Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. 'We shall find it convenient to use the word transformation in the special sense of a one-to-one correspondence P → P ′ {displaystyle P o P'} among all points in the plane (or in space), that is, a rule for associating pairs of points, with the understanding that each pair has a first member P and a second member P' and that every point occurs as the first member of just one pair and also as the second member of just one pair...3.51 Any direct isometry is either a translation or a rotation. Any opposite isometry is either a reflection or a glide reflection.3.11 Any two congruent triangles are related by a unique isometry. In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry; the isometry that relates them is either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection. Isometries are often used in constructions where one space is embedded in another space. For instance, the completion of a metric space M involves an isometry from M into M', a quotient set of the space of Cauchy sequences on M. The original space M is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space. An isometric surjective linear operator on a Hilbert space is called a unitary operator. Let X and Y be metric spaces with metrics dX and dY. A map f : X → Y is called an isometry or distance preserving if for any a,b ∈ X one has An isometry is automatically injective; otherwise two distinct points, a and b, could be mapped to the same point, thereby contradicting the coincidence axiom of the metric d. This proof is similar to the proof that an order embedding between partially ordered sets is injective. Clearly, every isometry between metric spaces is a topological embedding. A global isometry, isometric isomorphism or congruence mapping is a bijective isometry. Like any other bijection, a global isometry has a function inverse. The inverse of a global isometry is also a global isometry. Two metric spaces X and Y are called isometric if there is a bijective isometry from X to Y. The set of bijective isometries from a metric space to itself forms a group with respect to function composition, called the isometry group. There is also the weaker notion of path isometry or arcwise isometry: