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. 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. In some applications in statistics and optimization, the square of the Euclidean distance is used instead of the distance itself.ĭ ( p, q ) = ( p 1 − q 1 ) 2 + ( p 2 − q 2 ) 2 + ⋯ + ( p i − q i ) 2 + ⋯ + ( p n − q n ) 2. 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. These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation was not made until the 18th century. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. Referencing the above figure and using the Pythagorean Theorem,ĪC 2 = (x 2 - x 1) 2 + (y 2 - y 1) 2.Using the Pythagorean theorem to compute two-dimensional Euclidean distance ![]() In a 3D coordinate plane, the distance between two points, A and B, with coordinates (x 1, y 1, z 1) and (x 2, y 2, z 2), can also be derived from the Pythagorean Theorem. Which is the distance formula between two points on a coordinate plane. We can rewrite this using the letter d to represent the distance between the two points as The horizontal and vertical distances between the two points form the two legs of the triangle and have lengths |x 2 - x 1| and |y 2 - y 1|. The hypotenuse of the right triangle, labeled c, is the distance between points A and B. Given two points, A and B, with coordinates (x 1, y 1) and (x 2, y 2) respectively on a 2D coordinate plane, it is possible to connect the points with a line and draw vertical and horizontal extensions to form a right triangle: Referencing the right triangle sides below, the Pythagorean theorem can be written as: ![]() The Pythagorean Theorem says that the square of the hypotenuse equals the sum of the squares of the two legs of a right triangle. The distance formula can be derived from the Pythagorean Theorem. The Pythagorean Theorem and the distance formula Other coordinate systems exist, but this article only discusses the distance between points in the 2D and 3D coordinate planes. Where (x 1, y 1, z 1) and (x 2, y 2, z 2) are the 3D coordinates of the two points involved. d =ĭistance formula for a 3D coordinate plane: ![]() Find the length of line segment AB given that points A and B are located at (3, -2) and (5, 4), respectively.
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