Editing 2D computer graphics (section)

===Scaling===
{{Refimprove|date=April 2008}}
In [[Euclidean geometry]], '''uniform scaling''' ('''[[isotropic]] scaling''',<ref>{{cite web|format=PowerPoint|last1=Durand|last2=Cutler|url=http://groups.csail.mit.edu/graphics/classes/6.837/F03/lectures/04_transformations.ppt |title=Transformations|publisher=Massachusetts Institute of Technology|access-date =12 September 2008}}</ref> '''homogeneous dilation''', [[Homothetic transformation|homothety]]) 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 [[Similarity (geometry)|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. (Some school text books specifically exclude this possibility, just as some exclude squares from being rectangles or circles from being ellipses.)

More general is '''scaling''' with a separate scale factor for each axis direction. '''Non-uniform scaling''' ('''[[anisotropic]] scaling''', '''inhomogeneous dilation''') 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]] 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).

A scaling can be represented by a scaling matrix.  To scale an object by a [[Vector (geometric)|vector]] ''v'' = (''v<sub>x</sub>, v<sub>y</sub>, v<sub>z</sub>''), each point ''p'' = (''p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>'') would need to be multiplied with this [[scaling matrix]]:
:<math> S_v = 
\begin{bmatrix}
v_x & 0 & 0  \\
0 & v_y & 0  \\
0 & 0 & v_z  \\
\end{bmatrix}.
</math>

As shown below, the multiplication will give the expected result:
:<math>
S_vp =
\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_xp_x \\ v_yp_y \\ v_zp_z
\end{bmatrix}.
</math>

Such a scaling changes the [[diameter]] of an object by a factor between the scale factors, the [[area]] 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<sub>x</sub> = v<sub>y</sub> = v<sub>z</sub>''). If all except one of the scale factors are equal to 1, we have directional scaling.

In the case where ''v<sub>x</sub> = v<sub>y</sub> = v<sub>z</sub> = k'', the scaling is also called an '''enlargement''' or '''[[Dilation (metric space)|dilation]]''' by a factor k, increasing the area by a factor of k<sup>2</sup> and the volume by a factor of k<sup>3</sup>.

Scaling in the most general sense is any [[affine transformation]] with a [[diagonalizable 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 (linear algebra)|projection]]), 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.