Transformations - Transformations and Projections in Computer Graphics - page 3

Exercise 1.2: Is the operation of combining transformations commutative?
Another important example of a group of transformations is the set of linear trans-
formations that map a point P =( x, y, z )toapoint P ∗ =( x ∗ ,y ∗ ,z ∗ ), where

x ∗ = a 11 x + a 12 y + a 13 z + a 14 ,
y ∗ = a 21 x + a 22 y + a 23 z + a 24 ,
z ∗ = a 31 x + a 32 y + a 33 z + a 34 .

(1.1)

Each new coordinate depends on all three original coordinates, and the dependence
is linear. Such transformations are called a ne and are defined more rigorously on
page 22.
A little thinking shows that the coe cients a i 4 of Equation (1.1) represent quanti-
ties that are added to the transformed coordinates ( x ∗ ,y ∗ ,z ∗ ) regardless of the original
coordinates, thereby simply translating P ∗ in space. This is why we start the detailed
discussion here by temporarily ignoring these coe cients, which leads to the simple
system of equations

x ∗ = a 11 x + a 12 y + a 13 z,
y ∗ = a 21 x + a 22 y + a 23 z,
z ∗ = a 31 x + a 32 y + a 33 z.

(1.2)

If the 3 3 coe cient matrix of this system of equations is nonsingular or, equivalently,

if the determinant of the coe cient matrix is nonzero (see any text on linear algebra for
a refresher on matrices and determinants), then the system is easy to invert and can be
expressed in the form

×

x = b 11 x ∗ + b 12 y ∗ + b 13 z ∗ ,
y = b 21 x ∗ + b 22 y ∗ + b 23 z ∗ ,
z = b 31 x ∗ + b 32 y ∗ + b 33 z ∗ ,

(1.3)

where the b ij 's are expressed in terms of the a ij 's. It is now easy to see that, for
example, the two-dimensional line Ax + By + C = 0 is transformed by Equation (1.3)
to the two-dimensional line

( Ab 11 + Bb 21 ) x ∗ +( Ab 12 + Bb 22 ) y ∗ + C =0 .

Exercise 1.3: Show that Equation (1.3) maps the general second-degree curve

Ax 2 + Bxy + Cy 2 + Dx + Ey + F =0

to another second-degree curve.

In general, an a ne transformation maps any curve of degree n to another curve
of the same degree.