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. Taxicab/City-block Distance.Using the Pythagorean theorem to compute two-dimensional Euclidean distance As one of the mechanical proofs of the Pythagorean theorem shows, the same is also true in physics, although in either science it's not the only distance formula used. In mathematics, the Euclidean distance is most fundamental. In a city, one often finds that the taxicab distance formula There are buildings, streets busy with traffic, fences and what not, to be accounted for. Indeed, in a city - just to take one example - it is often impossible to move from one point straight to another. If the question is, How fast you can get from one point to another while moving at a given speed, the Euclidean formula may not be very useful providing the answer. In the plane - since the Earth is round, this means within relatively small areas of Earth's surface - it is pretty good, provided the distance is exactly what you want to estimate. How good is the (Euclidean) distance formula for measuring real distances? This depends on the circumstances. This is of course always the case: the straight line segment whose length is taken to be the distance between its endpoints always serves as a hypotenuse of a right triangle (in fact, of infinitely many of them. Which gives the length of the hypotenuse as 5, same as the distance between the two points according to the distance formula. By the Pythagorean theorem, the square of the hypotenuse is (hypotenuse)² = 3² + 4². With this small addition we get a right-angled triangle with legs 3 and 4. The horizontal distance between the points is 4 and the vertical distance is 3. The source of this formula is in the Pythagorean theorem. According to the Euclidean distance formula, the distance between two points in the plane with coordinates (x, y) and (a, b) is given byĪs an example, the (Euclidean) distance between points (2, -1) and (-2, 2) is found to be Very often, especially when measuring the distance in the plane, we use the formula for the Euclidean distance.
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