Computer Aided Engineering Design

34 COMPUTER AIDED ENGINEERING DESIGN

(aaa 112 + + ) = 1^221231 (a 11 a 13 + a 21 a 23 + a 31 a 33 ) = 0

(aaa 122 + + ) = 1^222232 (a 13 a 12 + a 23 a 22 + a 33 a 32 ) = 0

(a 11 a 12 + a 21 a 22 + a 31 a 32 ) = 0 (a 11 a 12 + a 21 a 22 + a 31 a 32 ) = 0

(a 11 a 22 – a 12 a 21 ) = 1

which suggests that A must be orthogonal having the property A–1 = AT so that AAT = ATA = I, where
I is the identity matrix of the same size as A. Further, (a 11 a 22 – a 12 a 21 ) = 1 implies that the determinant
ofAshould be 1. In pure rotation, the above conditions are completely met where for R in Eq. (2.4),
a 13 =a 23 =a 31 =a 32 = 0, and a 33 = 1. In reflection, the determinant of the transformation matrix is –
1; hence, although the matrix is orthogonal, the angle is not preserved and that it changes to (2π−θ)
though the absolute angle between the adjacent sides of the polygon remains θ. The magnitudes of
the vectors are preserved. The angle between the intersecting vectors is also preserved in case of
translation, that is

v

v

v

v

vv

vv

vv

vv

1

*

2

*

1

2

11

22

11

22

=

10

01

001

=

10

01

001

0

0

=

0

0

=

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

T T

xy

xy

T

xy

xy

p T

q

p

q

1

2

v

v

⎡

⎣

⎢

⎤

⎦

⎥

T

which implies that the translation does not alter vectors.

2.3 Deformations

Previous sections dealt with transformations wherein the object was relocated and/or reoriented
without the change in its shape or size. In this section, one would deal with transformations that
would alter the size and/or shape of the object. Examples involve those of scaling and shear.

2.3.1 Scaling

A point P (x,y, 1) belonging to the object S can be scaled to a new position vector P (x,y*, 1) using
factorsμx and μy such that

x* = μxx and y* = μyy

Or in matrix form

x

y

x

y

x

y

*

*

1

=

00

00

0011

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

μ

μ = SP (2.18)

whereS =

μ

μ

x

y

00

00

001

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

is the scaling matrix. Scale factors μx and μy are always non-zero and

positive. For both μx and μy less than 1, the geometric model gets shrunk. In case of uniform scaling
whenμx = μy = μ, the model gets changed uniformly in size (Figure 2.11) and there is no distortion.