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(Redirected from Transformation (mathematics))
In mathematics, a 'transformation' in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. These can be carried out in Euclidean space, particularly in dimensions 2 and 3. They are also operations that can be performed using linear algebra, and explicitly using matrix theory.
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).
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 ''T'''v' will work as ''T'''v'('p') = 'p' + 'v'.
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.
A 'glide reflection' is a type of isometry of the Euclidean plane: the combination of a reflection 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.
'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). Shapes not aligned with the axes may be subject to shear (see below) as a side effect: although the angles between lines parallel to the axes are preserved, other angles are not.
Main articles: Shear (mathematics)
'Shear' is a transform that effectively rotates one axis so that the axes are no longer perpendicular. Under shear, a rectangle becomes a parallelogram, and a circle becomes an ellipse. Even if lines parallel to the axes stay the same length, others do not.
Main articles: Linear transformation
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).
★ Coordinate transformation
★ Data transformation (statistics)
★ Infinitesimal transformation
★ Linear transformation
★ Transformation geometry
★ Transformation group
In mathematics, a 'transformation' in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. These can be carried out in Euclidean space, particularly in dimensions 2 and 3. They are also operations that can be performed using linear algebra, and explicitly using matrix theory.
| Contents |
| Rotation |
| Translation |
| Reflection |
| Glide reflection |
| Scaling |
| Shear |
| More generally |
| See also |
Rotation
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
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 ''T'''v' will work as ''T'''v'('p') = 'p' + 'v'.
Reflection
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 reflection
Example of a glide reflection
A 'glide reflection' is a type of isometry of the Euclidean plane: the combination of a reflection 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
'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). Shapes not aligned with the axes may be subject to shear (see below) as a side effect: although the angles between lines parallel to the axes are preserved, other angles are not.
Shear
Main articles: Shear (mathematics)
'Shear' is a transform that effectively rotates one axis so that the axes are no longer perpendicular. Under shear, a rectangle becomes a parallelogram, and a circle becomes an ellipse. Even if lines parallel to the axes stay the same length, others do not.
More generally
Main articles: Linear transformation
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).
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| Before | After |
See also
★ Coordinate transformation
★ Data transformation (statistics)
★ Infinitesimal transformation
★ Linear transformation
★ Transformation geometry
★ Transformation group
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