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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 ...
This is a glide reflection, except in the special case that the translation is perpendicular to the line of reflection, in which case the combination is itself just a reflection in a parallel line. The identity isometry, defined by I ( p ) = p for all points p is a special case of a translation, and also a special case of a rotation.
Effect of applying various 2D affine transformation matrices on a unit square. Note that the reflection matrices are special cases of the scaling matrix. Affine transformations on the 2D plane can be performed in three dimensions. Translation is done by shearing parallel to the xy plane, and rotation is performed around the z axis.
Rotations about an axis combined with translation along that axis are in the same class if the angle is the same and the translation distance is the same. Reflections in a plane are in the same class; Reflections in a plane combined with translation in that plane by the same distance are in the same class.
Rotations and translations do preserve handedness, which in 3D Plane-based GA implies that they can be written as a composition of an even number of reflections. A rotations can thought of as a reflection in a plane followed by a reflection in another plane which is not parallel to the first (the quaternions, which are set in the context of PGA ...
(A reflection would not preserve handedness; for instance, it would transform a left hand into a right hand.) To avoid ambiguity, a transformation that preserves handedness is known as a rigid motion, a Euclidean motion, or a proper rigid transformation. In dimension two, a rigid motion is either a translation or a rotation.
This is because the conjugate of the translation by a glide reflection is the same as by the corresponding reflection: the translation vector is reflected. If the isometry group contains an n -fold rotation then the lattice has n -fold symmetry for even n and 2 n -fold for odd n .
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.