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Distance
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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.
In the social sciences, distance can refer to a qualitative measurement of separation, such as social distance or psychological distance.
Distances in physics and geometry
[edit]The distance between physical locations can be defined in different ways in different contexts.
Straight-line or Euclidean distance
[edit]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 . In coordinate geometry, Euclidean distance is computed using the Pythagorean 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-dimensional Euclidean spaces.
Measurement
[edit]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
[edit]
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
[edit]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
[edit]
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
[edit]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
[edit]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
[edit]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
[edit]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
[edit]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
[edit]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 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
[edit]
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: 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
[edit]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
[edit]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 or the distance marked by a milepost or an odometer. 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 2π × radius; if the radius is 1, each revolution of the wheel causes a vehicle to travel 2π radians.
Displacement and directed distance
[edit]
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
[edit]
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.
See also
[edit]
Wikiquote has quotations related to Distance.
- Absolute difference
- Astronomical system of units
- Color difference
- Closeness (mathematics)
- Distance geometry problem
- Dijkstra's algorithm
- Distance matrix
- Distance set
- Engineering tolerance
- Multiplicative distance
- Optical path length
- Orders of magnitude (length)
- Proper length
- Proxemics – physical distance between people
- Signed distance function
- Similarity measure
- Social distancing
- Vertical distance
Library support
[edit]- Python (programming language)
- SciPy -Distance computations (
scipy.spatial.distance)
- SciPy -Distance computations (
- Julia (programming language)
- Julia Statistics Distance -A Julia package for evaluating distances (metrics) between vectors.
References
[edit]- ^ 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. ISBN 978-1-4338-1579-9.
- ^ a b 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" (PDF). IEEE Transactions on Pattern Analysis and Machine Intelligence. 17 (2): 158–175. doi:10.1109/34.368173.
Bibliography
[edit]- Deza E, Deza M (2006). Dictionary of Distances. Elsevier. ISBN 0-444-52087-2.
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Distance
View on Grokipediafrom Grokipedia
Distance is a scalar quantity representing the length of the shortest path between two points in a space, fundamental to geometry, physics, and measurement.[1][2] In Euclidean geometry, the distance between two points and in the plane is calculated as , derived from the Pythagorean theorem.[3] This extends to higher dimensions and forms the basis for metric spaces, where a distance function must satisfy non-negativity, symmetry, the identity of indiscernibles, and the triangle inequality: .[1] In physics, distance quantifies the total path length traversed by an object, distinguishing it from displacement, which accounts for direction as a vector.[4] Applications span navigation, such as great-circle distances on spheres for aviation routes, to abstract spaces in data analysis and computer science.[5] Variations like Manhattan distance, which sums absolute differences along axes, arise in contexts prioritizing grid-like paths over straight lines.[6]
Definition and Historical Context
Core Definition and Intuition
Distance, in its most fundamental sense, quantifies the extent of spatial separation between two points or objects, representing the length of the path connecting them. In physics, it is defined as a scalar quantity measuring the total ground covered by an object during motion, independent of direction.[4] This contrasts with displacement, which accounts for the straight-line vector from initial to final position, highlighting distance's path-dependent nature.[7] For instance, an object traveling 5 kilometers eastward and then 5 kilometers westward covers a distance of 10 kilometers, though its displacement is zero.[4] The intuitive core of distance arises from everyday experience: it gauges "how far" entities are apart, enabling navigation, estimation of travel time, and comprehension of scale in the physical world. In Euclidean geometry, this intuition formalizes as the straight-line length between points, derived from the Pythagorean theorem.[8] For two points and in a plane, the distance is , embodying the shortest path in flat space.[1] This measure underpins calculations in surveying, engineering, and basic kinematics, where empirical verification through tools like rulers or odometers confirms its accuracy. Mathematically, distance extends beyond physical paths to abstract spaces via metric functions, which assign non-negative values to pairs of elements, satisfying properties like symmetry and the triangle inequality.[9] The Euclidean metric serves as the prototypical example, capturing the direct, causal separation in observable reality, while deviations in non-Euclidean contexts reveal how geometry influences perceived distances. This foundational concept drives empirical sciences by linking observable separations to quantifiable models, ensuring predictions align with measured outcomes.[1]Historical Evolution of the Concept
The concept of distance originated in practical necessities of ancient civilizations, where it was quantified using anthropometric units derived from human anatomy to facilitate trade, construction, and navigation. In Sumeria and ancient Egypt around 3000–2000 BCE, early systems employed measures such as the cubit—defined as the length from elbow to fingertip, approximately 45–52 cm depending on regional variations—and smaller subdivisions like the palm or finger.[10] These units enabled precise surveying for monumental architecture, as evidenced by cubit rods inscribed with markings found in Egyptian tombs, reflecting an empirical approach to linear separation without abstract formalization.[11] By the classical Greek period, around 300 BCE, Euclid's Elements elevated distance from mere measurement to a geometric primitive, implicit in the postulate that a straight line can be drawn between any two points, with length determined via constructive proofs and the Pythagorean theorem for right triangles. This framework treated distance as the invariant length of the shortest path in flat space, calculable as in coordinate terms later formalized, though Euclid avoided coordinates. Roman engineers extended practical application, using the passus (double pace, about 1.48 m) and mille passus (thousand paces, precursor to the mile) for road-building, achieving accuracies within 5% over long distances via chained measurements.[12][13] Medieval and Renaissance efforts focused on standardization amid inconsistent local units, with the 1791 French Academy proposal for the metric system grounding distance in natural invariants like Earth's quadrant meridian (1/10,000,000 defining the meter). In 19th-century mathematics, distance gained rigor through real analysis, as in Cauchy's 1821 work on convergence implying bounded separations, paving the way for abstraction. The modern formalization emerged in the early 20th century, with Maurice Fréchet's 1906 introduction of écart (a semi-metric satisfying non-negativity and symmetry) and Felix Hausdorff's 1914 definition of metric spaces, axiomatizing distance via positivity, symmetry, and the triangle inequality , decoupling it from Euclidean embedding.[14][15] This evolution shifted distance from empirical artifact to a foundational structure in topology and analysis, enabling non-intuitive metrics like those in taxicab geometry or function spaces.[15]Geometric and Physical Distances
Euclidean Distance and Measurement
The Euclidean distance between two points in a plane is the length of the straight-line segment connecting them, computed via the square root of the sum of the squared differences in their Cartesian coordinates.[16] This measure originates from the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides, directly yielding the distance formula for points and as .[8] In three-dimensional space, the formula extends to include the z-coordinate: for points and .[17] This generalization preserves the geometric intuition of the shortest path in flat space. For arbitrary n-dimensional Euclidean space, the distance is , forming the basis for the -norm in vector spaces.[18] Physical measurement of Euclidean distances relies on instruments calibrated in standardized units assuming local flatness, where general relativity effects are insignificant. The metre, the SI unit of length, is defined as the distance light travels in vacuum during 1/299792458 of a second, fixing the speed of light at exactly 299792458 m/s.[19] For short ranges, rigid rods or tape measures enforce this metric through material stiffness, approximating straight-line paths.[20] Precision techniques, such as laser interferometry, determine distances by counting interference fringes from coherent light, with each fringe corresponding to half a wavelength, typically around 532 nm for green lasers, enabling sub-micrometre accuracy under Euclidean assumptions.[21]Non-Euclidean and Curved-Space Distances
Non-Euclidean geometries deviate from Euclidean parallelism and incorporate constant non-zero curvature, altering distance measurements along geodesics rather than straight lines. Elliptic geometry, equivalent to spherical geometry on a unit sphere, defines distance as the great circle arc length between points, given by for position vectors and on the sphere.[22] This yields shorter paths than Euclidean chords, with total circumference and excess triangle angles summing positively.[22] Hyperbolic geometry, featuring constant negative curvature (often normalized to -1), employs models like the Poincaré disk or upper half-plane for distance computation. In the upper half-plane model, the hyperbolic distance between points and satisfies .[23] Geodesics appear as circular arcs orthogonal to the boundary, and distances grow exponentially, leading to negative angle excess in triangles.[24] In broader curved spaces modeled by Riemannian manifolds, distances arise from a metric tensor assigning inner products to tangent spaces. The length of a curve is , and the distance between points and is the infimum of such lengths over connecting curves.[25] This generalizes non-Euclidean cases, where curvature tensor components dictate geodesic deviation from Euclidean norms, enabling precise measurement on manifolds like surfaces of revolution.[26] Local Euclidean approximation holds via the exponential map, but global distances reflect intrinsic geometry.[27]Relativistic and Cosmological Distances
In special relativity, the proper length of an object is defined as the distance between its endpoints measured simultaneously in the object's rest frame, remaining invariant for all observers under Lorentz transformations. For an observer moving at velocity parallel to the object's length, the measured length contracts according to , where is the speed of light, reflecting the relativity of simultaneity and the spacetime interval's invariance.[28] This contraction applies only to the dimension parallel to the motion, with perpendicular dimensions unaffected, as derived from the Lorentz transformation of coordinates./28%3A_Special_Relativity/28.03%3A_Length_Contraction) In general relativity, distances are frame-dependent and path-dependent due to spacetime curvature, with proper distance computed as the length of a spacelike geodesic—the shortest path between two events in the curved geometry defined by the metric tensor . The proper distance along such a curve is given by , where and the negative sign distinguishes spacelike intervals from timelike ones; this integral accounts for gravitational effects like those near massive bodies, where light deflection and Shapiro delay alter measured paths. Free-falling observers follow geodesics, but measured distances incorporate the metric's local variations, as confirmed by experiments such as the 1919 solar eclipse observation of starlight bending.[29] Cosmological distances in an expanding universe, modeled by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, distinguish between proper distance (physical separation at a fixed cosmic time) and comoving distance (fixed coordinate separation scaled by the expansion factor ). The proper distance to an object at redshift is , where is the Hubble parameter at redshift and is the present epoch; this evolves with time due to cosmic expansion, unlike comoving coordinates which remain fixed.[30] Luminosity distance , inferred from flux dimming, satisfies where is the transverse comoving distance, accounting for both expansion and redshift effects on photon energy and arrival rate.[30] Angular diameter distance relates observed angular size to physical extent, peaking at intermediate redshifts before declining in standard CDM models due to the interplay of matter density and dark energy.[30] These measures enable consistency checks via the distance duality relation , tested against supernovae and cosmic microwave background data.[30]Empirical Measurement Challenges
In terrestrial surveying and geodesy, empirical distance measurements encounter random errors from stochastic variations in instrument readings or environmental noise, systematic errors from consistent biases such as instrument misalignment or uncompensated refraction, and gross errors from human blunders like misrecording data.[31] Random errors follow probabilistic distributions and can be mitigated through repeated measurements and statistical averaging, while systematic errors require identification and correction via calibration or modeling, as their unaddressed propagation amplifies inaccuracies in networks of interconnected measurements.[31] [32] Techniques like electronic distance measurement (EDM) using infrared or laser pulses suffer from atmospheric refraction, which bends light paths and introduces errors up to several parts per million over kilometer baselines, necessitating real-time corrections based on meteorological data such as temperature, pressure, and humidity.[33] Tape measurements face thermal expansion, sag due to gravity, and tension inconsistencies, with errors scaling quadratically with distance; for a 100-meter invar tape at standard conditions, uncompensated temperature shifts of 1°C can yield offsets of about 0.1 mm.[33] Global Navigation Satellite Systems (GNSS) like GPS achieve sub-meter precision but contend with multipath reflections, ionospheric delays (up to 10-20 meters equivalent path length), and satellite clock drifts, compounded by the need for differential corrections or precise point positioning algorithms.[34] At relativistic speeds, length contraction shortens measured distances for objects moving relative to the observer, with the observed length given by , where is the proper length, is relative velocity, and is the speed of light; this effect, negligible below (about 30,000 km/s), necessitates frame-dependent protocols for high-velocity experiments like particle accelerators.[35] [36] In satellite-based systems such as GPS, general relativistic gravitational time dilation (clocks run slower in weaker fields) and special relativistic velocity effects combine to require upward adjustments of about 38 microseconds per day to orbital clocks, preventing positional errors exceeding 10 km without correction.[37] Cosmological distance measurements via the cosmic distance ladder accumulate uncertainties across rungs, from trigonometric parallax (limited to ~1 kpc with Gaia mission precision of ~0.02% at 100 pc) to standard candles like Type Ia supernovae, where calibration inconsistencies contribute to the Hubble tension—a 4-6 sigma discrepancy between local (~73 km/s/Mpc) and CMB-derived (~67 km/s/Mpc) expansion rates, potentially signaling systematic biases in luminosity distance assumptions or unmodeled evolution in indicators.[38] Empirical verification remains challenged by light-travel time delays, redshift-distortion confounds, and the inability to directly observe intervening media, demanding cross-validation with multiple independent methods like baryon acoustic oscillations, which yield concordant but hierarchically dependent results.[38]Mathematical Formalization
Metric Spaces and Axioms
A metric space formalizes the notion of distance in an abstract setting, consisting of a nonempty set equipped with a function called a metric that satisfies specific axioms.[21] This structure generalizes Euclidean distance to arbitrary sets, enabling the study of convergence, continuity, and topology without reference to embedding in a vector space./03:_Vector_Spaces_and_Metric_Spaces/3.07:_Metric_Spaces) The concept was introduced by Maurice Fréchet in his 1906 doctoral thesis, where he unified notions from function spaces and point-set topology.[39][40] The axioms defining a metric are as follows:- Non-negativity: for all . This ensures distances are positive real numbers, mirroring physical distances.[21]
- Identity of indiscernibles: if and only if . This distinguishes distinct points by positive distance and assigns zero distance to a point with itself./03:_Vector_Spaces_and_Metric_Spaces/3.07:_Metric_Spaces)
- Symmetry: for all . This reflects the bidirectional nature of distance in isotropic spaces.[21]
- Triangle inequality: for all . This captures the efficiency of direct paths over indirect ones, preventing "shortcuts" that violate intuitive geometry./03:_Vector_Spaces_and_Metric_Spaces/3.07:_Metric_Spaces)