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In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system .
In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations, rotations, reflections, and glide reflections (see below § Classification).
The translations by a given distance in any direction form a conjugacy class; the translation group is the union of those for all distances. In 1D, all reflections are in the same class. In 2D, rotations by the same angle in either direction are in the same class. Glide reflections with translation by the same distance are in the same class. In 3D:
Combining two equal glide plane operations gives a pure translation with a translation vector that is twice that of the glide reflection, so the even powers of the glide reflection form a translation group. In the case of glide-reflection symmetry, the symmetry group of an object contains a glide reflection and the group generated by it. For ...
A reflection in a line is an opposite isometry, like R 1 or R 2 on the image. Translation T is a direct isometry: a rigid motion. [1] In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
The set of all reflections in lines through the origin and rotations about the origin, together with the operation of composition of reflections and rotations, forms a group. The group has an identity: Rot(0). Every rotation Rot(φ) has an inverse Rot(−φ). Every reflection Ref(θ) is its own inverse. Composition has closure and is ...
Point Q is the reflection of point P through the line AB. In a plane (or, respectively, 3-dimensional) geometry, to find the reflection of a point drop a perpendicular from the point to the line (plane) used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure.
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. [1] [self-published source] [2] [3] The rigid transformations include rotations, translations, reflections, or any