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Concept# Distance

Summary

Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). Since spatial cognition is a rich source of conceptual metaphors in human thought, the term is also frequently used metaphorically to mean a measurement of the amount of difference between two similar objects (such as statistical distance between probability distributions or edit distance between strings of text) or a degree of separation (as exemplified by distance between people in a social network). Most such notions of distance, both physical and metaphorical, are formalized in mathematics using the notion of a metric space.
In the social sciences, distance can refer to a qualitative measurement of separation, such as social distance or psychological distance.
The distance between physical locations can be defined in different ways in different contexts.
Euclidean distance
The distance between two points in physical space is the length of a straight line between them, which is the shortest possible path. This is the usual meaning of distance in classical physics, including Newtonian mechanics.
Straight-line distance is formalized mathematically as the Euclidean distance in two- and three-dimensional space. In Euclidean geometry, the distance between two points A and B is often denoted . In coordinate geometry, Euclidean distance is computed using the Pythagorean theorem. The distance between points (x1, y1) and (x2, y2) in the plane is given by:
Similarly, given points (x1, y1, z1) and (x2, y2, z2) in three-dimensional space, the distance between them is:
This idea generalizes to higher-dimensional Euclidean spaces.
Distance measurement
There are many ways of measuring straight-line distances. For example, it can be done directly using a ruler, or indirectly with a radar (for long distances) or interferometry (for very short distances).

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Euclidean distance

In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation was not made until the 18th century.

Line (geometry)

In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist embedded in two, three, or higher dimensional spaces. The word line may also refer to a line segment in everyday life that has two points to denote its ends (endpoints). A line can be referred to by two points that lie on it (e.g. ) or by a single letter (e.g. ).

Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation: The theorem is named for the Greek philosopher Pythagoras, born around 570 BC.

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