Using the Pythagorean theorem to compute two-dimensional Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance.
These names come from the ancient Greek mathematicians Euclid and Pythagoras. In the Greek deductive geometry exemplified by Euclid's Elements, distances were not represented as numbers but line segments of the same length, which were considered "equal". The notion of distance is inherent in the compass tool used to draw a circle, whose points all have the same distance from a common center point. The connection from the Pythagorean theorem to distance calculation was not made until the 18th century.
The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances between different types of objects, such as the distance from a point to a line. In advanced mathematics, the concept of distance has been generalized to abstract metric spaces, and other distances than Euclidean have been studied. In some applications in statistics and optimization, the square of the Euclidean distance is used instead of the distance itself.
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In mathematics, the Euclideandistance between two points in Euclidean space is the length of the line segment between them. It can be calculated from...
In mathematics, a Euclideandistance matrix is an n×n matrix representing the spacing of a set of n points in Euclidean space. For points x 1 , x 2 ,...
cosine distance is half of the squared Euclideandistance of the L 2 {\displaystyle L_{2}} normalization of the vectors, and squared Euclideandistance does...
analogous to Euclidean geometry but without uniquely determined parallel lines Euclideandistance, the distance between pairs of points in Euclidean spaces...
meaning of distance in classical physics, including Newtonian mechanics. Straight-line distance is formalized mathematically as the Euclideandistance in two-...
space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces...
Manhattan distance. Common metrics are: Euclideandistance Taxicab geometry, also known as City block distance or Manhattan distance. Chebyshev distance There...
Minkowski distance or Minkowski metric is a metric in a normed vector space which can be considered as a generalization of both the Euclideandistance and the...
Manhattan geometry is geometry where the familiar Euclideandistance is ignored, and the distance between two points is instead defined to be the sum...
variance, then the Mahalanobis distance corresponds to standard Euclideandistance in the transformed space. The Mahalanobis distance is thus unitless, scale-invariant...
thousands of years. When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclideandistance satisfies the Pythagorean...
squared Euclidean (which unlike Euclidean, does not have triangle inequality) distance at its core. The common learning goal is to minimize a distance metric...
transformations of that space that preserve the Euclideandistance between any two points (also called Euclidean transformations). The group depends only on...
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements...
populations having the same allele Similar to Euclideandistance, Czekanowski distance involves calculated the distance between points of allele frequency that...
matrix. A pre-distance matrix that can be embedded in a Euclidean space is called a Euclideandistance matrix. For mixed-type data that contain numerical as...
include Euclideandistance, Manhattan distance, Minkowski distance, and Chebyshev distance. The Euclideandistance formula is used to find the distance between...
called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclideandistance between...
3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic...
TSPs for various metrics. In the Euclidean TSP (see below), the distance between two cities is the Euclideandistance between the corresponding points...
in our city). For most cities, the distance between points can be measured using the familiar Euclideandistance: ℓ 2 = d [ ( a 1 , a 2 ) , ( b 1 , b...
interest while wanting to find a representative using some distance other than squared euclideandistance (for instance in movie-ratings). For some data sets...
definitions make use of the Euclideandistance in a device-independent color space. As most definitions of color difference are distances within a color space...
also metrical properties induced by a distance, which allows to define circles, and angle measurement. A Euclidean plane with a chosen Cartesian coordinate...
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