information:filtering, quality:correction, structural:expansion
Euclidean symmetries in general

The most common group of transforms applied to objects are termed the Euclidean group of " isometries ", which are distance-preserving transformations in space commonly referred to as two-dimensional or three-dimensional (i.e., in plane geometry or solid geometry Euclidean spaces). These isometries consist of reflections, rotations, translations, and combinations of these basic operations. Under an isometric transformation, a geometric object is said to be symmetric if, after transformation, the object is indistinguishable from the object before the transformation. A geometric object is typically symmetric only under a subset or " subgroup " of all isometries. The kinds of isometry subgroups are described below, followed by other kinds of transform groups, and by the types of object invariance that are possible in geometry.

By the Cartan–Dieudonné theorem, an orthogonal transformation in n -dimensional space can be represented by the composition of at most n reflections.