Difference between revisions of "Homothety"
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In [[mathematics]], a '''homothety''' (or '''homothecy''') is a transformation of space which dilates distances with respect to a fixed point. Such a transformation is also called an '''enlargement'''. A homothety with center <math>H</math> and factor <math>k</math> sends point <math>A</math> to a point <math>A' \ni HA'=k\cdot HA</math> This is denoted by <math>\mathcal{H}(H, k)</math>. | In [[mathematics]], a '''homothety''' (or '''homothecy''') is a transformation of space which dilates distances with respect to a fixed point. Such a transformation is also called an '''enlargement'''. A homothety with center <math>H</math> and factor <math>k</math> sends point <math>A</math> to a point <math>A' \ni HA'=k\cdot HA</math> This is denoted by <math>\mathcal{H}(H, k)</math>. | ||
| + | The following observations are noteworthy: | ||
| + | Under a homothety, parallel lines map onto parallel lines. | ||
| + | Homothety preserves angles. | ||
| + | Homothety preserves orientation. | ||
| + | Every homothety has an inverse, viz., HO, k-1 = HO, 1/k. In other words | ||
| + | |||
| + | HO, k(HO, 1/k(P)) = P = HO, 1/k(HO, k(P)). | ||
| + | |||
| + | Every homothety with k different from 1 has one and only one fixed point - the center O. Every line through O is also fixed although not pointwise. For k = 1, homothety is the identity transformation. | ||
| + | Successive applications of homotheties with coefficients k1 and k2 is either a homothety with coefficient k1k2, if the latter differs from 1, or a parallel translation, otherwise. In the former case, the centers of the three homotheties are collinear. In the latter case, the translation is parallel to the line joining the centers of the two homotheties. | ||
| + | The product of homotheties is not in general commutative. Two homotheties with a common center commute as a matter of course. | ||
== See Also == | == See Also == | ||
Revision as of 09:57, 25 October 2018
In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point. Such a transformation is also called an enlargement. A homothety with center 
and factor 
sends point 
to a point 
This is denoted by 
.
The following observations are noteworthy:
Under a homothety, parallel lines map onto parallel lines.
Homothety preserves angles.
Homothety preserves orientation.
Every homothety has an inverse, viz., HO, k-1 = HO, 1/k. In other words
HO, k(HO, 1/k(P)) = P = HO, 1/k(HO, k(P)).
Every homothety with k different from 1 has one and only one fixed point - the center O. Every line through O is also fixed although not pointwise. For k = 1, homothety is the identity transformation. Successive applications of homotheties with coefficients k1 and k2 is either a homothety with coefficient k1k2, if the latter differs from 1, or a parallel translation, otherwise. In the former case, the centers of the three homotheties are collinear. In the latter case, the translation is parallel to the line joining the centers of the two homotheties. The product of homotheties is not in general commutative. Two homotheties with a common center commute as a matter of course.
See Also
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