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Encyclopedia > Distance

Distance is a numerical description of how far apart objects are at any given moment in time. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria (e.g. "two counties over"). In mathematics, distance must meet more rigorous criteria. Physics (Greek: (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the science concerned with the fundamental laws of the universe. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...

People figures standing at a distance from one another

Image File history File linksMetadata Download high resolution version (2400x1466, 2363 KB) Summary Edit of for the FPC. Licensing Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ... Image File history File linksMetadata Download high resolution version (2400x1466, 2363 KB) Summary Edit of for the FPC. Licensing Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ...

### Geometry

In neutral geometry, the minimum distance between two points is the length of the line segment between them. Absolute geometry is a geometry that does not assume the parallel postulate or any of its alternatives. ... Three lines â€” the red and blue lines have same slope, while the red and green ones have same y-intercept. ...

In algebraic geometry, one can find the distance between two points of the xy-plane using the distance formula. The distance between (x1, y1) and (x2, y2) is given by Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Fig. ...

$d=sqrt{(Delta x)^2+(Delta y)^2}=sqrt{(x_1-x_2)^2+(y_1-y_2)^2}.,$

This formula could also be used as follows:

$d=sqrt{(-Delta x)^2+(-Delta y)^2}=sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.$

Similarly, given points (x1, y1, z1) and (x2, y2, z2) in three-space, the distance between them is Fig. ...

$d=sqrt{(Delta x)^2+(Delta y)^2+(Delta z)^2}=sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}.$

Which is easily proven by constructing a right triangle with a leg on the hypotenuse of another (with the other leg orthogonal to the plane that contains the 1st triangle) and applying the Pythagorean theorem. For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ... In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry among the three sides of a right triangle. ...

In the study of complicated geometries, we call this (most common) type of distance Euclidean distance, as it is derived from the Pythagorean theorem, which does not hold in Non-Euclidean geometries. This distance formula can also be expanded into the arc-length formula. In mathematics, the Euclidean distance or Euclidean metric is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application 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. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... In mathematics and in the sciences, a formula (plural: formulae, formulÃ¦ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... For other uses, see Curve (disambiguation). ...

### Distance in Euclidean space

In the Euclidean space Rn, the distance between two points is usually given by the Euclidean distance (2-norm distance). Other distances, based on other norms, are sometimes used instead. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, the Euclidean distance or Euclidean metric is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

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

 1-norm distance $= sum_{i=1}^n left| x_i - y_i right|$ 2-norm distance $= left( sum_{i=1}^n left| x_i - y_i right|^2 right)^{1/2}$ p-norm distance $= left( sum_{i=1}^n left| x_i - y_i right|^p right)^{1/p}$ infinity norm distance $= lim_{p to infty} left( sum_{i=1}^n left| x_i - y_i right|^p right)^{1/p}$ $= max left(|x_1 - y_1|, |x_2 - y_2|, ldots, |x_n - y_n| right).$

p need not be an integer, but it cannot be less than 1, because otherwise the triangle inequality does not hold.

The 2-norm distance is the Euclidean distance, a generalization of the Pythagorean theorem to more than two coordinates. It is what would be obtained if the distance between two points were measured with a ruler: the "intuitive" idea of distance. In mathematics, the Euclidean distance or Euclidean metric is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application 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. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... A variety of rulers A 2 metre carpenters rule Retractable flexible rule A ruler or rule is an instrument used in geometry, technical drawing and engineering/building to measure distances and/or to rule straight lines. ...

The 1-norm distance is more colourfully called the taxicab norm or Manhattan distance, because it is the distance a car would drive in a city laid out in square blocks (if there are no one-way streets). Manhattan versus Euclidean distance: The red, blue, and yellow lines representing the Manhattan distance all have the same length (12), whereas the green line representing the Euclidian distance has length 6Ã—âˆš2 â‰ˆ 8. ...

The infinity norm distance is also called Chebyshev distance. In 2D it represents the distance kings must travel between two squares on a chessboard. In a plane, the Chebyshev distance between the point P1 with coordinates (x1, y1) and the point P2 at (x2, y2) is This concept is named after Pafnuty Chebyshev. ... Staunton chess pieces, left to right: pawn, rook, knight, bishop, queen, and king. ... A chessboard is often painted or engraved on a chess table. ...

The p-norm is rarely used for values of p other than 1, 2, and infinity, but see super ellipse. A superellipse is a geometrical figure which in a cartesian coordinate system can be described as the set of all points (x, y) with where and and are the radii of the oval shape. ...

In physical space the Euclidean distance is in a way the most natural one, because in this case the length of a rigid body does not change with rotation. In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. ... A sphere rotating around its axis. ...

### General case

In mathematics, in particular geometry, 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: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Table of Geometry, from the 1728 Cyclopaedia. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Partial plot of a function f. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

• 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. It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a. ... In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ... In mathematics a metric or distance function is a function which defines a distance between elements of a set. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

For example, the usual definition of distance between two real numbers x and y is: d(x,y) = |xy|. 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. A MÃ¶bius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... In mathematics, the real line is simply the set of real numbers. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...

### Distances between non-empty sets

One might attempt to define the distance between two non-empty subsets of a given set as the infimum of the distances between any two of their respective points, which would agree with the every-day use of the word. However, this does not define a metric, since with this definition the distance between two different but overlapping sets is zero. A definition that does work defines the distance as 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 is a metric, called the Hausdorff metric. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all other elements of the subset. ... In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ... Hausdorff distance measures how far two compact subsets of a metric space are from each other. ...

## Distinguish

Distance along a path compared with displacement

As opposed to a position coordinate, a distance cannot be negative. Distance is a scalar quantity, containing only a magnitude, whereas displacement is an equivalent vector quantity containing both magnitude and direction. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... A negative number is a number that is less than zero, such as &#8722;3. ... In physics, a scalar is a simple physical quantity that does not depend on direction, and therefore does not depend on the choice of a coordinate system. ... The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ... In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ... In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... The term direction can be applied to various topics. ...

The distance covered by a vehicle (often recorded by an odometer), person, animal, object, etc. should be distinguished from the distance from starting point to end point, even if latter is taken to mean e.g. the shortest distance along the road, because a detour could be made, and the end point can even coincide with the starting point. A modern non-digital odometer A Smiths speedometer from the 1920s showing odometer and trip meter An odometer is a device used for indicating distance traveled by an automobile or other vehicle. ...

## Other "distances"

In statistics, Mahalanobis distance is a distance measure introduced by P. C. Mahalanobis in 1936. ... Template:Otherusescccc A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... In information theory, the Hamming distance, named after Richard Hamming, is the number of positions in two strings of equal length for which the corresponding elements are different. ... Coding theory is a branch of mathematics and computer science dealing with the error-prone process of transmitting data across noisy channels, via clever means, so that a large number of errors that occur can be corrected. ... In information theory and computer science, the Levenshtein distance is a string metric which is one way to measure edit distance. ...

## References

• Abdi, H. "[1] ((2007). Distance. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.".

Results from FactBites:

 Distance education - Wikipedia, the free encyclopedia (1325 words) Distance education, or distance learning, is a field of education that focuses on the pedagogy/andragogy, technology, and instructional systems design that are effectively incorporated in delivering education to students who are not physically "on site" to receive their education. In short then, though a range of technology presupposes a distance education 'inventory' it is technological appropriateness and connectivity, such as computer, or for that matter electrical connectivity that should be considered, when we think of the world as a whole, while fitting in technological applications to distance education. Distance education programs are sometimes called correspondence courses, an older term that originated in nineteenth-century vocational education programs that were conducted through postal mail.
 Distance (755 words) In the case of two locations on Earth, usually the distance along the surface is meant: either "as the crow flies" (along a great circle) or by road, railroad, etc. Distance is sometimes expressed in terms of the time to cover it, for example walking or by car. A distance between two points P and Q in a metric space is d(P,Q), where d is the distance function that defines the given metric space. Alternatively, the distance between sets may indicate "how different they are", by taking the supremum over one set of the distance from a point in that set to the other set, and conversely, and taking the larger of the two values (Hausdorff distance).
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