Computer Aided Engineering Design

62 COMPUTER AIDED ENGINEERING DESIGN

Figure 2.32 Cavalier projections

–6–4–20246

3

2.5

2

1.5

1

0.5

0

ψ = 0°

–6 –4 –2 0 2 4 6

3

2.5

2

1.5

1

0.5

0

• 0.5
  –1
  ψ = 15°

–4–3–2–10123456

3

2

1

0

–1

–2

–3

ψ = 45°

–4–2024 6

5 4 3 2 1 0

–1

–2

ψ = 30°

Exercises

1. For the points p 1 (1, 1), p 2 (3, 1), p 3 (4, 2), p 4 (2, 3), that defines a 2-D polygon, develop a single transformation
   matrix that
   (a) reflects about the line x = 0,
   (b) translates by –1 in both x and y directions, and
   (c) rotates about the z-axis by 180°
   Using the transformations, determine the new position vectors.


2. Develop an algorithm to find a set of vertices making a regular 2-D polygon. You may use only transformations
   on points. Input parameters are the starting point p 0 (0, 0), number of edges n, and length of edge l.


3. Prove that the transformation matrix

R =

1 –

1 +

2

1 +

0

–2

1 +

1 –

1 +

0

001

2

22

2

2

2

t

t

t

t

t

t

t

t

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

produces pure rotation. Find the equivalent rotation angle.

Parts (b) and (c) are shown in Figures 2.32 and 2.33, respectively.