Transformations - Transformations and Projections in Computer Graphics - page 29

rotate 58 0

clockwise

(a) (b)

rotate 32 0
counterclockwise

scale by
1.618 and 0.618

(c)

(d)

Figure 1.17: Shearing Decomposed into Rotation and Scaling.

Geometry has two great treasures: one the Theorem of Pythagoras; the other,
the division of a line into extreme and mean ratio. The first we may compare
to a measure of gold; the second we may name a precious jewel.
—Johannes Kepler

Exercise 1.33: Given the transformation

x ∗ =3 x ∗ =4 x +5 y − 2 y +1 , − 6 ,

calculate the transformation matrix and decompose it into a product of four matrices
as shown in Equation (1.21).

1.2.10 Reconstructing Transformations
Given a sequence of two-dimensional transformations, we normally write the 3 × 3 matrix

for each and then multiply the matrices. The result is another 3 3 matrix which is used

to transform all the points of an object. An interesting question is: Given the points
of an object before and after a transformation, can we reconstruct the transformation
matrix from them?
The answer is yes! The general two-dimensional transformation matrix depends
on six numbers, so all we need are six equations involving transformed points. Since

×