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

${\displaystyle M\mapsto S+\lambda {\overrightarrow {SM}},}$

Two similar geometric figures related by a homothetic transformation with respect to a homothetic center S. The angles at corresponding points are the same and have the same sense; for example, the angles ABC and A'B'C' are both clockwise and equal in magnitude.

in other words it fixes S, and sends each 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 λ². Here |λ| is 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.

The term, coined by French mathematician Michel Chasles, is derived from two Greek elements: the prefix homo- (όμο), meaning "similar", and thesis (Θέσις), meaning "position". It describes the relationship between two figures of the same shape and orientation. For example, two Russian dolls looking in the same direction can be considered homothetic.

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 ratio λ is equivalent to a uniform scaling by the same factor, which sends

${\displaystyle {\overrightarrow {OM}}\mapsto \lambda {\overrightarrow {OM}}.}$

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 ratio 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

1. Hadamard, p. 145)
2. Tuller (1967, p. 119)

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.