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ROTATION See Also: Rotation (mathematics) An arbitrary rotation with a given point fixed is given by the Formula For A Rotation About An Axis Through The Origin ; just add an arbitrary translation to get an arbitrary move of a rigid object. It can be decomposed into rotations about three fixed axes through that point, in terms of Flight Dynamics pitch, roll and yaw. See also Degrees Of Freedom (engineering) . TRANSLATION See Also: Translation (geometry) A translation, or '''translation operator''', is an Affine Transformation of Euclidean Space which moves every point by a fixed distance in the same direction. It can also be interpreted as the addition of a constant Vector to every point, or as shifting the Origin of the Coordinate System . In other words, if v is a fixed vector, then the translation Tv will work as Tv(p) = p + v. REFLECTION See Also: Reflection (mathematics) A reflection is a Map that transforms an object into its Mirror Image . For example, a reflection of the small English letter p in respect to a vertical line would look like q. In order to reflect a planar figure one needs the "mirror" to be a line ("axis of reflection"), while for reflections in the three-dimensional space one would use a Plane for a mirror. Reflection sometimes is considered as a special case of Inversion with infinite radius of the reference circle. GLIDE TRANSLATION See Also: Glide reflection A glide reflection is a type of in a line and a Translation along that line. Reversing the order of combining gives the same result. Depending on context, we may consider a reflection a special case, where the translation vector is the zero vector. SCALING See Also: Scaling (geometry) Uniform scaling is a Linear Transformation that enlarges or diminishes objects; the Scale Factor is the same in all directions; it is also called a Homothety . The result of uniform scaling is Similar (in the geometric sense) to the original. More general is scaling with a separate scale factor for each axis direction; a special case is '''directional scaling''' (in one direction). Shape s may change; e.g. a rectangle may change into a rectangle of a different shape, but also in a parallelogram (the angles between lines parallel to the axes are preserved, but not all angles). MORE GENERALLY More generally, a transformation in mathematics is one facet of the Mathematical Function ; the term '' Mapping '' is also used in ways that are quite close synonyms. A transformation is, most often, an Invertible Function from a set ''X'' to itself; but this is not always assumed. In a sense the term ''transformation'' only flags that a function's more geometric aspects are being considered (for example, with attention paid to Invariants ). SEE ALSO
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