Computer Aided Engineering Design

36 COMPUTER AIDED ENGINEERING DESIGN

0 5 10 15

16

14

12

10

8

6

4

2

Figure 2.12 Shear along the y direction

2.4 Generic Transformation in Two-Dimensions

Observing the transformation matrices developed previously for translation, rotation, reflection,
scaling and shear, we may realize that the matrices may be expressed generically in the partitioned
form as

A =

|

|

————

|

11 12 13

21 22 23

31 32 33

aa a

aa a

aa a

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

(2.22)

The top left 2 × 2 sub-matrix represents: (a) rotation when the elements are the sine and cosine
terms of the rotation angle about the z-axis, (b) reflection when the diagonal elements are +1 or –
1, and the off diagonal terms are zero, (c) scaling when the diagonal elements are positive μx and
μy with the off diagonal terms as zero and (d) shear when the off diagonal elements are non-zero
and diagonal elements are 1. The second top-right partition of 2 × 1 sub-matrix represents translation.
The bottom-left partition of 1 × 2 sub-matrix represents perspective transformation discussed later
and the bottom right matrix, the diagonal element a 33 = 1 represents the homogeneous coordinate
scalar. Like a point in the x-y plane is represented as (x,y, 1) using the homogenous system, in a
three-dimensional space, the representation can be extended to (x,y,z, 1). Accordingly, the matrix
A in Eq. (2.22) gets modified to

A =

|

a|

|

———|—

|

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

aaa a

aaa

aaa a

aaa a

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

(2.23)

The partitions now consist of 3 × 3, 3 × 1, 1 × 3, and 1 × 1 sub-matrices having the same role as
discussed for the respective partitions above for a two-dimensional case.