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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a homothety (or homothecy, or homogeneous dilation) is a
transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Trans ...
of an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
determined by a point ''S'' called its ''center'' and a nonzero number ''k'' called its ''ratio'', which sends point X to a point X' by the rule : \overrightarrow=k\overrightarrow for a fixed number k\ne 0. Using position vectors: :\mathbf x'=\mathbf s + k(\mathbf x -\mathbf s). In case of S=O (Origin): :\mathbf x'=k\mathbf x, which is a
uniform scaling In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a ''scale factor'' that is the same in all directions. The result of uniform scaling is similar ...
and shows the meaning of special choices for k: :for k=1 one gets the ''identity'' mapping, :for k=-1 one gets the ''reflection'' at the center, For 1/k one gets the ''inverse'' mapping defined by k. In
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
homotheties are the similarities that fix a point and either preserve (if k>0) or reverse (if k<0) the direction of all vectors. Together with the
translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
, all homotheties of an affine (or Euclidean) space form a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
, the group of dilations or homothety-translations. These are precisely the
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...
s with the property that the image of every line ''g'' is a line
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of IBM ...
to ''g''. In
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...
, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) that leaves the line at infinity pointwise
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
. In Euclidean geometry, a homothety of ratio k multiplies ''distances'' between points by , k, , ''areas'' by k^2 and volumes by , k, ^3. Here k 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 In geometry, a homothetic center (also called a center of similarity or a center of similitude) is a point from which at least two geometrically similar figures can be seen as a dilation or contraction of one another. If the center is ''externa ...
'' or ''center of similarity'' or ''center of similitude''. The term, coined by French mathematician
Michel Chasles Michel Floréal Chasles (; 15 November 1793 – 18 December 1880) was a French mathematician. Biography He was born at Épernon in France and studied at the École Polytechnique in Paris under Siméon Denis Poisson. In the War of the Sixth Coali ...
, is derived from two Greek elements: the prefix ''homo-'' (), meaning "similar", and ''thesis'' (), 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. 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: * A homothety is a '' similarity''. ''Derivation of the properties:'' In order to make calculations easy it is assumed that the center S is the origin: \mathbf x \to k\mathbf x. A line g with parametric representation \mathbf x=\mathbf p +t\mathbf v is mapped onto the point set g' with equation \mathbf x=k(\mathbf p+t\mathbf v)= k\mathbf p+tk\mathbf v, which is a line parallel to g. The distance of two points P:\mathbf p,\;Q:\mathbf q is , \mathbf p -\mathbf q, and , k\mathbf p -k\mathbf q, =, k, , \mathbf p-\mathbf q, the distance between their images. Hence, the ''ratio'' (quotient) of two line segments remains unchanged . In case of S\ne O the calculation is analogous but a little extensive. Consequences: A triangle is mapped on a similar one. The homothetic image of a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
is a circle. The image of an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
is a similar one. i.e. the ratio of the two axes is unchanged.


Graphical constructions


using the intercept theorem

If for a homothety with center S the image Q_1 of a point P_1 is given (see diagram) then the image Q_2 of a second point P_2, which lies not on line SP_1 can be constructed graphically using the intercept theorem: Q_2 is the common point th two lines \overline and \overline. The image of a point collinear with P_1,Q_1 can be determined using P_2,Q_2.


using a pantograph

Before computers became ubiquitous, scalings of drawings were done by using a
pantograph A pantograph (, from their original use for copying writing) is a mechanical linkage connected in a manner based on parallelograms so that the movement of one pen, in tracing an image, produces identical movements in a second pen. If a line dr ...
, a tool similar to a
compass A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with ...
. ''Construction and geometrical background:'' #Take 4 rods and assemble a mobile ''parallelogram'' with vertices P_0,Q_0,H,P such that the two rods meeting at Q_0 are prolongued at the other end as shown in the diagram. Choose the ''ratio'' k. #On the prolongued rods mark the two points S,Q such that , SQ_0, =k, SP_0, and , QQ_0, =k, HQ_0, . This is the case if , SQ_0, =\tfrac, P_0Q_0, . (Instead of k the location of the center S can be prescribed. In this case the ratio is k=, SQ_0, /, SP_0, .) #Attach the mobile rods rotatable at point S. #Vary the location of point P and mark at each time point Q. Because of , SQ_0, /, SP_0, =, Q_0Q, /, PP_0, (see diagram) one gets from the ''intercept theorem'' that the points S,P,Q are collinear (lie on a line) and equation , SQ, =k, SP, holds. That shows: the mapping P\to Q is a homothety with center S and ratio k.


Composition

*The composition of two homotheties with the ''same center'' S is again a homothety with center S. The homotheties with center S form a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
. *The composition of two homotheties with ''different centers'' S_1,S_2 and its ratios k_1,k_2 is ::in case of k_1k_2\ne 1 a ''homothety'' with its center on line \overline and ratio k_1k_2 or ::in case of k_1k_2= 1 a ''
translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
'' in direction \overrightarrow. Especially, if k_1=k_2=-1 (
point reflection In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
s). ''Derivation:'' For the composition \sigma_2\sigma_1 of the two homotheties \sigma_1,\sigma_2 with centers S_1,S_2 with :\sigma_1: \mathbf x \to \mathbf s_1+k_1(\mathbf x -\mathbf s_1), :\sigma_2: \mathbf x \to \mathbf s_2+k_2(\mathbf x -\mathbf s_2)\ one gets by calculation for the image of point X:\mathbf x: :(\sigma_2\sigma_1)(\mathbf x)= \mathbf s_2+k_2\big(\mathbf s_1+k_1(\mathbf x-\mathbf s_1)-\mathbf s_2\big) :\qquad \qquad \ =(1-k_1)k_2\mathbf s_1+(1-k_2)\mathbf s_2 + k_1k_2\mathbf x. Hence, the composition is :in case of k_1k_2= 1 a translation in direction \overrightarrow by vector \ (1-k_2)(\mathbf s_2-\mathbf s_1). :in case of k_1k_2\ne 1 point :S_3: \mathbf s_3=\frac =\mathbf s_1+\frac(\mathbf s_2-\mathbf s_1) is a ''fixpoint'' (is not moved) and the composition :\sigma_2\sigma_1: \ \mathbf x \to \mathbf s_3 + k_1k_2(\mathbf x -\mathbf s_3)\quad . is a ''homothety'' with center S_3 and ratio k_1k_2. S_3 lies on line \overline. *The composition of a homothety and a translation is a homothety. ''Derivation:'' The composition of the homothety :\sigma: \mathbf x \to \mathbf s +k(\mathbf x-\mathbf s),\; k\ne 1,\; and the translation :\tau: \mathbf x \to \mathbf x +\mathbf v is :\tau\sigma: \mathbf x \to \mathbf s +\mathbf v +k(\mathbf x-\mathbf s) :::=\mathbf s +\frac+k\left(\mathbf x-(\mathbf s+\frac)\right) which is a homothety with center \mathbf s'=\mathbf s +\frac and ratio k.


In homogenous coordinates

The homothety \sigma: \mathbf x \to \mathbf s+k(\mathbf x -\mathbf s) with center S=(u,v) can be written as the composition of a homothety with center O and a translation: :\mathbf x \to k\mathbf x + (1-k)\mathbf s. Hence \sigma can be represented in
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
by the matrix: :\begin k & 0 & (1-k)u\\ 0 & k & (1-k)v\\ 0 & 0 & 1 \end .


See also

*
Scaling (geometry) In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a ''scale factor'' that is the same in all directions. The result of uniform scaling is similarit ...
a similar notion in vector spaces *
Homothetic center In geometry, a homothetic center (also called a center of similarity or a center of similitude) is a point from which at least two geometrically similar figures can be seen as a dilation or contraction of one another. If the center is ''externa ...
, the center of a homothetic transformation taking one of a pair of shapes into the other *The
Hadwiger conjecture There are several conjectures known as the Hadwiger conjecture or Hadwiger's conjecture. They include: * Hadwiger conjecture (graph theory), a relationship between the number of colors needed by a given graph and the size of its largest clique mino ...
on the number of strictly smaller homothetic copies of a convex body that may be needed to cover it *
Homothetic function (economics) In consumer theory, a consumer's preferences are called homothetic if they can be represented by a utility function which is homogeneous of degree 1. For example, in an economy with two goods x,y, homothetic preferences can be represented by a ut ...
, a function of the form ''f''(''U''(''y'')) in which ''U'' is a
homogeneous function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
and ''f'' is a monotonically increasing function.


Notes


References

* H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961), p. 94 * * * {{ citation , last1 = Tuller , first1 = Annita , author-link = Annita Tuller , 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 Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
. Transformation (function)