Homothetic transformation
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 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 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 λ^{2}. 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
References
- Hadamard, J., Lessons in Plane Geometry.
- Meserve, Bruce E. (1955), "Homothetic transformations", Fundamental Concepts of Geometry, Addison-Wesley, pp. 166–169.
- Tuller, Annita (1967), A Modern Introduction to Geometries, University Series in Undergraduate Mathematics, Princeton, NJ: D. Van Nostrand Co..
External links
- Homothety, interactive applet from Cut-the-Knot.