Scaling (geometry) - Matrix Representation

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:

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_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}.

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 (vx = vy = vz). If all except one of the scale factors are equal to 1, we have directional scaling.

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.

Read more about this topic:  Scaling (geometry)

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