Transformations - Transformations and Projections in Computer Graphics - page 31

1.2.11 A Note
All the expressions derived so far for transformations are based on the basic relation
P ∗ = PT . Some authors prefer the equivalent relation P ∗ = TP , which changes the
mathematics somewhat. If we want the coordinates of the transformed point to be the
same as before (i.e., x ∗ = ax + cy + m , y ∗ = bx + dy + n ), we have to write the relation
P ∗ = TP in the form

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

x ∗
y ∗
1

acm
bdn
00 1

x
y
1

⎝ ⎠ = ⎝ ⎠ ⎝ ⎠ .

The first difference is that both P and P ∗ are columns instead of rows. This is because of
the rules of matrix multiplication. The second difference is that the new transformation
matrix T is the transpose of the original one. Hence, rotation, for example, is achieved
by the matrices

⎛ ⎞

cos θ sin θ 0

⎝ ⎠

− sin θ cos θ 0

0 0 1

for a clockwise rotation, and

⎛ ⎞

cos θ − sin θ 0

⎝ ⎠

sin θ cos θ 0

0 0 1

for a counterclockwise rotation.

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠ .

10 m
01 n
00 1

100
010
mn 1

Similarly, translation is done by instead of

1.2.12 Summary
The general two-dimensional a ne transformation is given by x ∗ = ax + cy + m , y ∗ =
bx + dy + n . This section shows the values or constraints that should be assigned to the
four coe cients a , b , c ,and d in order to obtain certain types of transformations (we
ignore translations).

A general a ne transformation is obtained when ad − bc =0. For ad − bc =+1,

the transformation is rotation, and for ad − bc = − 1, it is reflection.

The case ad − bc = 0 corresponds to a singular transformation.

The identity transformation is obtained when a = d =1and b = c =0.
An isometry is obtained by a 2 + b 2 = c 2 + d 2 =1and ac + bd =0. Anisometryisa
transformation that preserves distances. If P and Q are two points on an object, then
the distance d between them is preserved, meaning that the distance d between P ∗ and
Q ∗ is the same. Rotations, reflections, and translations are isometries.