Homothety: Difference between revisions
modified defintion to require having a center, with reference (all literature agrees on this), removed confused and incorrect stuff, and redundancies |
|||
| Line 1: | Line 1: | ||
In [[mathematics]], a '''homothety''' (or '''homothecy''' or non-rotating '''dilation''') is a [[Transformation (mathematics)|transformation]] of an [[affine space]] determined by a point ''S'' called its ''center'' and a nonzero number ''λ'' called its ''ratio'', which sends |
|||
{{unreferenced|date=August 2009}} |
|||
:<math> M \mapsto S + \lambda \overrightarrow{SM}, </math> |
|||
In [[mathematics]], a '''homothety''' (or '''homothecy''' or non-rotating '''dilation''') is a [[Transformation (mathematics)|transformation]] of space which takes each line into a parallel line (in essence, a [[Similarity (geometry)|similarity]] that allows reflection in a single point, but otherwise preserves orientation). All homotheties form a [[group (mathematics)|group]] in either [[affine geometry|affine]] or [[Euclidean geometry]]. [[isometry|Congruent]] examples of homotheties are translations, reflections, and the identity transformation. |
|||
in other words it fixes ''S'', and sends any ''M'' to another point ''N'' such that the segment ''SN'' is on the same line as ''SM'', but scaled by a factor ''λ''.<ref>J. Hadamard, Lessons in Plane Geometry, p. 145</ref> In Euclidean geometry homotheties are the [[Similarity (geometry)|similarities]] that fix a point and either preserve (if {{nowrap|''λ'' > 0}}) or reverse (if {{nowrap|''λ'' < 0}}) the direction of all vectors. Together with the [[Translation (geometry)|translations]], all homotheties of an affine (or Euclidean) space form a group, the group of '''homothety-translations'''. These are precisely the [[affine transformation]]s with the property that the image of every line ''L'' is a line [[parallel (geometry)|parallel]] to ''L''. |
|||
In Euclidean geometry, |
In Euclidean geometry, a homothety of ratio ''λ'' multiplies distances between points are by |''λ''| and all areas by ''λ''<sup>2</sup>. This number is is called the ''ratio of magnification'' or ''dilation factor'' or ''scale factor'' or ''similitude ratio''. Such a transformation can be called an '''enlargement''' if the scale factor exceeds 1. |
||
{{Reflist}} |
|||
Choose an ''origin'' or ''center'' ''A'' and a [[real number]] <math>c</math> (possibly negative). The homothety <math>h_{A,c}</math> maps any point ''M'' to a point <math>M'</math> such that |
|||
: <math>A-M'=c(A-M)\!</math> |
|||
(as vectors). |
|||
A homothety is an [[affine transformation]] (if the fixed point is the origin: a [[linear transformation]]) and also a [[similarity (geometry)|similarity transformation]]. It multiplies all distances by |''c''|, all [[surface]] areas by <math>c^2</math>, etc. |
|||
==See also== |
==See also== |
||
Revision as of 10:41, 1 March 2011
In mathematics, a homothety (or homothecy or non-rotating dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends
in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ.[1] In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L.
In Euclidean geometry, a homothety of ratio λ multiplies distances between points are by |λ| and all areas by λ2. This number is is called the ratio of magnification or dilation factor or scale factor or similitude ratio. Such a transformation can be called an enlargement if the scale factor exceeds 1.
- ↑ J. Hadamard, Lessons in Plane Geometry, p. 145
See also
- Homothetic center, the center of a homothetic transformation taking one of a pair of shapes into the other
- Dilation, transformations that allow rotation as well as scaling and translation
- The Hadwiger conjecture on the number of homothetic copies of a convex body that may be needed to cover it
- Homothetic function (economics), a function of the form U(f(y)) in which f is a homogeneous function and U is a monotonically increasing function.
External links
- Homothety, interactive applet from Cut-the-Knot.
