euclidiennes

Euclidiennes is a term used to describe Euclidean spaces and, more broadly, Euclidean geometry as developed from the ideas of Euclid of Alexandria. In mathematics, Euclidean spaces are real coordinate spaces equipped with the Euclidean metric, the standard measure of distance derived from the dot product. The n-dimensional Euclidean space, denoted R^n or E^n, consists of all n-tuples of real numbers with the inner product ⟨x,y⟩ = Σ x_i y_i. The Euclidean norm is ||x|| = sqrt(⟨x,x⟩), and the distance between points x and y is d(x,y) = ||x−y|| = sqrt(Σ (x_i−y_i)^2).

These spaces provide the natural setting for classical geometry, linear algebra, and analysis. The geometry is flat, and notions of length, angle, and orthogonality arise from the inner product. Lines, planes, and higher-dimensional affine subspaces are defined in the usual way, and straight lines serve as geodesics. The Pythagorean theorem remains a fundamental relation in Euclidean spaces.

Euclidean spaces are contrasted with non-Euclidean geometries, such as spherical or hyperbolic geometries, where the parallel postulate does not hold. They also serve as the standard backdrop for many practical applications, including physics, computer science, and engineering, where Euclidean distance and norms are used for measurement, clustering, and optimization.

Historically, Euclidean geometry is attributed to Euclid's Elements. The term “Euclidean” distinguishes this flat geometric framework from broader geometrical theories that generalize or modify its basic postulates.