Izometri

Izometri, or isometry, is a distance-preserving transformation between metric spaces. In Euclidean geometry, an izometri maps points in the plane or space so that the distance between any two points remains unchanged after the mapping. Formally, a map f from R^n to R^n is an izometri if for all x and y, the Euclidean distance d(f(x), f(y)) equals d(x, y). Because distances are preserved, the size and shape of figures are preserved under izometries.

In the plane and in three-dimensional space, common examples of izometries include translations (shifting every point by the same vector), rotations (turning around a fixed point or axis), reflections (flipping across a line or plane), and glide reflections (a reflection followed by a parallel translation). In three dimensions, izometries also include more complex motions such as screw motions, which combine rotation with translation along an axis. The identity transformation is trivially an izometry.

Key properties of izometries include that the composition of two izometries is an izometry, and the inverse of an izometry is also an izometry. The set of all izometries of a space forms a group under composition, known as the isometry group (for example, Isom(R^n)).

A common distinction is between orientation-preserving and orientation-reversing izometries. In two dimensions, orientation-preserving izometries are exactly translations and rotations, while orientation-reversing ones include reflections and glide reflections.

Applications of izometries appear in computer graphics (rigid motions of objects), computer vision and pattern recognition, robotics and animation, and crystallography, where symmetry operations are modeled as isometries.

Etymology derives from Greek iso meaning equal and metron meaning measure. In Turkish, izometri is the term commonly used for isometry in geometry.