euclidien

Euclidien is an adjective used in French and other languages to describe concepts derived from Euclid of Alexandria or from Euclidean geometry. The term stems from Euclid’s Elements and is used to distinguish flat, standard geometry from non-Euclidean geometries. In common mathematical language, euclidien properties pertain to the classical, flat space geometry described by Euclid’s postulates.

Euclidean geometry concerns the geometry of flat spaces, such as the plane and ordinary three-dimensional space. It studies points, lines, angles, polygons, and solids, and it assumes the parallel postulate, which leads to familiar results like the angle sum of a triangle and the existence of unique parallel lines through a given point.

Euclidean space and distance are central to the concept. An n-dimensional Euclidean space, denoted R^n, is equipped with the Euclidean metric. The distance between two points x and y is d(x,y) = sqrt(sum_i (x_i − y_i)^2). This metric arises from the standard inner product and induces the L2 norm. Distances measured in this way satisfy non-negativity, identity, symmetry, and the triangle inequality.

Non-Euclidean geometries modify the parallel postulate, leading to alternative geometries such as spherical or hyperbolic geometry. In these spaces, distances and angle measures can differ from Euclidean expectations, and the familiar results from Euclidean geometry may not hold.

Euclidean concepts have wide applications, including in computational geometry, clustering and nearest-neighbor methods, physics in flat spacetime approximations, computer graphics, and data analysis, where the Euclidean distance is a common measure of similarity or difference.