Matrices and Linear Transformations - 3D Math Primer for Graphics and Game Development - page 145

Figure 5.6
Scaling a 2D object
with various factors
for k x and k y

As is intuitively obvious, the basis vectors p and q are independently
affected by the corresponding scale factors:

′

p = k x p = k x 1 0 = k x 0 ,

′

q = k y q = k y 0 1 = 0 k y .

Constructing the 2D scale matrix S (k x ,k y ) from these basis vectors, we get

′ −

− p k x 0

0 k y

S (k x ,k y ) = = .

′ −

2D matrix to scale on
cardinal axes

− q

For 3D, we add a third scale factor k z , and the 3D scale matrix is then
given by

2 3

k x 0 0

3D matrix to scale on
cardinal axes

4 5

S (k x ,k y ,k z ) = 0 k y 0 .

0 0 k z