Distance

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Distance is a numerical measure of how far the object is. In physics or everyday use, distance can refer to physical length or estimation based on other criteria (eg "the two districts above"). In many cases, "distance from A to B" can be exchanged with "distance from B to A". In mathematics, distance or metric functions are generalizations of the concept of physical distance. Metrics are functions that behave in accordance with a certain set of rules, and are a way of describing what it means for elements of some space to "close" or "away from" each other.

Video Distance

Overview and definitions

Physical distance

Physical distance can mean several different things:

• The length of a particular path being crawled between two points, such as walking distance while navigating the labyrinth

• The shortest possible path length through space between two points can be taken if there are no obstacles (usually formalized as Euclidean distance)
• The shortest path length between two points while remaining on multiple surfaces, such as large-circle spacing along the curve of the Earth

"Circular distance" is the distance traveled by the wheel, which can be useful when designing a vehicle or mechanical gear. The wheel circumference is 2 ? Ã, â € "radius, and assuming radius to 1, then every wheel revolution equals distance 2 ? radians. In the technique? Ã, = Ã, 2 ?? is often used, where ? is the frequency.

The definition of unusual distance can help to model a particular physical situation, but it is also used in mathematical theory:

• "Manhattan distance" is a straight line distance, named after the number of blocks north, south, east or west of a taxi must travel to reach its destination on the road network in New York City.
• "Chessboard spacing", formalized as Chebyshev distance, is the minimum number of moves a king should make on a chessboard to travel between two boxes.

The size of the distance in cosmology is complicated by the expansion of the universe, and by the effects described by the theory of relativity as the contraction of the length of a moving object.

Theoretical distance

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

In computer science, there is the idea of ​​"edit distance" between two strings. For example, the words "dog" and "point", which varies only by one letter, closer than "dog" and "cat", which are different from the 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's distance
• Jaro-Winkler distance

In mathematics, the space metric is a set whose distances between all members of the set are defined. In this way, many different types of "distances" can be calculated, such as for chart traversal, distribution and curve comparison, and using unusual "space" definitions (eg using manifolds or reflections). The idea of ​​distance in graph theory has been used to describe social networks, for example by the Erd number or Bacon number, the number of collaborative relationships of a person is from productive mathematician Paul Erd or actor Kevin Bacon, respectively..

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

Maps Distance

Distance versus distance and redirected moves

Distance can not be negative, and the distance traveled never decreases. Distance is scalar quantity or quantity, while displacement is vector quantity with magnitude and direction. The range directed is a positive, zero, or negative scalar quantity.

The distance traveled by the vehicle (eg as recorded by the odometer), persons, animals, or objects along the curved path from point A to point B should be distinguished from a straight line distance from A to B . For example, whatever the distance traveled during the round trip from A to B and back to A , the displacement is zero as the starting and ending points coincide. In general the straight line distance is not the same distance traveled, except for traveling in a straight line.

The redirected distance

Directed distance is the distance with the directional meaning. They can be determined along the straight line and along the curved line. The distance directed from the point of C from the point A towards B at the line AB in Euclidean vector space is the distance from C if C falls on the rays of AB , but negative from that distance if < i> falls on the light BA (That is, if C is not on the same side A .

The distance directed along the curve line is not a vector and is represented by a segment of the curve determined by the endpoints A and B , with some specific information indicating the meaning (or direction) of the motion ideal or real from one endpoint segment to another (see figure). For example, simply labeling two end points as A and B can indicate meaning, if the order is ordered ( A , B ) is assumed, implying that A is the starting point.

Moving

The shift (see above) is a specially directed range of distances defined in mechanics. The directed distance is called displacement when it is the distance along the straight line (minimum distance) of A and B , and when A and B is the position occupied by the same particle at two different times time. This implies the movement of particles. The distance traveled by the particle must always be greater or equal to its displacement, with equations only occurring when particles move along a straight path.

Another type of directional distance is that between two particles or different mass points at a given time. For example, the distance from the center of gravity of Earth A and the center of gravity of the Moon B (which does not strictly imply movement from A to B ) fall into this category.

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Math

Geometry

Dalam geometri analitik, jarak antara dua titik dari xy-plane dapat ditemukan menggunakan rumus jarak. Jarak antara ( x ₁ , y ₁ ) dan ( x ₂ , y ₂ ) diberikan oleh:

${\ displaystyle d = {\ sqrt {(\ Delta x) ^ {2} (\ Delta y) ^ {2}}} = {\ sqrt {(x_ { 2} -x_ {1}) ^ {2} (y_ {2} -y_ {1}) ^ {2}}}.}  ÂÂ$

Demikian pula, poin yang diberikan ( x ₁ , y ₁ , z ₁ ) dan ( x ₂ , y ₂ , z ₂ ) dalam tiga ruang, jarak antara keduanya adalah:

${\ displaystyle d = {\ sqrt {(\ Delta x) ^ {2} (\ Delta y) ^ {2} (\ Delta z) ^ {2} }} = {\ sqrt {(x_ {2} -x_ {1}) ^ {2} (y_ {2} -y_ {1}) ^ {2} (z_ {2} -z_ {1}) ^ {2}}}.}  ÂÂ$

This formula is easily obtained by constructing a right triangle with the foot on the other side (with the other orthogonal legs to the field containing triangle 1) and applying the Pythagoras theorem. In a complex geometry study, we call this (the most common) type of Euclidean distance spacing, as derived from the Pythagorean theorem, which does not apply in non-Euclidean geometry. This distance formula can also be extended to the arc length formula.

Distance in Euclidean space

In Euclidean space R ⁿ , the distance between two points is usually given by Euclidean distance (2-norm distance). Other distance, based on other norms, is sometimes used instead.

For a point ( x ₁ , x _{2 n} ) and a dot ( y ₁ /sub>,..., y _n ), Minkowski distance i> ( p -norm distance ) is defined as:

p does not need to be an integer, but should not be less than 1, because otherwise triangle inequality does not apply.

The 2-norm distance is the Euclidean distance, a generalization of the Pythagorean theorem to more than two coordinates. This is what will be obtained if the distance between two points is measured by a ruler: the idea of ​​"intuitive" distance.

The more colorful 1-norm spacing is called the taxicab norm or Manhattan distance , because it is the distance the car will drive in a city laid out in a square block (if there is no one direction).

The distance of the infinity norm is also called the Chebyshev distance. In 2D, it is the minimum number required by a moving king between two boxes on a chess board.

The p -norm is rarely used for p values ​​other than 1, 2, and unlimited, but see superlipses.

In physical space the Euclidean distance is the most natural way, because in this case the stiff body length does not change with rotation.

Variational distance formulation

Jarak Euclidean antara dua titik dalam ruang ( ${\ displaystyle A = {\ vec {r}} (0)} Â$ dan ${\ displaystyle B = {\ vec {r}} (T)} Â Â$ ) dapat ditulis dalam bentuk variasional di mana jarak adalah nilai minimum integral:

${\ displaystyle D = \ int _ {0} ^ {T} {\ sqrt {\ kiri ({\ parsial {\ vec {r}} (t ) \ over \ parsial t} \ kanan) ^ 2}}} \, dt} Â Â$

Di sini ${\ displaystyle {\ vec {r}} (t)} Â$ adalah lintasan (jalur) antara second titik. Nilai integral (D) mewakili panjang lintasan ini. Jarak adalah nilai minimal integral ini dan diperoleh ketika ${\ displaystyle r = r ^ {*}} Â$ di mana ${\ displaystyle r ^ {*}} Â Â$ adalah lintasan optimal. Dalam kasus Euclidean yang familier (integral di atas) Lintasan optimal ini hanyalah garis lurus. Sudah diketahui dengan baik bahwa jalur terpendek former second title adalah garis lurus. Garis lurus secara formal dapat diperoleh dengan memecahkan persamaan Euler-Lagrange untuk fungsional di atas. Dalam manifold non-Euclidean (ruang lengkung) di mana sifat ruang diwakili oleh metrik tensor ${\ displaystyle g_ {ab}} Â$ integand harus dimodifikasi menjadi ${\ displaystyle {\ sqrt {g ^ {ac} {\ dot {r}} _ {c} g_ {ab} {\ dot {r}} ^ {b}}}} Â Â$ , di mana konvensi penjumlahan Einstein telah digunakan.

Generalisasi ke objek dengan dimensi lebih tinggi

Ini adalah metrik yang sering digunakan dalam visi komputer yang dapat diminimalkan dengan estimasi kuadrat terkecil. [1] [2] Untuk kurva atau permukaan yang diberikan oleh persamaan ${\ displaystyle x ^ {\ text {T}} Cx = 0} Â$ (seperti konik dalam koordinat homogen), jarak aljabar dari titik ${\ displaystyle x '} Â Â$ every kurva hanya ${\ displaystyle x '^ {\ text {T}} Cx'} Â Â$ . Ini dapat berfungsi sebagai "tebakan awal" untuk jarak geometrik untuk memperbaiki estimasi kurva dengan metode yang lebih akurat, seperti kuadrat terkecil non-linear.

Metrik umum

In mathematics, in particular geometry, the distance function of the given set M is the function d /i> -> R , where R indicates a set of real numbers, which meet the following conditions:

• d ( x , y )> = 0 , and ( x , y ) = 0 if and only if y . (The positive distance between two different points, and zero exactly from one point to the point itself.)
• It's symmetrical: d ( x , y ) = d ( y , x ) . (The distance between x and y equals both directions.)
• It meets the triangle inequality: d ( x , z i> x , y ) d ( y , z ) . (The distance between two points is the shortest distance along the path). Such proximity functions are known as metrics. Along with the set, it makes the metric space.

For example, the definition of the distance between two real numbers x and y is: d ( x , < i> y ) = | x - y | . This definition meets the above three conditions, and conforms to the standard topology of the real line. But the distance on the given set is a definite choice. Another possible option is to define: d ( x , y ) = 0 if x = y , and 1 vice versa. It also defines metrics, but gives an entirely different topology, "discrete topology"; with this definition number can not be arbitrarily shut down.

The distance between set and between point and set

Various definitions of distance are possible between objects. For example, among the heavenly bodies one should not confuse the surface distance to the surface and the center distance to the center. If the former is much less than the last, such as for low Earth orbit, the first tends to be quoted (altitude), if not, for example for the Earth-Moon distance, the latter.

Ada dua definisi umum untuk jarak antara dua subset yang tidak kosong dari ruang metrik yang diberikan:

• Satu versi jarak antara dua set yang tidak kosong adalah jarak maksimum antara dua titik masing-masing, yang merupakan arti kata sehari-hari, yaitu.

${\ displaystyle d (A, B) = \ inf _ {x \ dalam A, y \ dalam B} d (x, y).}  ÂÂ$

Ini adalah premet simetris. Pada kumpulan set yang beberapa sentuhan atau tumpang tindih satu sama lain, itu tidak "memisahkan", karena jarak antara dua set yang berbeda tetapi menyentuh atau tumpang tindih adalah nol. Juga tidak hemimetric, yaitu, ketimpangan segitiga tidak berlaku, kecuali dalam kasus-kasus khusus. Oleh karena itu hanya dalam kasus khusus jarak ini membuat kumpulan set ruang metrik.

• Jarak Hausdorff lebih besar dari dua nilai, satu sebagai supremum, untuk titik yang berkisar lebih dari satu set, dari infimum, untuk titik kedua yang berkisar dari set yang lain, dari jarak antara titik-titik, dan nilai lain yang juga didefinisikan tetapi dengan peran dari dua set bertukar. Jarak ini membuat himpunan himpunan ringkas yang tidak kosong dari ruang metrik itu sendiri merupakan ruang metrik.

The distance between a point and a set is the infimum distance between the point and the point in the set. This corresponds to the distance, corresponding to the above-mentioned definition above from the distance between sets, from sets containing only this point to another set.

In this case, the definition of the Hausdorff distance can be simplified: it is greater than two values, one being supremum, for points ranging from more than one set, the spacing between points and sets, and other values. also defined but with the role of two sets of exchanges.

Graph theory

In graph theory, the distance between two vertices is the shortest path length between the two nodes.

Other mathematical distances "

• Canberra Distance - a weighted version of Manhattan distance, used in computer science
• Energy distance, statistical distance between probability distributions
• Kullback-Leibler Divergences, which measures the difference between two probability distributions
• Mahalanobis distance is used in statistics

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See also

• System of astronomical units
• Cosmic distance ladder
• Remote geometry issues
• Dijkstra Algorithm
• Distance matrix
• Gauge (DME)
• Remote line-based out number
• Technical tolerance
• Length
• Meridian bow
• Milestone
• Order size (length)
• The exact length
• Proxemics - physical distance between people
• Spy
• The marked distance function

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References

Notes

Source

• Deza, E ;; Deza, M. (2006), Distance Dictionary , Elsevier, ISBNÃ, 0-444-52087-2 .

Source of the article : Wikipedia