Scaling (geometry)

Each iteration of theSierpinski trianglecontains triangles related to the next iteration by a scale factor of 1/2

Inaffine geometry,uniform scaling(orisotropicscaling^[1]) is alinear transformationthat enlarges (increases) or shrinks (diminishes) objects by ascale factorthat is the same in all directions. The result of uniform scaling issimilar(in the geometric sense) to the original. A scale factor of 1 is normally allowed, so thatcongruentshapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing aphotograph,or when creating ascale modelof a building, car, airplane, etc.

More general isscalingwith a separate scale factor for each axis direction.Non-uniform scaling(anisotropicscaling) is obtained when at least one of the scaling factors is different from the others; a special case isdirectional scalingorstretching(in one direction). Non-uniform scaling changes theshapeof 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 anoblique angle,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 calleddilationorenlargement.When the scale factor is a positive number smaller than 1, scaling is sometimes also calledcontractionorreduction.

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 areflection).

Scaling is alinear transformation,and a special case ofhomothetic transformation(scaling about a point). In most cases, the homothetic transformations are non-linear transformations.

Uniform scaling

Ascale factoris usually a decimal which scales, or multiplies, some quantity. In the equationy=Cx,Cis the scale factor forx.Cis also thecoefficientofx,and may be called theconstant of proportionalityofytox.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 scalingmatrix.To scale an object by avectorv= (v_x,v_y,v_z), each pointp= (p_x,p_y,p_z) would need to be multiplied with this scaling matrix:

S_{v}={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}.

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

S_{v}p={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\end{bmatrix}}.

Such a scaling changes thediameterof an object by a factor between the scale factors, theareaby a factor between the smallest and the largest product of two scale factors, and thevolumeby the product of all three.

The scaling is uniformif and only ifthe 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 wherev_x= v_y= v_z= k,scaling increases the area of any surface by a factor ofk²and the volume of any solid object by a factor ofk³.

Scaling in arbitrary dimensions

In $n$ -dimensional space $\mathbb {R} ^{n}$ ,uniform scaling by a factor $v$ is accomplished byscalar multiplicationwith $v$ ,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 adiagonal matrixwhose entries on the diagonal are all equal to $v$ ,namely $vI$ .

Non-uniform scaling is accomplished by multiplication with anysymmetric matrix.Theeigenvaluesof the matrix are the scale factors, and the correspondingeigenvectorsare 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 aneigenspacewill 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

Inprojective geometry,often used incomputer graphics,points are represented usinghomogeneous coordinates.To scale an object by avectorv= (v_x,v_y,v_z), each homogeneous coordinate vectorp= (p_x,p_y,p_z,1) would need to be multiplied with thisprojective transformationmatrix:

S_{v}={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}.

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

S_{v}p={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\\1\end{bmatrix}}.

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 factors(uniform scaling) can be accomplished by using this scaling matrix: