Distance

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Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria (e.g. "two counties over"). In mathematics, a distance function or metric 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".

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Mathematics

Geometry

In neutral geometry, the distance between (x1) and (x2) is the length of the line segment between them:

In analytic geometry, the distance between two points of the xy-plane 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, the distance between them is:

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.

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