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approaches the boundary of Ω where the signed distance function is zero, and it takes negative values outside of Ω. However, the alternative convention is also sometimes taken instead (i.e., negative inside Ω and positive outside). The concept also sometimes goes by the name oriented distance
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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
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or an estimation based on other criteria (e.g. "two counties over"). The term is also frequently used metaphorically to mean a measurement of the amount of difference between two similar objects (such as
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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
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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.
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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
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318:
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is "the different ways in which an object might be removed from" the self along dimensions such as "time, space, social distance, and hypotheticality". In
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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
2258:
2234:
1211:, distance is often theorized not as an objective numerical measurement, but as a qualitative description of a subjective experience. For example,
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is also used for related concepts that are not encompassed by the description "a numerical measurement of how far apart points or objects are".
1236:
2413:
1706:, whichever is larger. (Here "farthest point" must be interpreted as a supremum.) The Hausdorff distance defines a metric on the set of
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766:{\displaystyle 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}}}.}
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1066:
and a
Bregman divergence (and in fact the only example which is both). Statistical manifolds corresponding to Bregman divergences are
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126:
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Unusual definitions of distance can be helpful to model certain physical situations, but are also used in theoretical mathematics:
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of an airplane or spacecraft is its distance from the Earth. The same sense of distance is used in
Euclidean geometry to define
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In practice, one is often interested in the travel distance between two points along roads, rather than as the crow flies. In a
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by an object is the length of a specific path travelled between two points, such as the distance walked while navigating a
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Most of the notions of distance between two points or objects described above are examples of the mathematical idea of a
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1997:
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can be interpreted mathematically as saying that the distance between any two vertices is at most six. Similarly, the
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in classical physics measures the change in position of an object during an interval of time. While distance is a
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Distance along a path compared with displacement. The
Euclidean distance is the length of the displacement vector.
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1962:
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60:
31:
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Malladi, R.; Sethian, J.A.; Vemuri, B.C. (1995). "Shape modeling with front propagation: a level set approach".
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2406:
2013:
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which starts and ends at the same point, such as a ball thrown straight up, or the Earth when it completes one
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839:(bottom) when heading eastwards. The shortest route appears as a curve rather than a straight line because the
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1191:, respectively—are distances in the graphs whose edges represent mathematical or artistic collaborations.
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510:{\displaystyle d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}}}={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}
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965:: the number of east–west and north–south blocks one must traverse to get between those two points.
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The distance between physical locations can be defined in different ways in different contexts.
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One may measure the distance between the closest points of the two objects; in this sense, the
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between them, which is the shortest possible path. This is the usual meaning of distance in
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does not scale all distances equally compared to the real spherical surface of the Earth.
1778:, which can be useful to consider when designing vehicles or mechanical gears (see also
1183:—the number of collaborative relationships away a person is from prolific mathematician
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Animation visualizing the function (abs(x)^r + abs(y)^r)^(1/r) for various values of r.
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This condition can be described informally as "intermediate stops can't speed you up."
922:. On galactic and larger scales, the measurement of distance is also affected by the
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2002:
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191:). Most such notions of distance, both physical and metaphorical, are formalized in
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17:
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1977:
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There are multiple ways of measuring the physical distance between objects that
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2298:. IEEE Computer Society Conference on Computer Vision and Pattern Recognition.
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1832:. In general, the vector measuring the difference between two locations (the
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Even more generally, this idea can be used to define the distance between two
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The distances between these three sets do not satisfy the triangle inequality:
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1768:: a "forward" distance is positive and a "backward" distance is negative.
1019:. There are many kinds of statistical distances, typically formalized as
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of geodesics gives a way of measuring distance from the perspective of an
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One may measure the distance between representative points such as the
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1023:; these allow a set of probability distributions to be understood as a
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straight-line distances. For example, it can be done directly using a
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2066:-A Julia package for evaluating distances (metrics) between vectors.
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between two vertices is measured by the length of the shortest
861:. Instead, one typically measures the shortest path along the
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IEEE Transactions on
Pattern Analysis and Machine Intelligence
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is the distance traveled by a point on the circumference of a
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can refer to a qualitative measurement of separation, such as
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The power of metaphor: Examining its influence on social life
961:, the travel distance between street corners is given by the
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More generally, the shortest path between two points along a
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of the distances between any two of their respective points:
1074:(which holds for squared Euclidean distance) to be used for
559:) in three-dimensional space, the distance between them is:
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Straight-line distance is formalized mathematically as the
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of how far apart objects, points, people, or ideas are. In
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The distance between distinct objects is always positive.
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The distance between an object and itself is always zero.
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in the corresponding geometry, allowing an analog of the
2090:. American Psychological Association. pp. 225–247.
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is a set of ways of measuring extremely long distances.
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Schnall, Simone (2014). "Are there basic metaphors?".
1537:; this is used for astronomical distances such as the
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between them. For example, if the graph represents a
1906:(such as the surface of a geometric shape), with the
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or other flightless creature living on that surface.
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or everyday usage, distance may refer to a physical
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1665:{\displaystyle d(A,B)=\inf _{x\in A,y\in B}d(x,y).}
67:. Unsourced material may be challenged and removed.
2187:"Construal-level theory of psychological distance"
1664:
1514:
1409:
1085:Other important statistical distances include the
918:, distances between objects depend on a choice of
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509:
312:
1614:
1223:describes the separation between individuals or
1840:. For example, the directed distance from the
183:) or a degree of separation (as exemplified by
1573:of a metric space. The distance between sets
869:. This is approximated mathematically by the
2407:
1059:geometrically; this is an example of both an
1015:measure the degree of difference between two
8:
1757:. This is formalized mathematically as the
773:This idea generalizes to higher-dimensional
240:The distance between two points in physical
324:, Euclidean distance is computed using the
154:is a numerical or occasionally qualitative
2414:
2400:
2392:
1284:which takes pairs of points or objects to
815:Shortest-path distance on a curved surface
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2210:
1617:
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1410:{\displaystyle d(x,z)\leq d(x,y)+d(y,z).}
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1055:), which allows one to analogously study
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127:Learn how and when to remove this message
2296:Level set based shape prior segmentation
2010:– physical distance between people
1310:is always the same as the distance from
941:
818:
138:
2078:
1515:{\displaystyle d(A,B)>d(A,C)+d(C,B)}
980:in order to travel between two squares.
2268:. University of Kansas. Archived from
1886:or signed distance field (SDF) is the
1782:). The circumference of the wheel is
7:
2107:
2105:
1425:used in statistics are not metrics.
65:adding citations to reliable sources
1926:inside Ω, it decreases in value as
1852:A starting point: library flag pole
1764:The distance travelled may also be
1288:and satisfies the following rules:
972:, is the minimum number of moves a
968:Chessboard distance, formalized as
230:Straight-line or Euclidean distance
2185:Trope Y, Liberman N (April 2010).
1794:Displacement and directed distance
621:
599:
577:
404:
382:
279:, the distance between two points
25:
2235:"What is displacement? (article)"
1855:An ending point: statue flag pole
1828:quantity with both magnitude and
807:(for very short distances). The
222:Distances in physics and geometry
1871:This section is an excerpt from
1554:distance from a point to a plane
41:
1550:distance from a point to a line
1039:method; this is the most basic
910:, because of phenomena such as
143:A board showing distances near
52:needs additional citations for
1922:has positive values at points
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1644:
1607:
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1527:consist of more than one point
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1031:. The most elementary is the
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328:. The distance between points
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1:
2042:Python (programming language)
1910:determined by whether or not
1421:As an exception, many of the
1057:maximum likelihood estimation
835:going west (top) but use the
27:Separation between two points
2059:Julia (programming language)
1998:Orders of magnitude (length)
1948:Astronomical system of units
1690:is either the distance from
1035:, which is minimized by the
934:to quantify such distances.
926:. In practice, a number of
2138:"Distance Between 2 Points"
1053:Kullback–Leibler divergence
920:inertial frame of reference
2525:
2294:Chan, T.; Zhu, W. (2005).
1882:and its applications, the
1870:
1842:New York City Main Library
1836:) is sometimes called the
1805:
1718:
1258:
1255:Mathematical formalization
1148:
1033:squared Euclidean distance
996:
916:relativity of simultaneity
899:
846:
784:
364:in the plane is given by:
233:
29:
2448:
2429:
2064:Julia Statistics Distance
1963:Distance geometry problem
1702:to the farthest point of
1694:to the farthest point of
1231:along dimensions such as
1173:six degrees of separation
1017:probability distributions
924:expansion of the universe
517:Similarly, given points (
173:probability distributions
32:Distance (disambiguation)
2049:-Distance computations (
2014:Signed distance function
1884:signed distance function
1873:Signed distance function
1861:A distance: 8.72 km
1337:are three objects, then
1145:Distance in graph theory
1043:. The most important in
803:(for long distances) or
2375:Dictionary of Distances
2259:"The Directed Distance"
1988:Multiplicative distance
1958:Closeness (mathematics)
1698:, or the distance from
1321:Distance satisfies the
1151:Distance (graph theory)
1076:linear inverse problems
938:Other spatial distances
831:approximately follow a
823:Airline routes between
799:, or indirectly with a
791:There are many ways of
273:three-dimensional space
2051:scipy.spatial.distance
1803:
1745:. This can even be a
1666:
1558:perpendicular distance
1556:, or, more generally,
1522:
1516:
1433:
1411:
1213:psychological distance
1195:In the social sciences
985:Metaphorical distances
950:
844:
809:cosmic distance ladder
767:
511:
314:
216:psychological distance
195:using the notion of a
148:
2304:10.1109/CVPR.2005.212
2118:mathworld.wolfram.com
1983:Engineering tolerance
1808:Displacement (vector)
1801:
1784:2π × radius
1719:Further information:
1667:
1517:
1444:
1437:Distance between sets
1431:
1412:
1139:Jaro–Winkler distance
1013:statistical distances
993:Statistical distances
945:
896:Effects of relativity
871:great-circle distance
822:
768:
512:
315:
142:
18:Distance between sets
2191:Psychological Review
1968:Dijkstra's algorithm
1824:, displacement is a
1589:
1449:
1341:
1302:: the distance from
1127:Levenshtein distance
1087:Mahalanobis distance
1029:statistical manifold
1009:information geometry
999:Statistical distance
908:theory of relativity
863:surface of the Earth
787:Distance measurement
563:
368:
313:{\displaystyle |AB|}
291:
187:between people in a
169:statistical distance
61:improve this article
30:For other uses, see
2275:on 10 November 2016
2112:Weisstein, Eric W.
1993:Optical path length
1943:Absolute difference
1888:orthogonal distance
1539:Earth–Moon distance
1323:triangle inequality
1171:, then the idea of
1080:optimization theory
1072:Pythagorean theorem
849:Geographic distance
326:Pythagorean theorem
322:coordinate geometry
258:Newtonian mechanics
2488:Higher derivatives
2162:"SOCIAL DISTANCES"
2142:www.mathsisfun.com
2019:Similarity measure
1804:
1739:distance travelled
1733:Distance travelled
1710:of a metric space.
1680:Hausdorff distance
1662:
1640:
1523:
1512:
1434:
1407:
1045:information theory
1041:Bregman divergence
1025:geometrical object
970:Chebyshev distance
963:Manhattan distance
951:
947:Manhattan distance
912:length contraction
845:
763:
507:
310:
277:Euclidean geometry
265:Euclidean distance
236:Euclidean distance
149:
2496:
2495:
2341:10.1109/34.368173
2096:10.1037/14278-010
2029:Vertical distance
2024:Social distancing
1890:of a given point
1858:A direction: -38°
1846:Statue of Liberty
1844:flag pole to the
1838:directed distance
1834:relative position
1772:Circular distance
1613:
1275:distance function
928:distance measures
867:as the crow flies
758:
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287:is often denoted
254:classical physics
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16:(Redirected from
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1103:computer science
1078:in inference by
1049:relative entropy
902:Distance measure
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1973:Distance matrix
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1931:function/field.
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1866:Signed distance
1820:quantity, or a
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1209:social sciences
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976:must make on a
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1806:Main article:
1795:
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1761:of the curve.
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1535:center of mass
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1259:Main article:
1256:
1253:
1196:
1193:
1169:social network
1149:Main article:
1146:
1143:
1119:spell checkers
1098:
1097:Edit distances
1095:
1068:flat manifolds
997:Main article:
994:
991:
986:
983:
982:
981:
966:
939:
936:
900:Main article:
897:
894:
880:is known as a
878:curved surface
859:Earth's mantle
841:map projection
816:
813:
805:interferometry
785:Main article:
782:
779:
762:
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234:Main article:
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189:social network
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2384:0-444-52087-2
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2197:(2): 440–63.
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2178:
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2154:
2143:
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2039:
2035:
2030:
2027:
2025:
2022:
2020:
2017:
2015:
2012:
2009:
2006:
2004:
2003:Proper length
2001:
1999:
1996:
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1991:
1989:
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1981:
1979:
1976:
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1971:
1969:
1966:
1964:
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1715:Related ideas
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1140:
1136:
1132:
1128:
1124:
1123:coding theory
1120:
1116:
1112:
1111:string metric
1108:
1107:edit distance
1104:
1096:
1094:
1092:
1088:
1083:
1081:
1077:
1073:
1069:
1065:
1063:
1058:
1054:
1050:
1046:
1042:
1038:
1037:least squares
1034:
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1026:
1022:
1018:
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1000:
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921:
917:
913:
909:
903:
895:
893:
891:
887:
883:
879:
874:
873:on a sphere.
872:
868:
864:
860:
854:
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842:
838:
834:
830:
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333:
327:
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302:
299:
278:
274:
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261:
259:
255:
251:
250:straight line
247:
243:
237:
229:
227:
221:
219:
217:
213:
209:
205:
200:
198:
194:
190:
186:
182:
178:
177:edit distance
174:
170:
165:
161:
157:
153:
146:
145:Visakhapatnam
141:
131:
128:
120:
117:February 2020
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: –
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
2478:Acceleration
2461:
2458:Displacement
2377:. Elsevier.
2374:
2360:Bibliography
2322:
2318:
2312:
2295:
2289:
2279:18 September
2277:. Retrieved
2270:the original
2265:
2253:
2242:. Retrieved
2239:Khan Academy
2238:
2229:
2194:
2190:
2180:
2169:. Retrieved
2165:
2156:
2145:. Retrieved
2141:
2132:
2121:. Retrieved
2117:
2087:
2081:
1978:Distance set
1927:
1923:
1911:
1904:metric space
1891:
1837:
1814:displacement
1811:
1771:
1770:
1763:
1751:closed curve
1746:
1738:
1736:
1726:
1724:
1524:
1420:
1298:Distance is
1286:real numbers
1274:
1270:
1264:
1261:Metric space
1233:social class
1198:
1181:Bacon number
1177:Erdős number
1154:
1135:Lee distance
1113:between two
1100:
1084:
1061:
1002:
988:
952:
930:are used in
905:
875:
856:
833:great circle
790:
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338:
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256:, including
239:
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207:
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197:metric space
151:
150:
123:
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
1880:mathematics
1423:divergences
1189:Kevin Bacon
1064:-divergence
1021:divergences
825:Los Angeles
781:Measurement
193:mathematics
156:measurement
2423:Kinematics
2244:2020-07-20
2171:2020-07-20
2147:2020-09-01
2123:2020-09-01
2114:"Distance"
2073:References
1918:of Ω. The
1914:is in the
1759:arc length
1207:, and the
1201:psychology
1187:and actor
1185:Paul Erdős
1005:statistics
978:chessboard
886:arc length
837:jet stream
87:newspapers
76:"Distance"
2435:Integrate
2327:CiteSeerX
2008:Proxemics
1830:direction
1822:magnitude
1790:radians.
1725:The word
1634:∈
1622:∈
1363:≤
1300:symmetric
1249:sexuality
1241:ethnicity
1217:sociology
1165:edge path
1027:called a
959:grid plan
949:on a grid
932:cosmology
793:measuring
736:−
697:−
658:−
622:Δ
600:Δ
578:Δ
480:−
441:−
405:Δ
383:Δ
2509:Distance
2503:Category
2468:Velocity
2462:Distance
2453:Absement
2373:(2006).
2221:20438233
1936:See also
1920:function
1916:interior
1896:boundary
1780:odometry
1749:along a
1727:distance
1560:between
1546:altitude
1279:function
1179:and the
1161:distance
1089:and the
914:and the
882:geodesic
853:geodesic
208:distance
185:distance
179:between
171:between
152:Distance
2349:9505101
2212:3152826
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1894:to the
1788:2π
1673:union).
1583:infimum
1581:is the
1571:subsets
1229:society
1121:and in
1115:strings
1047:is the
906:In the
884:. The
538:) and (
244:is the
202:In the
160:physics
147:, India
101:scholar
2381:
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2367:Deza E
2347:
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2209:
1826:vector
1818:scalar
1766:signed
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1271:metric
1267:metric
1245:gender
1159:, the
1137:, and
320:. In
275:. In
246:length
164:length
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2472:Speed
2345:S2CID
2273:(PDF)
2262:(PDF)
2047:SciPy
1898:of a
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1277:is a
1269:. A
1157:graph
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1105:, an
829:Tokyo
801:radar
797:ruler
248:of a
242:space
108:JSTOR
94:books
2483:Jerk
2379:ISBN
2281:2018
2217:PMID
1908:sign
1812:The
1743:maze
1737:The
1686:and
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1577:and
1471:>
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1007:and
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283:and
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2199:doi
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2092:doi
1900:set
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