isometrioiden

Isometrioiden is the Finnish term used in mathematics to refer to isometries, the distance-preserving maps between metric spaces. An isometry f: X → Y between metric spaces (X, dX) and (Y, dY) preserves distances: for all x1, x2 in X, dY(f(x1), f(x2)) = dX(x1, x2). If X = Y, f is a self-map of X.

In Euclidean spaces, common isometries include translations, rotations, reflections, and glide reflections. The set of all isometries from a space to itself, Iso(X), forms a group under composition, known as the isometry group. In Euclidean n-space, every isometry can be expressed as an affine map x ↦ Ax + b, where A is an orthogonal matrix (AᵀA = I) and b is a vector. The determinant of A is ±1; orientation-preserving isometries correspond to translations and rotations (det A = 1), while reflections and glide reflections involve det A = -1.

Applications of isometrioiden include modeling rigid motions in computer graphics and animation, pose estimation in robotics, and symmetry analysis in crystallography. They also play a central role in geometry, geometric group theory, and the study of metric spaces.

Notes: the term isometrioiden is used in Finnish mathematical literature and may appear as a plural form or a heading in Finnish texts. In English-language contexts, the term isometries is standard. The concept has historical roots in classical geometry and was formalized within the broader study of metric spaces and transformation groups.