Computer Aided Engineering Design

54 COMPUTER AIDED ENGINEERING DESIGN

The vertices of the square ABCD are now transformed to get the perspective image ABCD in
Figure 2.24.

A*

B*

C*

D*

A

B

C

D

T T

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥

⎥

= =

0.5 – 0.866 0 0

0000

001–2

• 0.433 – 0.25 0 1

1001

1101

0101

0001

M ⎥⎥

⎥

⎥

⎥

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

≡

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

T

T

=

0.5 0 –2 0.567

• 0.366 0 – 2 0.317


• 0.866 0 – 2 0.750

00–21

0.882 0 –3.527 1

–1.115 0 –6.309 1

–1.155 0 –2.667 1

0 0 –2 1

Figure 2.24 Perspective image on the x-z plane for Example 2.11

Perspective

view

Original object

Object rotated

and translated

E

1

y x

0 –2

–1

0

1

0

–2

z–4

–6

–8

2

C B

D

A

D* A*

C*

B*

2.8 Orthographic Projections

Orthographic projections have been universally adopted for engineering drawings, especially for
machine parts. They are simplest among parallel projections and are popular in all manufacturing
industries because they accurately depict the true size and shape of a planar-faced object. In an
orthographic projection, the projectors are perpendicular to the view plane. Multi-view projections
are a set of orthographic images, usually on the coordinate planes, generated with direction of
projections perpendicular to different faces of the object. The following transformation matrices
obtain parallel projections on the x-y,y-z and z-x planes.

Prxy= Pryz Przx

1000

0100

0000

0001

, =

0000

0100

0010

0001

, =

1000

0000

0010

0001

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

(2.36)