Isometrically

Isometrically refers to doing something in a way that preserves distances between all pairs of points. In mathematics and related fields, an isometry is a map or embedding that preserves the metric structure of a space.

Formally, a function f between metric spaces X and Y is isometric if for all x1, x2 in X, the distance in Y satisfies d_Y(f(x1), f(x2)) = d_X(x1, x2). If f is also a bijection, it is called an isometry; if it is an injection that preserves distances, it is an isometric embedding. When a space is said to be isometrically embedded into another, its intrinsic distances are preserved by the embedding, yielding the same metric as the host space on the image.

In Euclidean geometry, isometries are rigid motions that preserve distances and angles. Common Euclidean isometries include translations, rotations, reflections, and, in some contexts, glide reflections. An object mapped isometrically into another space retains its shape and size, with no distortion. Conversely, scaling or stretching is not an isometry because it changes distances.

Common examples: the identity map is isometric; a map f(x) = Ax + b with A an orthogonal matrix (A^T A = I) is an isometry of Euclidean space. An example of a non-isometric transformation is f(x) = 2x, which doubles all distances.

Etymology: isometric comes from Greek isos, equal, and metron, measure. The concept underpins many areas, including geometry, computer graphics, and data embedding, where preserving pairwise distances is important. See also isometry, metric space, and embedding.