Растојање

Овај чланак је о растојању у математици, физици и рачунању. За друге употребе погледајте страницу Растојање (вишезначна одредница).

Растојање је бројно мерење удаљености објеката. У физици или свакодневној употреби, растојање се може односити на физичку дужину или процену засновану на другим критеријима (нпр. „две жупаније”). У већини случајева, „растојање између А и Б” је заменљиво са „растојањем између Б и А”. У математици, функција растојања или метрика је генерализација концепта физичког растојања. Метрика је функција која се понаша у складу са одређеним скупом правила и представља начин описивања шта то значи за елементе неког простора да буду „близу” или „далеко” један од другог.

Пређени пут

Не поистовећивати са Положајeм, Вектором положаја, Померајем или Вектором помераја.

Вектор помераја (зелена права испрекидана линија), пређени пут (љубичаста крива испрекидана линија) и путања (плава линија)

Пређени пут (енгл. distance travelled; SI ознака — ${\boldsymbol {s}}$) јест једнак интензитету (апсолутној вредности) векторa помераја:

${\boldsymbol {s}}=\left|\Delta {\boldsymbol {\vec {r}}}\,\right\vert =\left|{\boldsymbol {\vec {r}}}_{2}-{\boldsymbol {\vec {r}}}_{1}\right\vert ,$

односно:

${\boldsymbol {s}}_{\Delta t}=\left|\Delta {\boldsymbol {\vec {r}}}_{\Delta t}\right\vert =\left|{\boldsymbol {\vec {r}}}_{t_{2}}-{\boldsymbol {\vec {r}}}_{t_{1}}\right\vert ,$

где је ${\boldsymbol {\vec {r}}}_{t_{2}}$ вектор положаја у тренутку $t_{2}$ и ${\boldsymbol {\vec {r}}}_{t_{1}}$ вектор положаја у тренутку $t_{1}$.

Укупни пређени пут је једнак збиру интензитета појединих векторa помераја:

${\boldsymbol {s}}_{u}={\boldsymbol {s}}_{\Delta t_{1}+\Delta t_{2}+\Delta t_{3}+\,\cdots \,+\Delta t_{n}}=\left|\Delta {\boldsymbol {\vec {r}}}_{\Delta t_{1}}\right\vert +\left|\Delta {\boldsymbol {\vec {r}}}_{\Delta t_{2}}\right\vert +\left|\Delta {\boldsymbol {\vec {r}}}_{\Delta t_{3}}\right\vert +\cdots +\left|\Delta {\boldsymbol {\vec {r}}}_{\Delta t_{n}}\right\vert .$

Пређени пут је скaларна величина.

Преглед и дефиниције

Физичке удаљености

Airline routes between Los Angeles and Tokyo approximately follow a direct great circle route (top), but use the jet stream (bottom) when heading eastwards. Note that the shortest route appears as a curve rather than a straight line because this map is a Mercator projection, which does not scale all distances equally compared to the real spherical surface of the Earth.

"Manhattan distance" on a grid

A physical distance can mean several different things:

• Distance traveled: The length of a specific path traveled between two points, such as the distance walked while navigating a maze
• Straight-line (Euclidean) distance: The length of the shortest possible path through space, between two points, that could be taken if there were no obstacles (usually formalized as Euclidean distance)
• Geodesic distance: The length of the shortest path between two points while remaining on some surface, such as the great-circle distance along the curve of the Earth
• The length of a specific path that returns to the starting point, such as a ball thrown straight up, or the Earth when it completes one orbit.

A board showing distances near Visakhapatnam

"Circular distance" is the distance traveled by a wheel, which can be useful when designing vehicles or mechanical gears. The circumference of the wheel is 2π × radius, and assuming the radius to be 1, then each revolution of the wheel is equivalent of the distance 2π radians. In engineering ω = 2πƒ is often used, where ƒ is the frequency.

Unusual definitions of distance can be helpful to model certain physical situations, but are also used in theoretical mathematics:

• "Manhattan distance" is a rectilinear distance, named after the number of blocks (in the north, south, east or west directions) a taxicab must travel on, in order to reach its destination on the grid of streets in parts of New York City.
• "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.

Distance measures in cosmology are complicated by the expansion of the universe, and by effects described by the theory of relativity (such as length contraction of moving objects).

Теоријске удаљености

The term "distance" is also used by analogy to measure non-physical entities in certain ways.

In computer science, there is the notion of the "edit distance" between two strings. For example, the words "dog" and "dot", which vary by only one letter, are closer than "dog" and "cat", which differ by three letters. This idea is used in spell checkers and in coding theory, and is mathematically formalized in several different ways such as:

• Levenshtein distance
• Hamming distance
• Lee distance
• Jaro–Winkler distance

In mathematics, a metric space is a set for which distances between all members of the set are defined. In this way, many different types of "distances" can be calculated, such as for traversal of graphs, comparison of distributions and curves, and using unusual definitions of "space" (for example using a manifold or reflections). The notion of distance in graph theory has been used to describe social networks, for example with the Erdős number or the Bacon number—the number of collaborative relationships away a person is from prolific mathematician Paul Erdős and actor Kevin Bacon, respectively.

In psychology, human geography, and the social sciences, distance is often theorized not as an objective metric, but as a subjective experience.

Види још

• Склада удаљености у астрономији
• Дајкстрин алгоритам
• Положај
• Вектор положаја
• Померај
• Вектор помераја

Литература

• Deza E, Deza M (2006). Dictionary of Distances. Elsevier. ISBN 0-444-52087-2. 
• Aldrovandi, Ruben; Pereira, José Geraldo (2017), An Introduction to Geometrical Physics (2nd изд.), Hackensack, New Jersey: World Scientific, стр. 20, ISBN 978-981-3146-81-5, MR 3561561 
• Arkhangel'skii, A. V.; Pontryagin, L. S. (1990), General Topology I: Basic Concepts and Constructions Dimension Theory, Encyclopaedia of Mathematical Sciences, Springer, ISBN 3-540-18178-4 
• Buldygin, V. V.; Kozachenko, Yu. V. (2000), Metric Characterization of Random Variables and Random Orocesses, Translations of Mathematical Monographs, 188, Providence, Rhode Island: American Mathematical Society, стр. 129, ISBN 0-8218-0533-9, MR 1743716, doi:10.1090/mmono/188 
• Čech, Eduard (1969), Point Sets, New York: Academic Press, стр. 42 
• Cecil, Thomas E. (2008), Lie Sphere Geometry: With Applications to Submanifolds, Universitext (2nd изд.), New York: Springer, стр. 9, ISBN 978-0-387-74655-5, MR 2361414 
• Cohen, Andrew R.; Vitányi, Paul M. B. (2012), „Normalized compression distance of multisets with applications”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 37 (8): 1602—1614, PMC 4566858 , PMID 26352998, arXiv:1212.5711 , doi:10.1109/TPAMI.2014.2375175 
• Deza, Michel Marie; Laurent, Monique (1997), Geometry of Cuts and Metrics, Algorithms and Combinatorics, 15, Springer-Verlag, Berlin, стр. 27, ISBN 3-540-61611-X, MR 1460488, doi:10.1007/978-3-642-04295-9 
• Fraigniaud, P.; Lebhar, E.; Viennot, L. (2008), „The inframetric model for the internet”, 2008 IEEE INFOCOM - The 27th Conference on Computer Communications, стр. 1085—1093, CiteSeerX 10.1.1.113.6748 , ISBN 978-1-4244-2026-1, S2CID 5733968, doi:10.1109/INFOCOM.2008.163 
• Helemskii, A. Ya. (2006), Lectures and Exercises on Functional Analysis, Translations of Mathematical Monographs, 233, Providence, Rhode Island: American Mathematical Society, стр. 14, ISBN 978-0-8218-4098-6, MR 2248303, doi:10.1090/mmono/233 
• Lawvere, F. William (2002), „Metric spaces, generalized logic, and closed categories” (PDF), Reprints in Theory and Applications of Categories (1): 1—37, MR 1925933 ; reprinted with added commentary from Lawvere, F. William (1973), „Metric spaces, generalized logic, and closed categories”, Rendiconti del Seminario Matematico e Fisico di Milano, 43: 135—166 (1974), MR 352214, doi:10.1007/BF02924844 
• Parrott, Stephen (1987), Relativistic Electrodynamics and Differential Geometry, New York: Springer-Verlag, стр. 4, ISBN 0-387-96435-5, MR 867408, doi:10.1007/978-1-4612-4684-8 
• Rolewicz, Stefan (1987), Functional Analysis and Control Theory: Linear Systems, Springer, ISBN 90-277-2186-6 
• Smyth, M. (1987), „Quasi uniformities: reconciling domains with metric spaces”, Ур.: Main, M.; Melton, A.; Mislove, M.; Schmidt, D., 3rd Conference on Mathematical Foundations of Programming Language Semantics, Lecture Notes in Computer Science, 298, Springer-Verlag, стр. 236—253, doi:10.1007/3-540-19020-1_12 
• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology, Dover, ISBN 978-0-486-68735-3, MR 507446 
• Vitányi, Paul M. B. (2011), „Information distance in multiples”, IEEE Transactions on Information Theory, 57 (4): 2451—2456, S2CID 6302496, arXiv:0905.3347 , doi:10.1109/TIT.2011.2110130 
• Väisälä, Jussi (2005), „Gromov hyperbolic spaces” (PDF), Expositiones Mathematicae, 23 (3): 187—231, MR 2164775, doi:10.1016/j.exmath.2005.01.010  
• Vickers, Steven (2005), „Localic completion of generalized metric spaces, I”, Theory and Applications of Categories, 14 (15): 328—356, MR 2182680 
• Xia, Qinglan (2008), „The geodesic problem in nearmetric spaces”, Journal of Geometric Analysis, 19 (2): 452—479, arXiv:0807.3377  
• Xia, Q. (2009), „The geodesic problem in quasimetric spaces”, Journal of Geometric Analysis, 19 (2): 452—479, S2CID 17475581, arXiv:0807.3377 , doi:10.1007/s12220-008-9065-4 
• Smith, Karl (2013), Precalculus: A Functional Approach to Graphing and Problem Solving, Jones & Bartlett Publishers, стр. 8, ISBN 978-0-7637-5177-7 
• Cohen, David (2004), Precalculus: A Problems-Oriented Approach (6th изд.), Cengage Learning, стр. 698, ISBN 978-0-534-40212-9 
• Aufmann, Richard N.; Barker, Vernon C.; Nation, Richard D. (2007), College Trigonometry (6th изд.), Cengage Learning, стр. 17, ISBN 978-1-111-80864-8 

Спољашње везе

• Interspace -A package for finding the distance between two vectors, numbers, strings etc.
• SciPy -Distance computations (scipy.spatial.distance)
• Julia Statistics Distance -A Julia package for evaluating distances (metrics) between vectors.
• „The Directed Distance” (PDF). Information and Telecommunication Technology Center. University of Kansas. Архивирано из оригинала (PDF) 10. 11. 2016. г. Приступљено 18. 9. 2018. CS1 одржавање: Формат датума (веза)