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Wikipedia / Homothety

In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point called its center and a nonzero number called its ratio, which sends point to a point by the rule, for a fixed number

which is a uniform scaling and shows the meaning of special choices for :

  • one gets the identity mapping;

  • one gets the reflection at the center;

  • one gets the inverse mapping defined by .

In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if ) or reverse (if ) 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 is a line parallel to .

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.

multiplies distances between points by , areas and volumes by 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 exceeds1. The above-mentioned fixed point 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 ( ); and () ). 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.

Homotheties are used to scale the contents of computer screens; for example, smartphones, notebooks, and laptops.

Properties

The following properties hold in any dimension.

Mapping lines, line segments and angles

A homothety has the following properties:

  • A line is mapped onto a parallel line. Hence: angles remain unchanged.

  • The ratio of two line segments is preserved.

Both properties show that a homothety is a similarity.

Derivation of the properties

is the origin: which is a line parallel to .

and , the distance between their images. Hence, the ratio (quotient) of two line segments remains unchanged.

the calculation is analogous but a little extensive.

Consequences: A triangle is mapped on a similar one. The homothetic image of a circle is a circle. The image of an ellipse is a similar one. i.e. the ratio of the two axes is unchanged.

Graphical construction (using the intercept theorem)

of a second point can be constructed graphically using the intercept theorem and can be determined using .

Graphical construction (using a pantograph)

Before computers became ubiquitous, scalings of drawings were done by using a pantograph, a tool similar to a compass.

  1. Take 4 rods and assemble a mobile parallelogram are prolonged at the other end as shown in the diagram. Choose the ratio .

  2. .)

  3. Attach the mobile rods rotatable at point .

  4. and mark at each time point .

(see diagram) one gets from the intercept theorem and ratio .

Composition

Composition of two homotheties with the same center

The composition of two homotheties with the same center is again a homothety with center form a group.

Composition of two homotheties with different centers

The composition of two homotheties with different centers is:

Derivation

Hence, the composition is by vector ; is a fixpoint is a homothety lies on line .

Composition of a homothety and a translation

The composition of a homothety and a translation is a homothety.

Derivation

and ratio .

In homogeneous coordinates

can be represented in homogeneous coordinates

A pure homothety linear transformation is also conformal because it is composed of translation and uniform scale.

See also

Notes

References