30 lines
3.0 KiB
Markdown
30 lines
3.0 KiB
Markdown
---
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title: The Distance Formula
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---
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## The Distance Formula
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<a href="https://www.codecogs.com/eqnedit.php?latex=\mathrm{Distance}=\sqrt{(x_{2}+x_{1})^2&space;+&space;(y_{2}&space;+&space;y_{1})^2}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\mathrm{Distance}=\sqrt{(x_{2}+x_{1})^2&space;+&space;(y_{2}&space;+&space;y_{1})^2}" title="\mathrm{Distance}=\sqrt{(x_{2}+x_{1})^2 + (y_{2} + y_{1})^2}" /></a>
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In analytic geometry, the distance between two points of the [xy-plane](https://en.wikipedia.org/wiki/Cartesian_coordinate_system) can be found using the distance formula. The distance between (*x*<sub>1</sub>, *y*<sub>1</sub>) and (*x*<sub>2</sub>, *y*<sub>2</sub>) is given by:
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<img src="https://latex.codecogs.com/gif.latex?d=\sqrt{(\Delta&space;x)^2+(\Delta&space;y)^2}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}." title="d=\sqrt{(\Delta x)^2+(\Delta y)^2}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}." />
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This formula is 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.
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In the study of complicated geometries, we call this (most common) type of distance [Euclidean distance](https://en.wikipedia.org/wiki/Euclidean_distance), as it is derived from the [Pythagorean theorem](https://en.wikipedia.org/wiki/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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#### Distance in Euclidean space
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In the [Euclidean space](https://en.wikipedia.org/wiki/Euclidean_space) **R**<sup>n</sup>, 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.
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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.
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The 1-norm distance is more colourfully called the *taxicab norm* or taxicab geometry ([Manhattan distance](https://en.wikipedia.org/wiki/Taxicab_geometry)), because it is the distance a car would drive in a city laid out in square blocks (if there are no one-way streets).
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The infinity norm distance is also called [Chebyshev distance](https://en.wikipedia.org/wiki/Chebyshev_distance). In 2D, it is the minimum number of moves kings require to travel between two squares on a chessboard.
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The *p*-norm is rarely used for values of *p* other than 1, 2, and infinity, but see [super ellipse](https://en.wikipedia.org/wiki/Super_ellipse).
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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.
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#### More Information:
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<!-- Please add any articles you think might be helpful to read before writing the article -->
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[Wikipedia: Distance](https://en.wikipedia.org/wiki/Distance)
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