Scaling (geometry)

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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.

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. In most cases, the homothetic transformations are non-linear transformations.

Matrix representation[edit]


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



In the case where vx = vy = vz = k, scaling increases the area of any surface by a factor of k2 and the volume of any solid object by a factor of k3.

Scaling in arbitrary dimensions[edit]

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 eigenvalues 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[edit]


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



which would be homogenized to



See also[edit]

External links[edit]