TRANSFORMATIONS AND PROJECTIONS 61For a parallel ray passing through any general point P(x,y,z), we can obtain the image Q(qx,qy,0)
on the x-y plane as (x + fz cos ψ,y + fz sin ψ, 0). Or
q
qf
fx
y
zx
y
0
1=1 0 cos 0
0 1 sin 0
00 0 0
00 0 11⎡⎣⎢
⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥
⎥ψ
ψ
(2.39)This is incorporated by shifting C (0, 0, z) by (x,y, 0) to P (x,y,z) on a plane parallel to the x-y plane
and at a distance z above it. The consequence is the corresponding shift in the image from D(dx,dy,0)
toQ(qx,qy, 0) is computed above. The oblique projections are cavalier for f = 1 and cabinet for f =^12.
Example 2.13. For the block shown in Example 2.12,
(a) Draw its cabinet projections on x-y plane (for f =^12 ) for ψ = 0°,ψ = 15°,ψ = 30° and ψ = 45°
(b) Draw its cavalier projections on x-y plane (for f = 1) for ψ = 0°,ψ = 15°,ψ = 30° and ψ = 45°
(c) Draw projections for ψ = 45° when f = 1, f = 3/4, and f = 1/2.
Part (a) is shown in Figure 2.31.
Figure 2.31 Cabinet projections–3 –2 –1 0 1 2 3 4 5 6 74
3.5
3
2.5
2
1.5
1
0.5
0
–0.5
–1
ψ = 0°–3 –2 –1 0 1 2 3 4 5 63
2.5
2
1.5
1
0.5
0
–0.5
ψ = 15°–2–101 23 4563
2.5
2
1.5
1
0.5
0
–0.5
–1
–1.5
ψ = 45°–3 –2 –1 0 1 2 3 4 5 63
2.5
2
1.5
1
0.5
0
–0.5
–1
ψ = 30°