Euclidean distance is the straight shortcut physics usually means
Distance measures how far apart things are—physical length, a rough “two counties over,” or social and psychological separation in soft sciences. In classical physics the default is the straight-line Euclidean length, the shortest path between two points.
Most versions of the idea, physical or figurative, are formalised as a metric space. In classical physics, including Newtonian mechanics, the distance between two points is the length of the straight segment joining them, the shortest route there is. Coordinate geometry computes it with the Pythagorean theorem: take the differences in each coordinate, square them, add them and take the square root, in two dimensions, three, or any higher number. It can be measured with a ruler, with radar over long ranges or interferometry over tiny ones, and astronomers use a cosmic distance ladder for the largest gaps.
On Earth the straight line is often useless because nobody can tunnel through the mantle, so people use the as-the-crow-flies route over the surface, approximated by great-circle distance on a sphere. On any curved surface the shortest path is a geodesic, the distance an ant living on that surface would measure. Relativity adds complications: length contraction and the relativity of simultaneity make distances depend on the observer's frame, and on galactic scales cosmic expansion does too, so cosmologists use several distance measures.
Other rules suit other tasks. Road trips care about travel distance, and in a grid city the Manhattan distance counts the east–west and north–south blocks between two corners. Chebyshev or chessboard distance is the fewest moves a king needs to cross between two squares.
The word also measures difference between similar things. Edit distance compares strings of text, and network distance counts degrees of separation between people. In statistics, divergences compare probability distributions and turn a family of them into a geometric object called a statistical manifold; squared Euclidean distance, minimised by least squares, is the simplest, and the Kullback–Leibler divergence is the most important in information theory, the only one that is both an f-divergence and a Bregman divergence.
Source: Distance
