Euclidean distance

Euclidean distance, in Euclidean space, the length of a straight line segment that would connect two points. Euclidean space is a two- or three-dimensional space in which the axioms and postulates of Euclidean geometry apply. In such a space, the distance formulas for points in rectangular coordinates are based on the Pythagorean theorem. For example, take two points (a, b) and (c, d) in two-dimensional space. (Here the Cartesian coordinate system [named for René Descartes] is used, in which points are designated by their distance along a horizontal [x] axis and a vertical [y] axis from a reference point, the origin, designated [0, 0].) One can make a right triangle by adding the point (c, b). From the Pythagorean theorem, in which the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides, the distance between the points (a, b) and (c, d) is given by Square root of√(a − c)² + (b − d)². In three-dimensional space, the distance between the points (a, b, c) and (d, e, f) is Square root of√(a − d)² + (b − e)² + (c − f)². This formula can be extended to other coordinate systems, such as polar coordinates and spherical coordinates.