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It can be extended to infinite-dimensional vector spaces as the L 2 norm or L 2 distance. [25] The Euclidean distance gives Euclidean space the structure of a topological space, the Euclidean topology, with the open balls (subsets of points at less than a given distance from a given point) as its neighborhoods. [26]
The distance (or perpendicular distance) from a point to a line is the shortest distance from a fixed point to any point on a fixed infinite line in Euclidean geometry. It is the length of the line segment which joins the point to the line and is perpendicular to the line. The formula for calculating it can be derived and expressed in several ways.
The closest pair of points problem or closest pair problem is a problem of computational geometry: given points in metric space, find a pair of points with the smallest distance between them. The closest pair problem for points in the Euclidean plane [ 1 ] was among the first geometric problems that were treated at the origins of the systematic ...
The Euclidean distance obeys the triangle inequality, so the Euclidean TSP forms a special case of metric TSP. However, even when the input points have integer coordinates, their distances generally take the form of square roots , and the length of a tour is a sum of radicals , making it difficult to perform the symbolic computation needed to ...
For a fixed length n, the Hamming distance is a metric on the set of the words of length n (also known as a Hamming space), as it fulfills the conditions of non-negativity, symmetry, the Hamming distance of two words is 0 if and only if the two words are identical, and it satisfies the triangle inequality as well: [2] Indeed, if we fix three words a, b and c, then whenever there is a ...
Lloyd's algorithm is usually used in a Euclidean space. The Euclidean distance plays two roles in the algorithm: it is used to define the Voronoi cells, but it also corresponds to the choice of the centroid as the representative point of each cell, since the centroid is the point that minimizes the average squared Euclidean distance to the ...
A very simple example can be given between the two colors with RGB values (0, 64, 0) ( ) and (255, 64, 0) ( ): their distance is 255. Going from there to (255, 64, 128) ( ) is a distance of 128. When we wish to calculate distance from the first point to the third point (i.e. changing more than one of the color values), we can do this:
Then the Euclidean distance over the end-points of any two vectors is a proper metric which gives the same ordering as the cosine distance (a monotonic transformation of Euclidean distance; see below) for any comparison of vectors, and furthermore avoids the potentially expensive trigonometric operations required to yield a proper metric.