![]() ![]() Other common distances in real coordinate spaces and function spaces: Comparison of Chebyshev, Euclidean and taxicab distances for the hypotenuse of a 3-4-5 triangle on a chessboard 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. It can be extended to infinite-dimensional vector spaces as the L 2 norm or L 2 distance. By Dvoretzky's theorem, every finite-dimensional normed vector space has a high-dimensional subspace on which the norm is approximately Euclidean the Euclidean norm is the One of the important properties of this norm, relative to other norms, is that it remains unchanged under arbitrary rotations of space around the origin. In more advanced areas of mathematics, when viewing Euclidean space as a vector space, its distance is associated with a norm called the Euclidean norm, defined as the distance of each vector from the origin. The collection of all squared distances between pairs of points from a finite set may be stored in a Euclidean distance matrix, and is used in this form in distance geometry. Since squaring is a monotonic function of non-negative values, minimizing squared distance is equivalent to minimizing the Euclidean distance, so the optimization problem is equivalent in terms of either, but easier to solve using squared distance. The squared distance is thus preferred in optimization theory, since it allows convex analysis to be used. However it is a smooth, strictly convex function of the two points, unlike the distance, which is non-smooth (near pairs of equal points) and convex but not strictly convex. Squared Euclidean distance does not form a metric space, as it does not satisfy the triangle inequality. In cluster analysis, squared distances can be used to strengthen the effect of longer distances. The addition of squared distances to each other, as is done in least squares fitting, corresponds to an operation on (unsquared) distances called Pythagorean addition. Thus if p īeyond its application to distance comparison, squared Euclidean distance is of central importance in statistics, where it is used in the method of least squares, a standard method of fitting statistical estimates to data by minimizing the average of the squared distances between observed and estimated values, and as the simplest form of divergence to compare probability distributions. The distance between any two points on the real line is the absolute value of the numerical difference of their coordinates, their absolute difference. In some applications in statistics and optimization, the square of the Euclidean distance is used instead of the distance itself.ĭistance formulas One dimension In advanced mathematics, the concept of distance has been generalized to abstract metric spaces, and other distances than Euclidean have been studied. Formulas are known for computing distances between different types of objects, such as the distance from a point to a line. 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. The connection from the Pythagorean theorem to distance calculation was not made until the 18th century. 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. In the Greek deductive geometry exemplified by Euclid's Elements, distances were not represented as numbers but line segments of the same length were considered "equal". These names come from the ancient Greek mathematicians Euclid and Pythagoras. ![]() It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance. In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. Using the Pythagorean theorem to compute two-dimensional Euclidean distance ![]()
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