isometrisk

Isometrisk is an adjective describing properties or transformations that preserve distances between points. In mathematics, an isometry is a mapping f between metric spaces (X, d_X) and (Y, d_Y) such that d_Y(f(x1), f(x2)) = d_X(x1, x2) for all x1, x2 in X. In Euclidean space, isometries include translations, rotations, reflections, and the compositions thereof, collectively known as rigid motions. Such maps preserve lengths, angles, and overall shape.

In linear algebra, a linear isometry preserves the inner product: ⟨Ax, Ay⟩ = ⟨x, y⟩ for all x, y. This implies ||Ax|| = ||x|| and, when A is square, that A is orthogonal (A^T A = I).

An isometric embedding is a map that preserves distances between all pairs of points; it is an embedding of one metric space into another that retains its metric structure.

Applications include computer graphics, robotics, and geographic information systems, where distance preservation is important. In cartography and game design, isometric projections or views provide a way to render 3D structures on a 2D plane with equal scales along axes, producing a recognizable pseudo-3D appearance without perspective distortion.