![Sierpinski triangle evolution](https://upload.wikimedia.org/wikipedia/commons/0/05/Sierpinski_triangle_evolution.svg)
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 inde ...
, uniform scaling (or
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
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 pre ...
that enlarges (increases) or shrinks (diminishes) objects by a ''
scale factor
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 ...
'' that is the same in all directions. 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 mod ...
shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a
photograph
A photograph (also known as a photo, image, or picture) is an image created by light falling on a photosensitive surface, usually photographic film or an electronic image sensor, such as a CCD or a CMOS chip. Most photographs are now create ...
, or when creating a
scale model
A scale model is a physical model which 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 prototypes ...
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 property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
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 or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type.
A pl ...
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 is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles are ...
, 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, projections or projective 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, graphic ...
), and the case of one or more negative scale factors (a directional scaling by -1 is equivalent to a
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
).
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 pre ...
, and a special case of
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 ''k'' called its ''ratio'', which sends point X to a point X' by the ...
(scaling about a point). In most cases, the homothetic transformations are non-linear transformations.
Uniform scaling
![Sierpinski triangle evolution](https://upload.wikimedia.org/wikipedia/commons/0/05/Sierpinski_triangle_evolution.svg)
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 multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
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, which is called the coefficient of proportionality or proportionality constant ...
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 most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. To scale an object by a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
''v'' = (''v
x, v
y, v
z''), each point ''p'' = (''p
x, p
y, p
z'') would need to be multiplied with this scaling matrix:
:
As shown below, the multiplication will give the expected result:
:
Such a scaling changes the
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
of an object by a factor between the scale factors, the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...
by a factor between the smallest and the largest product of two scale factors, and the
volume
Volume is a measure of occupied 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). The de ...
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" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
the scaling factors are equal (''v
x = v
y = v
z''). If all except one of the scale factors are equal to 1, we have directional scaling.
In the case where ''v
x = v
y = v
z = 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
-dimensional space
, uniform scaling by a factor
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 by ...
with
, that is, multiplying each coordinate of each point by
. 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 diagonal ma ...
whose entries on the diagonal are all equal to
, namely
.
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 re ...
. The
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of the matrix are the scale factors, and the corresponding
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s are the axes along which each scale factor applies. A special case is a diagonal matrix, with arbitrary numbers
along the diagonal: the axes of scaling are then the coordinate axes, and the transformation scales along each axis
by the factor
.
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 of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
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, pro ...
, often used in
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, 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. T ...
. To scale an object by a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
''v'' = (''v
x, v
y, v
z''), each homogeneous coordinate vector ''p'' = (''p
x, p
y, p
z'', 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, s ...
matrix:
:
As shown below, the multiplication will give the expected result:
:
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:
:
For each vector ''p'' = (''p
x, p
y, p
z'', 1) we would have
:
which would be equivalent to
:
Function dilation and contraction
Given a point
, the dilation associates it with the point
through the equations
:
for
.
Therefore, given a function
, the equation of the dilated function is
:
Particular cases
If
, the transformation is horizontal; when
, it is a dilation, when
, it is a contraction.
If
, the transformation is vertical; when
it is a dilation, when
, it is a contraction.
If
or
, 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 or shear mapping.
For a fixed positive real number , th ...
.
See also
*
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 spa ...
*
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 ...
*
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 ''k'' called its ''ratio'', which sends point X to a point X' by the ...
*
Orthogonal coordinates In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
*
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 scala ...
*
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 w ...
**
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 relative 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 sta ...
*
Scales of scale models
*
Scaling in statistical estimation
*
Scaling in gravity
In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or pos ...
*
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 or shear mapping.
For a fixed positive real number , th ...
*
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
T( \mathbf x ) = A \mathbf x
for some m \times n matrix ...
Footnotes
External links
Understanding 2D Scalingan
Understanding 3D Scalingby Roger Germundsson,
The Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
{{DEFAULTSORT:Scaling (Geometry)
Transformation (function)