Distance is a numerical description of how far apart objects are. In physics Physics is a natural science that involves the study of matter and its motion through space-time, as well as all applicable concepts, such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria (e.g. "two counties over"). In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, a distance function or metric In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric. When a topological space has a topology that can be described by a metric, we say that the is a generalization of the concept of physical distance. A metric is a function that behaves according to a specific set of rules, and provides a concrete way of describing what it means for elements of some space to be "close to" or "far away from" each other.

In most cases, "distance from A to B" is interchangeable with "distance between B and A".

Mathematics

Geometry

In neutral geometry Absolute geometry is a geometry based on an axiom system that does not assume the parallel postulate or any of its alternatives. The term was introduced by János Bolyai in 1832. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate, the distance between (x₁) and (x₂) is the length of the line segment In Euclidean geometry, a line is a straight curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height. Lines are an idealisation of such objects and have no width or height at all and are usually considered to be infinitely long. Lines are a fundamental concept in some between them:

In analytic geometry Analytic geometry, also known as coordinate geometry, analytical geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning based on, the distance between two points of the xy-plane A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length can be found using the distance formula. The distance between (x₁, y₁) and (x₂, y₂) is given by:

Similarly, given points (x₁, y₁, z₁) and (x₂, y₂, z₂) in three-space A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length, the distance between them is:

These formulae are easily derived by constructing a right triangle with a leg on the hypotenuse A hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides of another (with the other leg orthogonal In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. The word comes from the Greek ὀρθός , meaning "straight", and γωνία (gonia), meaning "angle" to the plane In mathematics, a plane is any flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line (one-dimension) and a space (three-dimensions). Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of that contains the 1st triangle) and applying the Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle . In terms of areas, it states:.

In the study of complicated geometries, we call this (most common) type of distance Euclidean distance In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space. The associated norm is called the Euclidean norm. Older literature refers to the, as it is derived from the Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle . In terms of areas, it states:, which does not hold in Non-Euclidean geometries A non-Euclidean geometry is the study of shapes and constructions that do not map directly to any n-dimensional Euclidean system, characterized by a non-vanishing Riemann curvature tensor. Examples of non-Euclidean geometries include the hyperbolic and elliptic geometry, which are contrasted with a Euclidean geometry. The essential difference. This distance formula In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language can also be expanded into the arc-length formula Determining the length of an irregular arc segment—also called rectification of a curve—was historically difficult. Although many methods were used for specific curves, the advent of calculus led to a general formula that provides closed-form solutions in some cases.

Distance in Euclidean space

In the Euclidean space In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” is used to distinguish these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity Rⁿ, the distance between two points is usually given by the Euclidean distance In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space. The associated norm is called the Euclidean norm. Older literature refers to the (2-norm distance). Other distances, based on other norms In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector. A seminorm , on the other hand, is allowed to assign zero length to some non-zero vectors, are sometimes used instead.

For a point (x₁, x₂, ...,x_n) and a point (y₁, y₂, ...,y_n), the Minkowski distance of order p (p-norm distance) is defined as:

p need not be an integer, but it cannot be less than 1, because otherwise the triangle inequality In mathematics, the triangle inequality states that for any triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater than or equal to the difference between the two sides does not hold.

The 2-norm distance is the Euclidean distance In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space. The associated norm is called the Euclidean norm. Older literature refers to the, a generalization of the Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle . In terms of areas, it states: to more than two coordinates In geometry, a coordinate system is a system which uses a set of numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in 'the x-coordinate'. In elementary. It is what would be obtained if the distance between two points were measured with a ruler A ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing and engineering/building to measure distances and/or to rule straight lines. Strictly speaking, the ruler is essentially a straightedge used to rule lines[citation needed] and the calibrated instrument used for determining measurement is: the "intuitive" idea of distance.

The 1-norm distance is more colourfully called the taxicab norm or Manhattan distance Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the differences of their coordinates. The taxicab metric is also known as rectilinear distance, L1 distance or 1 norm (see Lp, because it is the distance a car would drive in a city laid out in square blocks (if there are no one-way streets).

The infinity norm distance is also called Chebyshev distance In mathematics, Chebyshev distance , or L∞ metric is a metric defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension. It is named after Pafnuty Chebyshev. It is also known as chessboard distance, since in the game of chess the minimum number of moves needed by a king. In 2D, it is the minimum number of moves kings In chess, the King is the most important piece. The object of the game is to trap the opponent's king so that he would not be able to avoid capture (checkmate). If a player's king is threatened with capture, he is said to be in check, and the player must move so as to remove the threat of capture. If he cannot escape capture on the next move, the require to travel between two squares on a chessboard A chessboard is the type of checkerboard used in the game of chess, and consists of 64 squares arranged in two alternating colors (light and dark). The colors are called "black" and "white" (or "light" and "dark"), although the actual colors are usually dark green and buff for boards used in competition, and.

The p-norm is rarely used for values of p other than 1, 2, and infinity, but see; super ellipse This formula defines a closed curve contained in the rectangle −a ≤ x ≤ +a and −b ≤ y ≤ +b. The parameters a and b are called the semi-diameters of the curve.

In physical space the Euclidean distance is in a way the most natural one, because in this case the length of a rigid body In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it. Even though such an object cannot physically exist due to relativity, objects can normally be does not change with rotation A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A three-dimensional object rotates always around an imaginary line called an axis as the Euler's rotation theorem shows. If the axis of rotation is within the body, the body is said to rotate upon itself, or spin—which.

Variational formulation of distance

The Euclidean distance between two points in space ( and ) may be written in a variational Calculus of variations is a field of mathematics that deals with functionals, as opposed to ordinary calculus which deals with functions. Such functionals can for example be formed as integrals involving an unknown function and its derivatives. The interest is in extremal functions – those making the functional attain a maximum or minimum value � form where the distance is the minimum value of an integral:

Here is the trajectory (path) between the two points. The value of the integral (D) represents the length of this trajectory. The distance is the minimal value of this integral and is obtained when r = r ^* where r ^* is the optimal trajectory. In the familiar Euclidean case (the above integral) this optimal trajectory is simply a straight line. It is well known that the shortest path between two points is a straight line. Straight lines can formally be obtained by solving the Euler-Lagrange equations for the above functional In mathematics, and particularly in functional analysis, a functional is traditionally a map from a vector space to the field underlying the vector space, which is usually the real numbers. In other words, it is a function that takes a vector as its argument or input and returns a scalar. Commonly the vector space is a space of functions, so the. In non-Euclidean A non-Euclidean geometry is the study of shapes and constructions that do not map directly to any n-dimensional Euclidean system, characterized by a non-vanishing Riemann curvature tensor. Examples of non-Euclidean geometries include the hyperbolic and elliptic geometry, which are contrasted with a Euclidean geometry. The essential difference manifolds (curved spaces) where the nature of the space is represented by a metric In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric. When a topological space has a topology that can be described by a metric, we say that the g_ab the integrand has be to modified to , where Einstein summation convention In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulas. It was introduced by Albert Einstein in 1916 has been used.

Generalization to higher dimensional objects

The Euclidean distance between two objects may also be generalized to the case where the objects are no longer points but are higher-dimensional manifolds In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold. Thus a line and a circle are one-dimensional manifolds, a plane and sphere are two-dimensional manifolds, and so forth, such as space curves, so in addition to talking about distance between two points one can discuss concepts of distance between two strings. Since the new objects that are dealt with are extended objects (not points anymore) additional concepts such as non-extensibility, curvature In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context. There is a key distinction between extrinsic constraints, and non-local interactions that enforce non-crossing become central to the notion of distance. The distance between the two manifolds is the scalar quantity that results from minimizing the generalized distance functional, which represents a transformation between the two manifolds:

The above double integral is the generalized distance functional between two plymer conformation. s is a spatial parameter and t is pseudo-time. This means that is the polymer/string conformation at time t_i and is parameterized along the string length by s. Similarly is the trajectory of an infinitesimal segment of the string during transformation of the entire string from conformation to conformation . The term with cofactor λ is a Lagrange multiplier In mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maximum/minimum of a function subject to constraints and its role is to ensure that the length of the polymer remains the same during the transformation. If two discrete polymers are inextensible, then the minimal-distance transformation between them no longer involves purely straight-line motion, even on a Euclidean metric. There is a potential application of such generalized distance to the problem of protein folding Protein folding is the physical process by which a polypeptide folds into its characteristic and functional three-dimensional structure from random coil. Each protein exists as an unfolded polypeptide or random coil when translated from a sequence of mRNA to a linear chain of amino acids. This polypeptide lacks any developed three-dimensional^[1][2] This generalized distance is analogous to the Nambu-Goto action in string theory String theory is a developing theory in particle physics which attempts to reconcile quantum mechanics and general relativity. String theory posits that the electrons and quarks within an atom are not 0-dimensional objects, but rather 1-dimensional oscillating lines , possessing only the dimension of length, but not height or width. The theory, however there is no exact correspondence because the Euclidean distance in 3-space is inequivalent to the space-time distance minimized for the classical relativistic string.

Algebraic distance

The algebraic distance is a metric often used in computer vision Computer vision is the science and technology of machines that see. As a scientific discipline, computer vision is concerned with the theory behind artificial systems that extract information from images. The image data can take many forms, such as video sequences, views from multiple cameras, or multi-dimensional data from a medical scanner that that can be minimized by least squares The method of least squares is applied to approximate solutions of overdetermined systems, i.e. systems of equations in which there are more equations than unknowns. Least squares is often applied in statistical contexts, particularly regression analysis estimation. [1] [2] For curves or surfaces given by the equation x^TCx = 0 (such as a conic in homogeneous coordinates In mathematics, a conic section is a curve obtained by intersecting a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. It can be defined as the locus of points whose distances are in a fixed ratio to some point, called the focus, and some line,), the algebraic distance from the point x' to the curve is simply x'^TCx'. It may serve as an "initial guess" for geometric distance In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. By using this formula as distance, Euclidean space becomes a metric space . The associated norm is called the Euclidean norm to refine estimations of the curve by more accurate methods, such as non-linear least squares Non-linear least squares is the form of least squares analysis which is used to fit a set of m observations with a model that is non-linear in n unknown parameters . It is used in some forms of non-linear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations. There are.

General case

In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, in particular geometry Geometry "Earth-measuring" is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by, a distance function on a given set M is a function d: M×M → R, where R denotes the set of real numbers, that satisfies the following conditions:

• d(x,y) ≥ 0, and d(x,y) = 0 if and only if x = y. (Distance is positive between two different points, and is zero precisely from a point to itself.)
• It is symmetric: d(x,y) = d(y,x). (The distance between x and y is the same in either direction.)
• It satisfies the triangle inequality: d(x,z) ≤ d(x,y) + d(y,z). (The distance between two points is the shortest distance along any path).

Such a distance function is known as a metric. Together with the set, it makes up a metric space.

For example, the usual definition of distance between two real numbers x and y is: d(x,y) = |x − y|. This definition satisfies the three conditions above, and corresponds to the standard topology of the real line. But distance on a given set is a definitional choice. Another possible choice is to define: d(x,y) = 0 if x = y, and 1 otherwise. This also defines a metric, but gives a completely different topology, the "discrete topology"; with this definition numbers cannot be arbitrarily close.

Distances between sets and between a point and a set

Various distance definitions are possible between objects. For example, between celestial bodies one should not confuse the surface-to-surface distance and the center-to-center distance. If the former is much less than the latter, as for a LEO, the first tends to be quoted (altitude), otherwise, e.g. for the Earth-Moon distance, the latter.

There are two common definitions for the distance between two non-empty subsets of a given set:

• One version of distance between two non-empty sets is the infimum of the distances between any two of their respective points, which is the every-day meaning of the word. This is a symmetric premetric. On a collection of sets of which some touch or overlap each other, it is not "separating", because the distance between two different but touching or overlapping sets is zero. Also it is not hemimetric, i.e., the triangle inequality does not hold, except in special cases. Therefore only in special cases this distance makes a collection of sets a metric space.
• The Hausdorff distance is the larger of two values, one being the supremum, for a point ranging over one set, of the infimum, for a second point ranging over the other set, of the distance between the points, and the other value being likewise defined but with the roles of the two sets swapped. This distance makes the set of non-empty compact subsets of a metric space itself a metric space.

The distance between a point and a set is the infimum of the distances between the point and those in the set. This corresponds to the distance, according to the first-mentioned definition above of the distance between sets, from the set containing only this point to the other set.

In terms of this, the definition of the Hausdorff distance can be simplified: it is the larger of two values, one being the supremum, for a point ranging over one set, of the distance between the point and the set, and the other value being likewise defined but with the roles of the two sets swapped.

Graph theory

In graph theory the distance between two vertices is the length of the shortest path between those vertices.

Distance versus directed distance and displacement

Distance cannot be negative. Distance travelled never decreases. Distance is a scalar quantity, containing only a magnitude, whereas a displacement vector is a vector quantity characterized by both magnitude and direction.Distance from starting point may not be equal to distance travelled.

The distance covered by a vehicle (often recorded by an odometer), person, animal, or object along a curved path from a point A to a point B should be distinguished from the respective displacement (the distance along a straight line from A to B). For instance, the distance covered during a round trip from A to B and back to A may be very long, while the displacement is always zero (because starting and ending points coincide).

Directed distance

Directed distances are distances with a direction or sense. They can be determined along straight lines and along curved lines. A directed distance along a straight line from A to B is a vector joining any two points in a n-dimensional Euclidean vector space. A directed distance along a curved line is not a vector and is represented by a segment of that curved line defined by endpoints A and B, with some specific information indicating the sense (or direction) of an ideal or real motion from one endpoint of the segment to the other (see figure). For instance, just labelling the two endpoints as A and B can indicate the sense, if the ordered sequence (A, B) is assumed, which implies that A is the starting point.

A displacement (see above) is a special kind of directed distance defined in mechanics. A directed distance is called displacement when it is the distance along a straight line (minimum distance) from A and B, and when A and B are positions occupied by the same particle at two different instants of time. This implies motion of the particle.

Another kind of directed distance is that between two different particles or point masses at a given time. For instance, the distance from the center of gravity of the Earth A and the center of gravity of the Moon B (which does not strictly imply motion from A to B).Shortest path length may be equal to displacement or may not be equal to.Distance from starting point is always equal to magnitude of displacement. For same particle distance travelled is always greater than or equal to magnitude of displacement

Other "distances"

• Mahalanobis distance is used in statistics.
• Hamming distance and Lee distance are used in coding theory.
• Levenshtein distance
• Chebyshev distance

Circular distance is the distance traveled by a wheel. The circumference of the wheel is 2*(pi)*(radius), and assuming the radius to be 1, then each revolution of the wheel is equivalent of the distance 2*(pi) radians. In engineering (omega)=2*(pi)*f is used a lot, where f is the frequency.

See also

References

1. ^ SS Plotkin, PNAS.2007; 104: 14899-14904,
2. ^ AR Mohazab, SS Plotkin,"Minimal Folding Pathways for Coarse-Grained Biopolymer Fragments" Biophysical Journal, Volume 95, Issue 12, Pages 5496-5507

• Deza, E.; Deza, M. (2006), Dictionary of Distances, Elsevier, ISBN 0444520872 .

Categories: Length | Elementary mathematics

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