Rneliö

Rneliö is the Finnish term for the two-dimensional Euclidean space, commonly denoted R^2 in mathematical notation. It consists of all ordered pairs (x, y) of real numbers, equipped with the Euclidean distance d((x1,y1),(x2,y2)) = sqrt((x1−x2)^2+(y1−y2)^2). As a real vector space, Rneliö has dimension 2 and is isomorphic to R^2 with the standard basis e1 = (1,0) and e2 = (0,1). It carries the standard inner product ⟨(x1,y1),(x2,y2)⟩ = x1x2 + y1y2, which induces the norm ∥(x,y)∥ = sqrt(x^2 + y^2) and the usual notions of length and angle.

In Rneliö a line is the set of points satisfying ax + by = c for some real a, b, c, and a circle is the set {(x,y) : (x−a)^2 + (y−b)^2 = r^2}. More general curves, polygons, and surfaces can be described by equations or parametrizations in Cartesian coordinates. The space is connected, simply connected, and complete as a metric space, with the standard topology induced by the Euclidean metric.

Transformations of Rneliö include translations, rotations, reflections, and scalings. Isometries preserve distances; linear transformations correspond to 2×2 matrices. Rneliö serves as the foundational setting for analytic geometry, vector calculus, and many areas of physics and computer graphics, where coordinates and geometric relations are expressed in the plane.

See also: Euclidean space, R^n, Cartesian coordinate system, distance and inner product.