Scaling (geometry)

In Euclidean geometry, uniform scaling or isotropic scaling [Durand and Cutler (n.d.). [http://groups.csail.mit.edu/graphics/classes/6.837/F03/lectures/04_transformations.ppt Transformations] . Massachusetts Institute of Technology. Retrieved 12 September 2008.] is a linear transformation that enlarges or diminishes objects; the scale factor 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 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" = ("v_x, v_y, v_z"), each point "p" = ("p_x, p_y, p_z") would need to be multiplied with this scaling matrix:: $S_v = egin{bmatrix}v_x & 0 & 0 \0 & v_y & 0 \0 & 0 & v_z \end{bmatrix}.$

As shown below, the multiplication will give the expected result:: $S_vp =egin{bmatrix}v_x & 0 & 0 \0 & v_y & 0 \0 & 0 & v_z \end{bmatrix}egin{bmatrix}p_x \ p_y \ p_z end{bmatrix}=egin{bmatrix}v_xp_x \ v_yp_y \ v_zp_zend{bmatrix}.$

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.

A 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), 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, since translation cannot be accomplished with a 3-by-3 matrix. To scale an object by a vector "v" = ("v_x, v_y, v_z"), each homogeneous vector "p" = ("p_x, p_y, p_z", 1) would need to be multiplied with this scaling matrix:: $S_v = egin{bmatrix}v_x & 0 & 0 & 0 \0 & v_y & 0 & 0 \0 & 0 & v_z & 0 \0 & 0 & 0 & 1 end{bmatrix}.$

As shown below, the multiplication will give the expected result:: $S_vp =egin{bmatrix}v_x & 0 & 0 & 0 \0 & v_y & 0 & 0 \0 & 0 & v_z & 0 \0 & 0 & 0 & 1 end{bmatrix}egin{bmatrix}p_x \ p_y \ p_z \ 1 end{bmatrix}=egin{bmatrix}v_xp_x \ v_yp_y \ v_zp_z \ 1 end{bmatrix}.$

The scaling is uniform iff 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:: $S_v = egin{bmatrix}1 & 0 & 0 & 0 \0 & 1 & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & frac{1}{s} end{bmatrix}.$

For each homogeneous vector "p" = ("p_x, p_y, p_z", 1) we would have: $S_vp =egin{bmatrix}1 & 0 & 0 & 0 \0 & 1 & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & frac{1}{s} end{bmatrix}egin{bmatrix}p_x \ p_y \ p_z \ 1 end{bmatrix}=egin{bmatrix}p_x \ p_y \ p_z \ frac{1}{s} end{bmatrix}$ which would be homogenized to: $egin{bmatrix}sp_x \ sp_y \ sp_z \ 1 end{bmatrix}.$

Footnotes

ee also

* Scale (ratio)
* Scale (map)
* Scales of scale models
* Scale (disambiguation)
*
* Transformation matrix

External links

* [http://demonstrations.wolfram.com/Understanding2DScaling/ Understanding 2D Scaling] and [http://demonstrations.wolfram.com/Understanding3DScaling/ Understanding 3D Scaling] by Roger Germundsson, The Wolfram Demonstrations Project.

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