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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 same sense of distance is used in Euclidean geometry to define distance from a point to a line, distance from a point to a plane, or, more generally, perpendicular distance between affine subspaces. Even more generally, this idea can be used to define the distance between two subsets of a metric space.
In mathematics, a Euclidean distance matrix is an n×n matrix representing the spacing of a set of n points in Euclidean space. For points x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} in k -dimensional space ℝ k , the elements of their Euclidean distance matrix A are given by squares of distances between them.
The distance (more precisely the Euclidean distance) between two points of a Euclidean space is the norm of the translation vector that maps one point to the other; that is d ( P , Q ) = ‖ P Q → ‖ .
As most definitions of color difference are distances within a color space, the standard means of determining distances is the Euclidean distance.If one presently has an RGB (red, green, blue) tuple and wishes to find the color difference, computationally one of the easiest is to consider R, G, B linear dimensions defining the color space.
In Euclidean space, the distance from a point to a plane is the distance between a given point and its orthogonal projection on the plane, ...
defining the distance between two points P = (p x, p y) and Q = (q x, q y) is then known as the Euclidean metric, and other metrics define non-Euclidean geometries. In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e.g., y = 2 ...
Euclidean distance map, a digital image in which each pixel value represents the Euclidean distance to an obstacle; Euclidean rhythm, a method of distributing beats across musical steps based on Euclid's algorithm; Euclidean zoning, a system of land use management modeled after the zoning code of Euclid, Ohio