Difference between revisions of "Affine mapping"
From Online Dictionary of Crystallography
BrianMcMahon (talk | contribs) m |
BrianMcMahon (talk | contribs) m (Style edits to align with printed edition) |
||
| Line 1: | Line 1: | ||
| − | <font color="blue">Transformation affine</font> (''Fr''). <font color="red">Affine Abbildung</font> (''Ge''). <font color="black">Trasformazione affine</font> (''It''). <font color="purple">アフィン写像</font> (''Ja'') | + | <font color="blue">Transformation affine</font> (''Fr''). <font color="red">Affine Abbildung</font> (''Ge''). <font color="black">Trasformazione affine</font> (''It''). <font color="purple">アフィン写像</font> (''Ja''). |
== Definition == | == Definition == | ||
| − | An '''affine mapping''' is any mapping that preserves collinearity and ratios of distances: if three points belong to the same straight line, their images under an affine transformation also belong to | + | An '''affine mapping''' is any mapping that preserves collinearity and ratios of distances: if three points belong to the same straight line, their images under an affine transformation also belong to a straight line. Moreover, the middle point is also conserved under the affine mapping. By contrast, angles and lengths in general are not kept constant by an affine mapping. |
Under an affine mapping: | Under an affine mapping: | ||
| Line 15: | Line 15: | ||
Geometric contraction, expansion, dilation, reflection, rotation, shear, similarity transformations, spiral similarities, and translation are all affine transformations, as are their combinations. | Geometric contraction, expansion, dilation, reflection, rotation, shear, similarity transformations, spiral similarities, and translation are all affine transformations, as are their combinations. | ||
| − | Affine mappings that keep also distances and angles are called [[Euclidean mapping]]s. | + | Affine mappings that keep also distances and angles are called [[Euclidean mapping]]s or isometries. |
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Revision as of 11:12, 12 May 2017
Transformation affine (Fr). Affine Abbildung (Ge). Trasformazione affine (It). アフィン写像 (Ja).
Definition
An affine mapping is any mapping that preserves collinearity and ratios of distances: if three points belong to the same straight line, their images under an affine transformation also belong to a straight line. Moreover, the middle point is also conserved under the affine mapping. By contrast, angles and lengths in general are not kept constant by an affine mapping.
Under an affine mapping:
- parallel lines remain parallel;
- concurrent lines remain concurrent (images of intersecting lines intersect);
- the ratio of length of segments of a given line remains constant;
- the ratio of areas of two triangles remains constant;
- ellipses, parabolas and hyperbolas remain ellipses, parabolas and hyperbolas respectively;
- barycentres of polygons map into the corresponding barycentres.
Geometric contraction, expansion, dilation, reflection, rotation, shear, similarity transformations, spiral similarities, and translation are all affine transformations, as are their combinations.
Affine mappings that keep also distances and angles are called Euclidean mappings or isometries.