curved surface]] is known as a [[geodesic]].  The [[arc length]] of geodesics gives a way of measuring distance from the perspective of an [[ant]] or other flightless creature living on that surface.

===Effects of relativity===
{{main|Distance measure}}
In the [[theory of relativity]], because of phenomena such as [[length contraction]] and the [[relativity of simultaneity]], distances between objects depend on a choice of [[inertial frame of reference]].  On galactic and larger scales, the measurement of distance is also affected by the [[expansion of the universe]].  In practice, a number of [[distance measure]]s are used in [[cosmology]] to quantify such distances.

===Other spatial distances===
[[File:Manhattan distance.svg|thumb|200px|[[Manhattan distance]] on a grid]]
Unusual definitions of distance can be helpful to model certain physical situations, but are also used in theoretical mathematics:
* In practice, one is often interested in the travel distance between two points along roads, rather than as the crow flies.  In a [[grid plan]], the travel distance between street corners is given by the [[Manhattan distance]]: the number of east–west and north–south blocks one must traverse to get between those two points.
* Chessboard distance, formalized as [[Chebyshev distance]], is the minimum number of moves a [[king (chess)|king]] must make on a [[chessboard]] in order to travel between two squares.