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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. In most cases there is symmetry and "distance from A to B" is interchangable with "distance between A and B". MATHEMATICS See Also: Metric (mathematics) Geometry In Neutral Geometry , the minimum distance between two points is the length of the Line Segment between them. In Algebraic Geometry , one can find the distance between two points of the Xy-plane using the distance formula. The distance between (''x''1, ''y''1) and (''x''2, ''y''2) is given by : Similarly, given points (''x''1, ''y''1, ''z''1) and (''x''2, ''y''2, ''z''2) in Three-space , the distance between them is : 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 . 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 . 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. For a point (''x''1, ''x''2, ...,''x''''n'') and a point (''y''1, ''y''2, ...,''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 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. 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). The infinity norm distance is also called Chebyshev Distance . In 2D it represents the distance King s must travel between two squares on a Chessboard . The ''p''-norm is rarely used for values of ''p'' other than 1, 2, and infinity, but see Super Ellipse . 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 . General case In s, that satisfies the following conditions:
Such a distance function is known as a Metric . Together with the set, it makes up a Metric Space . |
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