Distance is a numerical or occasionally qualitativemeasurement of how far apart objects, points, people, or ideas are. Inphysics or everyday usage, distance may refer to a physicallength or an estimation based on other criteria (e.g. "two counties over"). The term is also frequently used metaphorically[1] to mean a measurement of the amount of difference between two similar objects (such asstatistical distance betweenprobability distributions oredit distance betweenstrings of text) or a degree of separation (as exemplified bydistance between people in asocial network). Most such notions of distance, both physical and metaphorical, are formalized inmathematics using the notion of ametric space.
Straight-line distance is formalized mathematically as theEuclidean distance intwo- andthree-dimensional space. InEuclidean geometry, the distance between two pointsA andB is often denoted. Incoordinate geometry, Euclidean distance is computed using thePythagorean theorem. The distance between points(x1,y1) and(x2,y2) in the plane is given by:[2][3]Similarly, given points (x1,y1,z1) and (x2,y2,z2) in three-dimensional space, the distance between them is:[2]This idea generalizes to higher-dimensionalEuclidean spaces.
There are many ways ofmeasuring straight-line distances. For example, it can be done directly using aruler, or indirectly with aradar (for long distances) orinterferometry (for very short distances). Thecosmic distance ladder is a set of ways of measuring extremely long distances.
Airline routes betweenLos Angeles andTokyo approximately follow agreat circle going west (top) but use thejet stream (bottom) when heading eastwards. The shortest route appears as a curve rather than a straight line because themap projection does not scale all distances equally compared to the real spherical surface of the Earth.
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 theEarth's mantle. Instead, one typically measures the shortest path along thesurface of the Earth,as the crow flies. This is approximated mathematically by thegreat-circle distance on a sphere.
More generally, the shortest path between two points along acurved surface is known as ageodesic. Thearc length of geodesics gives a way of measuring distance from the perspective of anant or other flightless creature living on that surface.
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 agrid plan, the travel distance between street corners is given by theManhattan distance: the number of east–west and north–south blocks one must traverse to get between those two points.
Chessboard distance, formalized asChebyshev distance, is the minimum number of moves aking must make on achessboard in order to travel between two squares.
Many abstract notions of distance used in mathematics, science and engineering represent a degree of difference or separation between similar objects. This page gives a few examples.
In agraph, thedistance between two vertices is measured by the length of the shortestedge path between them. For example, if the graph represents asocial network, then the idea ofsix degrees of separation can be interpreted mathematically as saying that the distance between any two vertices is at most six. Similarly, theErdős number and theBacon number—the number of collaborative relationships away a person is from prolific mathematicianPaul Erdős and actorKevin Bacon, respectively—are distances in the graphs whose edges represent mathematical or artistic collaborations.
Animation visualizing the function for various values of r.
Most of the notions of distance between two points or objects described above are examples of the mathematical idea of ametric. Ametric ordistance function is afunctiond which takes pairs of points or objects toreal numbers and satisfies the following rules:
The distance between an object and itself is always zero.
The distance between distinct objects is always positive.
Distance issymmetric: the distance fromx toy is always the same as the distance fromy tox.
Distance satisfies thetriangle inequality: ifx,y, andz are three objects, then This condition can be described informally as "intermediate stops can't speed you up."
As an exception, many of thedivergences used in statistics are not metrics.
One may measure the distance between representative points such as thecenter of mass; this is used for astronomical distances such as theEarth–Moon distance.
Even more generally, this idea can be used to define the distance between twosubsets of a metric space. The distance between setsA andB is theinfimum of the distances between any two of their respective points: 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).
TheHausdorff 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 betweenA andB is either the distance fromA to the farthest point ofB, or the distance fromB to the farthest point ofA, whichever is larger. (Here "farthest point" must be interpreted as a supremum.) The Hausdorff distance defines a metric on the set ofcompact subsets of a metric space.
The worddistance is also used for related concepts that are not encompassed by the description "a numerical measurement of how far apart points or objects are".
Thedistance travelled by an object is the length of a specific path travelled between two points,[6] such as the distance walked while navigating amaze or the distance marked by amilepost or anodometer. This can even be aclosed distance along aclosed curve which starts and ends at the same point, such as a ball thrown straight up, or the Earth when it completes oneorbit. This is formalized mathematically as thearc length of the curve.
The distance travelled may also besigned: a "forward" distance is positive and a "backward" distance is negative.
Circular distance is the distance traveled by a point on the circumference of awheel, which can be useful to consider when designing vehicles or mechanical gears (see alsoodometry). The circumference of the wheel is2π × radius; if the radius is 1, each revolution of the wheel causes a vehicle to travel2π radians.
Thedisplacement in classical physics measures the change in position of an object during an interval of time. While distance is ascalar quantity, or amagnitude, displacement is avector quantity with both magnitude anddirection. In general, the vector measuring the difference between two locations (therelative position) is sometimes called thedirected distance.[7] For example, the directed distance from theNew York City Main Library flag pole to theStatue of Liberty flag pole has:
Inmathematics and its applications, thesigned distance function or signed distance field (SDF) is theorthogonal distance of a given pointx to theboundary of aset Ω in ametric space (such as the surface of a geometric shape), with thesign determined by whether or notx is in theinterior of Ω. Thefunction has positive values at pointsx inside Ω, it decreases in value asx approaches the boundary of Ω where the signed distance function is zero, and it takes negative values outside of Ω.[8] However, the alternative convention is also sometimes taken instead (i.e., negative inside Ω and positive outside).[9] The concept also sometimes goes by the name oriented distance function/field.
^"The Directed Distance"(PDF).Information and Telecommunication Technology Center. University of Kansas. Archived fromthe original(PDF) on 10 November 2016. Retrieved18 September 2018.
^Chan, T.; Zhu, W. (2005).Level set based shape prior segmentation. IEEE Computer Society Conference on Computer Vision and Pattern Recognition.doi:10.1109/CVPR.2005.212.
