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 ind ...
, 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 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 simil ...
'' 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 shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a
photograph, or when creating a
scale model 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 graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.
A plane shape or plane figure is constrained to lie ...
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 a ...
, 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), 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, and a special case of
homothetic transformation (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 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 ...
of ''x'', and may be called the
constant of proportionality 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. To scale an object by a
vector ''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 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 or planar lamina, while ''surface area'' refers to the area of an open s ...
by a factor between the smallest and the largest product of two scale factors, and the
volume by the product of all three.
The scaling is uniform
if and only if 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 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 whose entries on the diagonal are all equal to
, namely
.
Non-uniform scaling is accomplished by multiplication with any
symmetric matrix. 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 ...
s of the matrix are the scale factors, and the corresponding
eigenvectors 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 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, often used in
computer graphics, points are represented using
homogeneous coordinates. To scale an object by a
vector ''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 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)
*
Homogeneous function
*
Homothetic transformation
*
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)
*
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 st ...
*
Scales of scale models
*
Scaling in statistical estimation
*
Scaling in gravity
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
*
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
Footnotes
External links
Understanding 2D Scalingan
Understanding 3D Scalingby Roger Germundsson,
The Wolfram Demonstrations Project.
{{DEFAULTSORT:Scaling (Geometry)
Transformation (function)