Isometry

In geometry and mathematical analysis, an isometry is a distance-preserving mapping. This idea occurs in the theories of metric spaces, normed vector spaces and inner product spaces; also in quadratic forms and differential geometry.

In Euclidean space with the usual distance function, the isometries can be characterized: there are no more than the 'expected' examples generated by rotations, reflections and translations. To put this more accurately, the isometries form a group, that is the semidirect product of the orthogonal group and the group of translations. (This group is sometimes called the Galilean group, at least for three dimensions and in relation with its role in Newtonian mechanics as expressed by permissible changes of frame of reference. See Galilean transformation.)

Plane isometries; glide reflections

Within the isometry group of the plane, the product of a rotation and a translation can always be expressed as a single rotation (or translation). On the other hand the product of a reflection and a translation is usually not a reflection, but can produce a transformation with no everyday name: a glide reflection.

For example, there is an isometry consisting of the reflection on the x-axis, followed by translation of one unit parallel to it. In co-ordinates, it takes (x,y) to (x+1,-y). It fixes a system of parallel lines, but is a combination of a reflection in a line and a translation parallel to that line. If one considers the effect of a reflection combined with any translation, it is a glide reflection with respect to a line parallel to the the line of the reflection, as one sees by resolving the translation into components parallel and orthogonal to that line.

Angle-preserving mappings are called conformal.

Isometric projection or isometric view is the name given to a type of technical drawing / projection used in fields such as Mechanical Engineering or Architecture that makes an object/ building visible from three planes/co-ordinates.