In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a homothety (or homothecy or dilation) is a transformation

Transformation (mathematics)

In mathematics, a transformation could be any function mapping a set X on to another set or on to itself. However, often the set X has some additional algebraic or geometric structure and the term "transformation" refers to a function from X to itself that preserves this structure.Examples include...

 of an affine space

Affine space

In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...

 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. In Euclidean geometry homotheties are the similarities

Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...

 that fix a point and either preserve (if ) or reverse (if ) the direction of all vectors. Together with the translations

Translation (geometry)

In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...

, all homotheties of an affine (or Euclidean) space form a group, the group of homothety-translations. These are precisely the affine transformation

Affine transformation

In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...

s with the property that the image of every line L is a line parallel

Parallel (geometry)

Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...

 to L.

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.

See also

• Scaling (geometry)
  Scaling (geometry)
  In Euclidean geometry, uniform scaling is a linear transformation that enlarges or shrinks objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original...
  
   a similar notion in vector spaces
• Homothetic center
  Homothetic center
  In geometry, a homothetic center is a point from which at least two geometrically similar figures can be seen as a dilation/contraction of one another. If the center is external, the two figures are directly similar to one another; their angles have the same rotational sense...
  
  , the center of a homothetic transformation taking one of a pair of shapes into the other
• The Hadwiger conjecture
  Hadwiger conjecture (combinatorial geometry)
  In combinatorial geometry, the Hadwiger conjecture states that any convex body in n-dimensional Euclidean space can be covered by 2n or fewer smaller bodies homothetic with the original body, and that furthermore, the upper bound of 2n is necessary iff the body is a parallelpiped...
  
   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 U(f(y)) in which f is a homogeneous function
  Homogeneous function
  In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
  
   and U is a monotonically increasing function
  Monotonic function
  In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
  
  .

External links

• Homothety, interactive applet from Cut-the-Knot
  Cut-the-knot
  Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
  
  .