42 COMPUTER AIDED ENGINEERING DESIGN
For reflection about a generic plane Π having the unit normal vector as n = [nxnynz 0] and for
A [pqr 1] as any known point on it, the modus operandi is similar to the rotation about an arbitrary
axis discussed in section 2.5.1. The steps followed are
(a) Translate Π to the new position Π′ such that point A coincides with the origin usingTA=100–
010–
001 –
000 1p
q
r⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥(b) Rotate the unit vector n (passing through the origin) on Π′ to coincide with the z-axis. The new
position of Π′ will be Π′′ and the reflecting plane will coincide with the x-y plane (z = 0). We
would need the following transformations to acccomplish this step:RRxz
xy
x
y
xz
xyxxxxn
nn
n
n
nn
nnnnn=10 0 0
0
1 –- 1 –
00
1 – 1 –000 0 1, =1 – 0 – 0
0100
0 1 – 0
0001222222⎡⎣⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎤⎦⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎡⎣⎢
⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥
⎥Rfxy=1000
01 0 0
00–10
00 0 1⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥(c) After reflection, the reverse order transformations need to be performed. The complete
transformation would beRTR = A–1 –1xy( )ψφR–1(– )Rfxy Ry(−φ)Rx(ψ)TA (2.33)Example 2.7. The corners of wedge-shaped block are A(0, 0, 2), B(0, 0, 3), C(0, 2, 3), D(0, 2, 2),
E(−1, 2, 2) and F(−1, 2, 3), and the reflection plane passes through the y-axis at 45° between (−x) and
z-axis. Determine the reflection of the wedge.
First, no translation of the reflecting plane is required as it passes through the origin. The direction
cosines of the plane are (0.707, 0, 0.707). We may apply Eq. (2.33) directly to get the result.
Alternatively, rotate the plane about the y-axis for the reflecting plane to coincide with the y-z plane.
Perform reflection about the y-z plane and thereafter, rotate the plane back to its original location.
Ry(– 225 ) =cos (45 ) 0 sin (45 ) 0
0100
–sin (45 ) 0 cos (45 ) 0
0001=0.707 0 0.707 0
0100- 0.707 0 0.707 0
0001
°°°°°⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥