Distance
In most cases, "distance from A to B" is interchangeable with "distance between B and A". See also: Metric (mathematics)
Geometry In neutral geometry, the distance between ( In analytic geometry, the distance between two points of the xy-plane can be found using the distance formula. The distance between ( Similarly, given points ( These formulae are easily derived 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. In the Euclidean space For a point (
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 The infinity norm distance is also called Chebyshev distance. In 2D, it is the minimum number of moves kings require to travel between two squares on a chessboard. The 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. The Euclidean distance between two points in space ( and ) may be written in a variational 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 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, 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 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.
The In mathematics, in particular geometry, a distance function on a given set - 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 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. In graph theory the distance between two vertices is the length of the shortest path between those vertices. Distance cannot be negative and distance travelled never decreases. Distance is a scalar quantity or a magnitude, whereas displacement is a vector quantity with both magnitude and direction. The distance covered by a vehicle (for example as recorded by an odometer), person, animal, or object along a curved path from a point 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 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 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 - E-statistics, or energy statistics, are functions of distances between statistical observations.
- 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
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