Distance

Last updated October 30, 2024

Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length 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 as statistical distance between probability distributions or edit distance between strings of text) or a degree of separation (as exemplified by distance between people in a social network). Most such notions of distance, both physical and metaphorical, are formalized in mathematics using the notion of a metric space.

Distances in physics and geometry

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

Straight-line or Euclidean distance

The distance between two points in physical space is the length of a 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- and three-dimensional space. In Euclidean geometry, the distance between two points $A$ and $B$ is often denoted $|AB|$ . In coordinate geometry, Euclidean distance is computed using the Pythagorean theorem. The distance between points $(x 1, y 1)$ and $(x 2, y 2)$ in the plane is given by:^[2]^[3] $d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}}}={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.$ Similarly, given points (x₁, y₁, z₁) and (x₂, y₂, z₂) in three-dimensional space, the distance between them is:^[2] $d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}}}={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}.$ This idea generalizes to higher-dimensional Euclidean spaces.

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

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 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

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 measures are used in cosmology to quantify such distances.

Other spatial distances

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 must make on a chessboard in order to travel between two squares.

Metaphorical distances

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.

Statistical distances

In statistics and information geometry, statistical distances measure the degree of difference between two probability distributions. There are many kinds of statistical distances, typically formalized as divergences; these allow a set of probability distributions to be understood as a geometrical object called a statistical manifold. The most elementary is the squared Euclidean distance, which is minimized by the least squares method; this is the most basic Bregman divergence. The most important in information theory is the relative entropy (Kullback–Leibler divergence), which allows one to analogously study maximum likelihood estimation geometrically; this is an example of both an f-divergence and a Bregman divergence (and in fact the only example which is both). Statistical manifolds corresponding to Bregman divergences are flat manifolds in the corresponding geometry, allowing an analog of the Pythagorean theorem (which holds for squared Euclidean distance) to be used for linear inverse problems in inference by optimization theory.

Other important statistical distances include the Mahalanobis distance and the energy distance.

Edit distances

In computer science, an edit distance or string metric between two strings measures how different they are. For example, the words "dog" and "dot", which differ by just one letter, are closer than "dog" and "cat", which have no letters in common. This idea is used in spell checkers and in coding theory, and is mathematically formalized in a number of different ways, including Levenshtein distance, Hamming distance, Lee distance, and Jaro–Winkler distance.

Distance in graph theory

In a graph, the distance between two vertices is measured by the length of the shortest edge path between them. For example, if the graph represents a social network, then the idea of six degrees of separation can be interpreted mathematically as saying that the distance between any two vertices is at most six. Similarly, the Erdős number and the Bacon number—the number of collaborative relationships away a person is from prolific mathematician Paul Erdős and actor Kevin Bacon, respectively—are distances in the graphs whose edges represent mathematical or artistic collaborations.

In the social sciences

In psychology, human geography, and the social sciences, distance is often theorized not as an objective numerical measurement, but as a qualitative description of a subjective experience.^[4] For example, psychological distance is "the different ways in which an object might be removed from" the self along dimensions such as "time, space, social distance, and hypotheticality".^[5] In sociology, social distance describes the separation between individuals or social groups in society along dimensions such as social class, race/ethnicity, gender or sexuality.

Mathematical formalization

Most of the notions of distance between two points or objects described above are examples of the mathematical idea of a metric. A metric or distance function is a function $d$ which takes pairs of points or objects to real 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 is symmetric: the distance from $x$ to $y$ is always the same as the distance from $y$ to $x$ .
Distance satisfies the triangle inequality: if $x$ , $y$ , and $z$ are three objects, then $d(x,z)\leq d(x,y)+d(y,z).$ This condition can be described informally as "intermediate stops can't speed you up."

As an exception, many of the divergences used in statistics are not metrics.

Distance between sets

There are multiple ways of measuring the physical distance between objects that 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 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 subspaces.

Even more generally, this idea can be used to define the distance between two subsets of a metric space. The distance between sets

A

and

B

is the infimum of the distances between any two of their respective points:

d(A,B)=\inf _{x\in A,y\in B}d(x,y).

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 $A$ and $B$ is either the distance from $A$ to the farthest point of $B$ , or the distance from $B$ to the farthest point of $A$ , whichever is larger. (Here "farthest point" must be interpreted as a supremum.) The Hausdorff distance defines a metric on the set of compact subsets of a metric space.

Related ideas

The word distance is also used for related concepts that are not encompassed by the description "a numerical measurement of how far apart points or objects are".

Distance travelled

The distance travelled by an object is the length of a specific path travelled between two points,^[6] such as the distance walked while navigating a maze. This can even be a closed distance along a closed curve which starts and ends at the same point, such as a ball thrown straight up, or the Earth when it completes one orbit. This is formalized mathematically as the arc length of the curve.

The distance travelled may also be signed: a "forward" distance is positive and a "backward" distance is negative.

Circular distance is the distance traveled by a point on the circumference of a wheel, which can be useful to consider when designing vehicles or mechanical gears (see also odometry). The circumference of the wheel is $2 π \times radius$ ; if the radius is 1, each revolution of the wheel causes a vehicle to travel $2 π$ radians.

Displacement and directed distance

Distance along a path compared with displacement. The Euclidean distance is the length of the displacement vector. Distancedisplacement.svg — Distance along a path compared with displacement. The Euclidean distance is the length of the displacement vector.

The displacement in classical physics measures the change in position of an object during an interval of time. While distance is a scalar quantity, or a magnitude, displacement is a vector quantity with both magnitude and direction. In general, the vector measuring the difference between two locations (the relative position) is sometimes called the directed distance.^[7] For example, the directed distance from the New York City Main Library flag pole to the Statue of Liberty flag pole has:

A starting point: library flag pole
An ending point: statue flag pole
A direction: -38°
A distance: 8.72 km

Signed distance

In mathematics and its applications, the signed distance function or signed distance field (SDF) is the orthogonal distance of a given point x to the boundary of a set Ω in a metric space (such as the surface of a geometric shape), with the sign determined by whether or not x is in the interior of Ω. The function has positive values at points x inside Ω, it decreases in value as x 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.

Library support

Python (programming language)
- SciPy -Distance computations (scipy.spatial.distance)
Julia (programming language)
- Julia Statistics Distance -A Julia package for evaluating distances (metrics) between vectors.

Related Research Articles

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics.

$<span class="mw-page-title-main">Hausdorff dimension</span> Invariant measure of fractal dimension$

In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension.

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.

<span class="mw-page-title-main">Similarity (geometry)</span> Property of objects which are scaled or mirrored versions of each other

In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling, possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other.

In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance.

In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the $-sphere$ , hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them.

Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point. Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.

<span class="mw-page-title-main">Hyperbolic geometry</span> Non-Euclidean geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two points is instead defined to be the sum of the absolute differences of their respective Cartesian coordinates, a distance function called the taxicab distance, Manhattan distance, or city block distance. The name refers to the island of Manhattan, or generically any planned city with a rectangular grid of streets, in which a taxicab can only travel along grid directions. In taxicab geometry, the distance between any two points equals the length of their shortest grid path. This different definition of distance also leads to a different definition of the length of a curve, for which a line segment between any two points has the same length as a grid path between those points rather than its Euclidean length.

In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability measures defined on a common probability space. It can be used to calculate the informational difference between measurements.

In mathematics, the magnitude or size of a mathematical object is a property which determines whether the object is larger or smaller than other objects of the same kind. More formally, an object's magnitude is the displayed result of an ordering of the class of objects to which it belongs. Magnitude as a concept dates to Ancient Greece and has been applied as a measure of distance from one object to another. For numbers, the absolute value of a number is commonly applied as the measure of units between a number and zero.

In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations between points. The definition, introduced by Mikhael Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Hyperbolicity is a large-scale property, and is very useful to the study of certain infinite groups called Gromov-hyperbolic groups.

In mathematics, Hilbert's fourth problem in the 1900 list of Hilbert's problems is a foundational question in geometry. In one statement derived from the original, it was to find — up to an isomorphism — all geometries that have an axiomatic system of the classical geometry, with those axioms of congruence that involve the concept of the angle dropped, and `triangle inequality', regarded as an axiom, added.

In statistics, probability theory, and information theory, a statistical distance quantifies the distance between two statistical objects, which can be two random variables, or two probability distributions or samples, or the distance can be between an individual sample point and a population or a wider sample of points.

In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. The property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces.

In mathematics, the Gromov product is a concept in the theory of metric spaces named after the mathematician Mikhail Gromov. The Gromov product can also be used to define δ-hyperbolic metric spaces in the sense of Gromov.

In mathematics, an ultralimit is a geometric construction that assigns a limit metric space to a sequence of metric spaces $. The concept captures the limiting behavior of finite configurations in the spaces employing an ultrafilter to bypass the need for repeated consideration of subsequences to ensure convergence. Ultralimits generalize Gromov-Hausdorff convergence in metric spaces.$

In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multiplicative factor, quasisymmetric maps satisfy the weaker geometric property that they preserve the relative sizes of sets: if two sets A and B have diameters t and are no more than distance t apart, then the ratio of their sizes changes by no more than a multiplicative constant. These maps are also related to quasiconformal maps, since in many circumstances they are in fact equivalent.

In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance" where he calls the quadric the absolute. The construction was developed in further detail by Felix Klein in papers in 1871 and 1873, and subsequent books and papers. The Cayley–Klein metrics are a unifying idea in geometry since the method is used to provide metrics in hyperbolic geometry, elliptic geometry, and Euclidean geometry. The field of non-Euclidean geometry rests largely on the footing provided by Cayley–Klein metrics.

In geometry, a motion is an isometry of a metric space. For instance, a plane equipped with the Euclidean distance metric is a metric space in which a mapping associating congruent figures is a motion. More generally, the term motion is a synonym for surjective isometry in metric geometry, including elliptic geometry and hyperbolic geometry. In the latter case, hyperbolic motions provide an approach to the subject for beginners.

References

↑ Schnall, Simone (2014). "Are there basic metaphors?". The power of metaphor: Examining its influence on social life. American Psychological Association. pp. 225–247. doi:10.1037/14278-010.
1 2 Weisstein, Eric W. "Distance". mathworld.wolfram.com. Retrieved 2020-09-01.
↑ "Distance Between 2 Points". www.mathsisfun.com. Retrieved 2020-09-01.
↑ "SOCIAL DISTANCES". www.hawaii.edu. Retrieved 2020-07-20.
↑ Trope Y, Liberman N (April 2010). "Construal-level theory of psychological distance". Psychological Review. 117 (2): 440–63. doi:10.1037/a0018963. PMC 3152826 . PMID 20438233.
↑ "What is displacement? (article)". Khan Academy. Retrieved 2020-07-20.
↑ "The Directed Distance" (PDF). Information and Telecommunication Technology Center. University of Kansas. Archived from the original (PDF) on 10 November 2016. Retrieved 18 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.
↑ Malladi, R.; Sethian, J.A.; Vemuri, B.C. (1995). "Shape modeling with front propagation: a level set approach". IEEE Transactions on Pattern Analysis and Machine Intelligence. 17 (2): 158–175. CiteSeerX 10.1.1.33.2443 . doi:10.1109/34.368173. S2CID 9505101.

Bibliography

Deza E, Deza M (2006). Dictionary of Distances. Elsevier. ISBN 0-444-52087-2.

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[1] Schnall, Simone (2014). "Are there basic metaphors?". The power of metaphor: Examining its influence on social life. American Psychological Association. pp. 225–247. doi:10.1037/14278-010.

[:0-2] 1 2 Weisstein, Eric W. "Distance". mathworld.wolfram.com. Retrieved 2020-09-01.

[3] "Distance Between 2 Points". www.mathsisfun.com. Retrieved 2020-09-01.

[4] "SOCIAL DISTANCES". www.hawaii.edu. Retrieved 2020-07-20.

[psych-5] Trope Y, Liberman N (April 2010). "Construal-level theory of psychological distance". Psychological Review. 117 (2): 440–63. doi:10.1037/a0018963. PMC 3152826 . PMID 20438233.

[6] "What is displacement? (article)". Khan Academy. Retrieved 2020-07-20.

[ittcKU-7] "The Directed Distance" (PDF). Information and Telecommunication Technology Center. University of Kansas. Archived from the original (PDF) on 10 November 2016. Retrieved 18 September 2018.

[8] 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.

[9] Malladi, R.; Sethian, J.A.; Vemuri, B.C. (1995). "Shape modeling with front propagation: a level set approach". IEEE Transactions on Pattern Analysis and Machine Intelligence. 17 (2): 158–175. CiteSeerX 10.1.1.33.2443 . doi:10.1109/34.368173. S2CID 9505101.

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