TRANSFORMATIONS AND PROJECTIONS 45
can be determined aspp 12 ′× ′. The angle β between the planes PP P 123 ′′ ′ and Q 1 Q 2 Q 3 is now given
by cos β = p′ 3 · q 3. To orient the plane PP P 123 ′′ ′with respect to Q 1 Q 2 Q 3 at any desired angle θ, we
can rotate point P 3 ′ about PP 12 ′′ (or p 1 ′) through an angleθ−β as discussed in section 2.5.1.
Example 2.8.Given two triangular objects, S 1 {P 1 (0, 0, 1), P 2 (1, 0, 0), P 3 (0, 0, 0)} and S 2 {Q 1 (0,
0, 2), Q 2 (0, 2, 0), Q 3 (2, 0, 0)}, it is required that after assembly, point P 1 coincides with Q 1 and edge
P 1 P 2 lie on Q 1 Q 3. Determine the transformations if (i) S 1 is required to be in the same plane as S 2 and
(ii)S 1 is perpendicular to S 2.
Translation of P 1 P 2 P 3 to a new position PPP 1 * 2 * 3 * with P 1 to coincide with Q 1 is obtained by
P
P
P
P
P
P
T TT
1
*
2
*
3
*
1
2
3
= =
100 0
010 0
0 0 1 (2 – 1)
000 1
0011
1001
0001
=
0021
1011
0011
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
T ⎥
⎥⎥
⎥
T
It can be verified that P 2 lies on line Q 1 Q 3 and thus one does not need to perform step (b) above. It
is now required to determine the angle between the lamina PPP 1 2 3 and Q 1 Q 2 Q 3 which can be
obtained using step (c).
p
PP
PP
1
* 2
*
1
*
2
*
1
= * 22
- |– |
=
[(1 – 0), (0), (1 – 2), (1 – 1)]
1 + (– 1)
=
1
2
0 –^1
2
0
⎛
⎝
⎜
⎞
⎠
⎟
Similarly, p
PP
PP
p
PP PP
(^2) PP PP
*^3
- 1
3 1
3
^3
1 2 1
3 1 2 1 *
- |– |
= (0 0 –1 0), =
(– ) (– )
| ( – ) ( – ) |
= (0 –1 0 0)
×
×
q
QQ
(^2) QQ
31
31 22
|– |
[(2 – 0), (0), (0 – 2), (1 – 1)]
2 + (– 2)
=^1
2
0 –
1
2
0
⎛
⎝
⎜
⎞
⎠
⎟
qq
QQ QQ
(^13) QQ QQ
31 21
31 21
= 0^1
2
1
2
0 , =
( – ) ( – )
| ( – ) ( – ) |
=^1
3
1
3
1
3
0
⎛
⎝
⎜
⎞
⎠
⎟
×
×
⎛
⎝
⎜
⎞
⎠
⎟
Therefore,cos = =^1
3
1
3
1
3
0 (0 –1 0 0) = –^1
3
ββqp 33 ⋅ * = 125.26
⎛
⎝
⎜
⎞
⎠
⎟⋅⇒°
Angleβ (or, 180° – β) is the angle between the planes S 1 * and S 2 , and Q 1 Q 3 is the line about which
P 3 * is to be rotated to bring S 1 * to be either: (i) in plane with S 2 , or (ii) perpendicular to S 2.
The direction cosines of Q 1 Q 3 are given by
q 2 =
1
2
0 –
1
2
0 = ( 0)
⎛
⎝
⎜
⎞
⎠
⎟ nnnxyz
Following section 2.5.1, where rotation of a point about an arbitrary line is discussed, we shift Q 1 to
the origin, rotate line Q 1 Q 3 about the x-axis and then about y-axis
dn
n
d
n
x d
= 1 – =^1 z y
2
(^2) , = – 1, = 0