euclidean and non euclidean geometry solutions chapter 1

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. 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.

Using the Pythagorean theorem to compute two-dimensional Euclidean distance

Euclidean and non euclidean geometry solutions chapter 1