Euclideanavståndet

Euclidean distance, also known as the L2 norm or Euclidean norm, is the straight-line distance between two points in a Euclidean space. It is the most common way to measure distance in geometry and is named after the ancient Greek mathematician Euclid.

In a two-dimensional plane, the Euclidean distance between two points (x1, y1) and (x2, y2) is calculated using the Pythagorean theorem. The formula is: sqrt((x2 - x1)^2 + (y2 - y1)^2). This formula can be generalized to any number of dimensions. For two points in n-dimensional space, P = (p1, p2, ..., pn) and Q = (q1, q2, ..., qn), the Euclidean distance is given by: sqrt(sum from i=1 to n of (qi - pi)^2).

Euclidean distance is widely used in various fields. In mathematics, it is fundamental to the study of geometry, topology, and analysis. In computer science and data science, it is a crucial metric for algorithms such as k-nearest neighbors (KNN), clustering (e.g., K-means), and dimensionality reduction techniques like Principal Component Analysis (PCA). It is also employed in machine learning for tasks like classification and regression. In physics, it represents physical distance, and in other areas, it can be used to quantify the dissimilarity between data points. The simplicity and intuitive nature of Euclidean distance make it a cornerstone for measuring separation.