- Homothety and uniform scaling
- See also
- Notes
- References
- External links
In mathematics, a homothety (or homothecy, or homogeneous 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 {{nowrap|λ > 0}}) or reverse (if {{nowrap|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 dilations or 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 projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) that leaves the line at infinity pointwise invariant.[2]
In Euclidean geometry, a homothety of ratio λ multiplies distances between points by |λ| and all areas by λ2. The first number 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. The above-mentioned fixed point S is called homothetic center or center of similarity or center of similitude.
Homothety and uniform scaling
If the homothetic center S happens to coincide with the origin O of the vector space (S ≡ O), then every homothety with scale factor λ is equivalent to a uniform scaling by the same factor, which sends
As a consequence, in the specific case in which S ≡ O, the homothety becomes a linear transformation, which preserves not only the collinearity of points (straight lines are mapped to straight lines), but also vector addition and scalar multiplication.
The image of a point (x, y) after a homothety with center (a, b) and scale factor λ is given by (a + λ(x − a), b + λ(y − b)).
See also
- Scaling (geometry) a similar notion in vector spaces
- Homothetic center, the center of a homothetic transformation taking one of a pair of shapes into the other
- The Hadwiger conjecture on the number of strictly smaller homothetic copies of a convex body that may be needed to cover it
- Homothetic function (economics), a function of the form f(U(y)) in which U is a homogeneous function and f is a monotonically increasing function.
Notes
2. ^{{harvtxt|Tuller|p=119}}
References
- {{ citation | first1 = J. | last1 = Hadamard | title = Lessons in Plane Geometry }}.
- {{ citation | first1 = Bruce E. | last1 = Meserve | year =1955 | title = Fundamental Concepts of Geometry | chapter = Homothetic transformations | pages = 166–169 | publisher = Addison-Wesley }}.
- {{ citation | last1 = Tuller | first1 = Annita | title = A Modern Introduction to Geometries | date=1967 | location=Princeton, NJ | publisher=D. Van Nostrand Co. | series=University Series in Undergraduate Mathematics}}.
External links
- Homothety, interactive applet from Cut-the-Knot.
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