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In
affine geometry In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle. As the notion of '' parallel lines'' is one of the main properties that is i ...
, uniform scaling (or isotropic scaling) is a
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions ( isotropically). The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a
photograph A photograph (also known as a photo, or more generically referred to as an ''image'' or ''picture'') is an image created by light falling on a photosensitivity, photosensitive surface, usually photographic film or an electronic image sensor. Th ...
, or when creating a
scale model A scale model is a physical model that is geometrically similar to an object (known as the ''prototype''). Scale models are generally smaller than large prototypes such as vehicles, buildings, or people; but may be larger than small protot ...
of a building, car, airplane, etc. More general is scaling with a separate scale factor for each axis direction. Non-uniform scaling (
anisotropic Anisotropy () is the structural property of non-uniformity in different directions, as opposed to isotropy. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit ver ...
scaling) is obtained when at least one of the scaling factors is different from the others; a special case is directional scaling or stretching (in one direction). Non-uniform scaling changes the
shape A shape is a graphics, graphical representation of an object's form or its external boundary, outline, or external Surface (mathematics), surface. It is distinct from other object properties, such as color, Surface texture, texture, or material ...
of the object; e.g. a square may change into a rectangle, or into a parallelogram if the sides of the square are not parallel to the scaling axes (the angles between lines parallel to the axes are preserved, but not all angles). It occurs, for example, when a faraway billboard is viewed from an
oblique angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight lines at a point. Formally, an angle is a figure lying in a plane formed by two rays, called the '' sides'' of the angle, sharing ...
, or when the shadow of a flat object falls on a surface that is not parallel to it. When the scale factor is larger than 1, (uniform or non-uniform) scaling is sometimes also called dilation or enlargement. When the scale factor is a positive number smaller than 1, scaling is sometimes also called contraction or reduction. In the most general sense, a scaling includes the case in which the directions of scaling are not perpendicular. It also includes the case in which one or more scale factors are equal to zero (
projection Projection or projections may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphics, and carto ...
), and the case of one or more negative scale factors (a directional scaling by -1 is equivalent to a reflection). Scaling is a
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
, and a special case of
homothetic transformation In mathematics, a homothety (or homothecy, or homogeneous dilation) is a Transformation (mathematics), transformation of an affine space determined by a point called its ''center'' and a nonzero number called its ''ratio'', which sends point ...
(scaling about a point). In most cases, the homothetic transformations are non-linear transformations.


Uniform scaling

A scale factor is usually a decimal which scales, or multiplies, some quantity. In the equation ''y'' = ''Cx'', ''C'' is the scale factor for ''x''. ''C'' is also the
coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
of ''x'', and may be called the
constant of proportionality In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio. The ratio is called ''coefficient of proportionality'' (or ''proportionality c ...
of ''y'' to ''x''. For example, doubling distances corresponds to a scale factor of two for distance, while cutting a cake in half results in pieces with a scale factor for volume of one half. The basic equation for it is image over preimage. In the field of measurements, the scale factor of an instrument is sometimes referred to as sensitivity. The ratio of any two corresponding lengths in two similar geometric figures is also called a scale.


Matrix representation

A scaling can be represented by a scaling
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
. To scale an object by a
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
''v'' = (''vx, vy, vz''), each point ''p'' = (''px, py, pz'') would need to be multiplied with this scaling matrix: : S_v = \begin v_x & 0 & 0 \\ 0 & v_y & 0 \\ 0 & 0 & v_z \\ \end. As shown below, the multiplication will give the expected result: : S_vp = \begin v_x & 0 & 0 \\ 0 & v_y & 0 \\ 0 & 0 & v_z \\ \end \begin p_x \\ p_y \\ p_z \end = \begin v_xp_x \\ v_yp_y \\ v_zp_z \end. Such a scaling changes the
diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...
of an object by a factor between the scale factors, the
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
by a factor between the smallest and the largest product of two scale factors, and the
volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
by the product of all three. The scaling is uniform
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
the scaling factors are equal (''vx = vy = vz''). If all except one of the scale factors are equal to 1, we have directional scaling. In the case where ''vx = vy = vz = k'', scaling increases the area of any surface by a factor of ''k''2 and the volume of any solid object by a factor of ''k''3.


Scaling in arbitrary dimensions

In n-dimensional space \mathbb^n, uniform scaling by a factor v is accomplished by
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...
with v, that is, multiplying each coordinate of each point by v. As a special case of linear transformation, it can be achieved also by multiplying each point (viewed as a column vector) with a
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagon ...
whose entries on the diagonal are all equal to v, namely v I . Non-uniform scaling is accomplished by multiplication with any
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with ...
. The
eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s of the matrix are the scale factors, and the corresponding
eigenvector In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...
s are the axes along which each scale factor applies. A special case is a diagonal matrix, with arbitrary numbers v_1,v_2,\ldots v_n along the diagonal: the axes of scaling are then the coordinate axes, and the transformation scales along each axis i by the factor v_i. In uniform scaling with a non-zero scale factor, all non-zero vectors retain their direction (as seen from the origin), or all have the direction reversed, depending on the sign of the scaling factor. In non-uniform scaling only the vectors that belong to an
eigenspace In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...
will retain their direction. A vector that is the sum of two or more non-zero vectors belonging to different eigenspaces will be tilted towards the eigenspace with largest eigenvalue.


Using homogeneous coordinates

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 (''p ...
, often used in
computer graphics Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ...
, points are represented using
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. ...
. To scale an object by a
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
''v'' = (''vx, vy, vz''), each homogeneous coordinate vector ''p'' = (''px, py, pz'', 1) would need to be multiplied with this
projective transformation In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
matrix: : S_v = \begin v_x & 0 & 0 & 0 \\ 0 & v_y & 0 & 0 \\ 0 & 0 & v_z & 0 \\ 0 & 0 & 0 & 1 \end. As shown below, the multiplication will give the expected result: : S_vp = \begin v_x & 0 & 0 & 0 \\ 0 & v_y & 0 & 0 \\ 0 & 0 & v_z & 0 \\ 0 & 0 & 0 & 1 \end \begin p_x \\ p_y \\ p_z \\ 1 \end = \begin v_xp_x \\ v_yp_y \\ v_zp_z \\ 1 \end. Since the last component of a homogeneous coordinate can be viewed as the denominator of the other three components, a uniform scaling by a common factor ''s'' (uniform scaling) can be accomplished by using this scaling matrix: : S_v = \begin 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \frac \end. For each vector ''p'' = (''px, py, pz'', 1) we would have : S_vp = \begin 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \frac \end \begin p_x \\ p_y \\ p_z \\ 1 \end = \begin p_x \\ p_y \\ p_z \\ \frac \end , which would be equivalent to : \begin sp_x \\ sp_y \\ sp_z \\ 1 \end.


Function dilation and contraction

Given a point P(x,y), the dilation associates it with the point P'(x',y') through the equations : \beginx'=mx \\ y'=ny\end for m,n \in \R^+. Therefore, given a function y=f(x), the equation of the dilated function is : y=nf\left(\frac\right).


Particular cases

If n=1, the transformation is horizontal; when m > 1, it is a dilation, when m < 1, it is a contraction. If m=1, the transformation is vertical; when n>1 it is a dilation, when n<1, it is a contraction. If m=1/n or n=1/m, the transformation is a
squeeze mapping In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation (mathematics), rotation or shear mapping. For a fixed p ...
.


See also

* 2D computer graphics#Scaling * Digital zoom *
Dilation (metric space) In mathematics, a dilation is a function f from a metric space M into itself that satisfies the identity :d(f(x),f(y))=rd(x,y) for all points x, y \in M, where d(x, y) is the distance from x to y and r is some positive real number. In Euclidean s ...
*
Homogeneous function In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar (mathematics), scalar, then the function's value is multiplied by some p ...
*
Homothetic transformation In mathematics, a homothety (or homothecy, or homogeneous dilation) is a Transformation (mathematics), transformation of an affine space determined by a point called its ''center'' and a nonzero number called its ''ratio'', which sends point ...
*
Orthogonal coordinates In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. ...
*
Scalar (mathematics) A scalar is an element of a field which is used to define a ''vector space''. In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scal ...
*
Scale (disambiguation) Scale or scales may refer to: Mathematics * Scale (descriptive set theory), an object defined on a set of points * Scale (ratio), the ratio of a linear dimension of a model to the corresponding dimension of the original * Scale factor, a number ...
**
Scale (ratio) The scale ratio of a model represents the proportional ratio of a linear dimension of the model to the same feature of the original. Examples include a 3-dimensional scale model of a building or the scale drawings of the elevations or plans of a ...
**
Scale (map) The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation ...
* Scale factor (computer science) *
Scale factor (cosmology) The expansion of the universe is parametrized by a dimensionless scale factor a . Also known as the cosmic scale factor or sometimes the Robertson–Walker scale factor, this is a key parameter of the Friedmann equations. In the early stages of ...
* Scales of scale models * Scaling in statistical estimation * Scaling in gravity *
Transformation matrix In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then there exists an m \times n matrix A, called the transfo ...
*
Image scaling In computer graphics and digital imaging, image scaling refers to the resizing of a digital image. In video technology, the magnification of digital material is known as upscaling or resolution enhancement. When scaling a vector graphic image ...


Footnotes


External links


Understanding 2D Scaling
an
Understanding 3D Scaling
by Roger Germundsson,
The Wolfram Demonstrations Project The Wolfram Demonstrations Project is an open-source collection of interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown to over 10,000. The site won a Pa ...
.
Scale Factor Calculator
{{DEFAULTSORT:Scaling (Geometry) Transformation (function)