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Scaling (geometry)

 

 
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Scaling (geometry)
 
 
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In Euclidean geometry
Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
, uniform scaling or isotropic scaling is a linear transformation
Linear transformation

In mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication....
 that enlarges or increases or diminishes objects; the scale factor
Scale factor

A scale factor is a number which scaling, or multiplies, some quantity. In the equation, is the scale factor for . is also the coefficient of , and may be called the constant of proportionality of to ....
 is the same in all directions; it is also called a homothety. The result of uniform scaling is similar (in the geometric sense) to the original.

More general is scaling with a separate scale factor for each axis direction. Non-uniform or anisotropic scaling is obtained when at least one of the scaling factors is different from the others; a special case is directional scaling (in one direction).






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Encyclopedia


In Euclidean geometry
Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
, uniform scaling or isotropic scaling is a linear transformation
Linear transformation

In mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication....
 that enlarges or increases or diminishes objects; the scale factor
Scale factor

A scale factor is a number which scaling, or multiplies, some quantity. In the equation, is the scale factor for . is also the coefficient of , and may be called the constant of proportionality of to ....
 is the same in all directions; it is also called a homothety. The result of uniform scaling is similar (in the geometric sense) to the original.

More general is scaling with a separate scale factor for each axis direction. Non-uniform or anisotropic scaling is obtained when at least one of the scaling factors is different from the others; a special case is directional scaling (in one direction). Non-uniform scaling changes the shape
Shape

The shape of an object located in some space is the part of that space occupied by the object, as determined by its external boundary ? abstracting from other properties such as colour, content, and material composition, as well as from the object's other spatial properties ....
 of the object; e.g. a rectangle may change into a rectangle of a different shape, but also in a parallelogram (the angles between lines parallel to the axes are preserved, but not all angles).

Matrix representation


A scaling can be represented by a scaling matrix. To scale an object by a vector v = (vx, vy, vz), each point p = (px, py, pz) 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
Diameter

In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle....
 of an object by a factor between the scale factors, the area
Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
 by a factor between the smallest and the largest product of two scale factors, and the volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
 by the product of all three.

A scaling in the most general sense is any affine transformation
Affine transformation

In geometry, an affine transformation or affine map or an affinity between two vector spaces consists of a linear transformation followed by a translation :...
 with a diagonalizable matrix
Diagonalizable matrix

In linear algebra, a square matrix A is called diagonalizable if it is similar matrix to a diagonal matrix, i.e. if there exists an invertible matrix P such that P −1AP is a diagonal matrix....
. It includes the case that the three directions of scaling are not perpendicular. It includes also the case that one or more scale factors are equal to zero (projection
Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P....
), and the case of one or more negative scale factors. The latter corresponds to a combination of scaling proper and a kind of reflection: along lines in a particular direction we take the reflection in the point of intersection with a plane that need not be perpendicular; therefore it is more general than ordinary reflection in the plane.

Using homogeneous coordinates


Often, it is more useful to use homogeneous coordinates
Homogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand M?bius in his 1827 work Der barycentrische Calc?l, allow affine transformations to be easily represented by a matrix....
, since translation
Translation (geometry)

In Euclidean geometry, a translation is moving every point a constant distance in a specified direction. It is one of the Euclidean groups . A translation can also be interpreted as the addition of a constant vector space to every point, or as shifting the Origin of the coordinate system....
 cannot be accomplished with a 3-by-3 matrix. To scale an object by a vector v = (vx, vy, vz), each homogeneous
Homogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand M?bius in his 1827 work Der barycentrische Calc?l, allow affine transformations to be easily represented by a matrix....
 vector p = (px, py, pz, 1) would need to be multiplied with this scaling matrix:

As shown below, the multiplication will give the expected result:

The scaling is uniform if and only if
If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
 the scaling factors are equal. If all scale factors except one are 1 we have directional scaling.

Since the last component of a homogeneous coordinate can be viewed as the denominator of the other three components, a scaling by a common factor s can be accomplished by using this scaling matrix:

For each homogeneous
Homogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand M?bius in his 1827 work Der barycentrische Calc?l, allow affine transformations to be easily represented by a matrix....
 vector p = (px, py, pz, 1) we would have

which would be homogenized to

Footnotes


See also

  • Scale (ratio)
    Scale (ratio)

    The concept of scale is applicable if a system is represented Proportionality ly by another system. For example, for a scale model of an object, the ratio of corresponding lengths is a Dimensionless number scale, e.g....
  • Scale (map)
    Scale (map)

    Sorry, no overview for this topic
  • Scales of scale models
    Scale model

    Sorry, no overview for this topic
  • Scale
    Scale

    Scale can refer to:...
     (disambiguation)
  • Transformation matrix
    Transformation matrix

    In linear algebra, linear transformations can be represented by matrix . If T is a linear transformation mapping Rn to Rm and x is a column vector with n entries, then...


External links

  • and by Roger Germundsson, The Wolfram Demonstrations Project.