TRANSFORMATIONS AND PROJECTIONS 53
=
2 1 0 1.2
2 2 0 1.2
1 2 0 1.2
1 1 0 1.2
2 1 0 1.1
2 2 0 1.1
1 2 0 1.1
1 1 0 1.1
1.67 .833 0 1
1.67 1.67 0 1
.833 1.67 0 1
.833 .833 0 1
1.82 .91 0 1
1.82 1.82 0 1
.91 1.82 0 1
.91 .91 0 1
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
≡
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥
⎥
⎥
⎥
T
⎥⎥
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
T
We observe all twelve edges of the cube in its perspective projection in Figure 2.23.
2.7.2 Two Point Perspective Projection
Example 2.10 suggests that translating the object may show up its multiple faces giving a three-
dimensional effect on the plane of projection. Rotating an object about an axis also reveals two or
more faces. A rotation about z-axis by an angle θ followed by a single point perspective projection
ony = 0 plane with center of projection at y = yp gives the following transformation matrix:
M 1 =
1000
0000
0010
0–^1
cos –sin 0 0
sin cos
=
cos –sin 0 0
0000
0010
- sin –cos 01
yp yypp
01
00
0010
0001
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
θθ
θθ
θθ
θθ
(2.35a)
A rotation about the x-axis by an angle ψ followed by a single point perspective projection on y = 0
plane with center of projection at y = yp gives another transformation matrix M 2 , where
M 2 =
1000
0000
0010
0–^1
10 0 0
0 cos –sin
sin cos
=
10 00
00 00
0 sin cos 0
0–
cos sin
y^1
p yypp
01
0
00
00 0 1
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥
⎥
⎥
ψψ
ψψ ψψ
ψψ
⎥⎥
⎥
⎥
(2.35b)
Example 2.11. Given a square planar sheet ABCD in the x-y plane with A (1, 0, 0), B (1, 1, 0),
C (0, 1, 0) and D (0, 0, 0), find the perspective image of the sheet on y = 0 plane, with the view point
atyp = 2. The sheet is rotated 60° about the z-axis and translated –2 units along z-axis.
The transformation matrix M is given by Pers(yp = 2) T(z = –2) Rz(60°), that is
1000
0000
0010
0–.5
100 0
010 0
001–2
000 1
cos 60 –sin 60 0 0
sin 60 cos 60 0 0
0010
0001
=
0.5 – 0.866 0 0
0000
0
01
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
°°
°°
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
0 01–2
- 0.433 – 0.25 0 1
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥