Transformations - Transformations and Projections in Computer Graphics - page 30

each point consists of two numbers, three points are enough to reconstruct the trans-
formation matrix. Given three points both before ( P 1 , P 2 , P 3 )andafter( P 1 , P 2 ,
P 3 ) a transformation, we can write the three equations P 1 = P 1 T , P 2 = P 2 T ,and
P 3 = P 3 T and solve for the six elements of T .
Example . Thethreepoints(1 , 1), (1 , 0), and (0 , 1) are transformed to (3 , 4),

1), and (0 , 2), respectively. We write the general transformation ( x ∗ ,y ∗ )=( ax +
cy + m, bx + dy + n ) for the three sets

(2 , −

(3 , 4) = ( a + c + m, b + d + n ) ,
(2 , 1) = ( a + m, b + n ) ,

(0 , 2) = ( c + m, d + n ) ,

−

and this is easily solved to yield a =3, b =2, c =1, d =5, m = − 1, and n = − 3. The

transformation matrix is therefore

⎛ ⎞

320
150
−

⎝ ⎠ .

T =

1 − 31

Exercise 1.34: Inverse transformations .From P ∗ = PT ,weget P ∗ T − 1 = PTT − 1
or P = P ∗ T − 1 . We can therefore reconstruct an original point P from the transformed
one, P ∗ , if we know the inverse of the transformation matrix T . In general, the inverse
of the 3 × 3matrix

⎛ ⎞

ab 0
cd 0
mn 1

⎝ ⎠

T =

is

⎛ ⎞

d − b 0

1

T − 1 =

⎝ ⎠ .

− c a 0 (1.22)

ad − bc

cn − dm bm − an 1

Calculate the inverses of the transformation matrices for scaling, shearing, rotation, and
translation, and discuss their properties.

Exercise 1.35: Given that the four points

P 1 =(0 , 0) , P 2 =(0 , 1) , P 3 =(1 , 1) , and P 4 =(1 , 0)

are transformed to

P 1 =(0 , 0) , P 2 =(2 , 3) , P 3 =(8 , 4) , and P 4 =(6 , 1) ,

reconstruct the transformation matrix.