A transformation that preserves angles and changes all distances in the same ratio, called the ratio of magnification. A similarity can also be defined as a transformation that preserves ratios of distances.
A similarity therefore transforms figures into similar figures. When written explicitly in terms of transformation matrices in three dimensions, similarities are commonly referred to as similarity transformations.
Examples of similarities include the following.
1. Central dilation: a transformation of lines to parallel lines that is not merely a translation.
2. Geometric contraction: a transformation in which the scale is reduced.
3. Dilation: a transformation taking each line to a parallel line whose length is a fixed multiple of the length of the original line.
4. Expansion: a transformation in which the scale is increased.
5. Isometry: a transformation that preserves distances.
6. Reflection: a transformation in which all points are exchanged with their corresponding reflections in an infinite plane mirror.
7. Rotation: a transformation that preserves angles and distances.
8. Improper rotation: reflection through the origin combined with a rotation.
9. Translation: a transformation consisting of a constant offset with no rotation or distortion.
