TRANSFORMATIONS AND PROJECTIONS 41
Eq. (2.30) uses the equivalence [xyzs]T≡ ⎡
⎣⎢
⎤
⎦⎥
1
x
s
y
s
z
s
T
since both vectors represent the same
point in the four-dimensional homogeneous coordinate system.
2.5.3 Shear in Three-Dimensions
In the 3 × 3 sub-matrix of the general transformation matrix (2.23), if all diagonal elements including
a 44 are 1, and the elements of 1 × 3 row sub-matrix and 3 × 1 column sub-matrix are all zero, we get
the shear transformation matrix in three-dimensions, similar to the two-dimensional case. The generic
form is
Sh =
10
10
10
0001
12 13
21 23
31 32
sh sh
sh sh
sh sh
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
(2.31)
whose effect on point P is
x
y
z
sh sh
sh sh
sh sh
x
y
z
x sh y sh z
shxyshz
sh x sh y z
*
*
*
1
=
10
10
10
00011
=
+ +
+ +
+ +
1
12 13
21 23
31 32
12 13
21 23
31 32
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
Thus, to shear an object only along the y direction, the entries sh 12 = sh 13 = sh 31 = sh 32 would be 0
while either sh 21 and sh 23 or both would be non-zero.
2.5.4 Reflection in Three-Dimensions
Generic reflections about the x-y plane (z becomes –z),y-z plane (x becomes – x), and z-x plane (y
becomes –y) can be expressed using the following respective transformations:
Rfxy= Rfyz Rfzx
1000
01 0 0
00–10
00 0 1
, =
1000
0–100
0010
0001
and =
–1000
0100
0010
0001
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
(2.32)
Figure 2.17 A scaled cylinder using different factors: (a) original size, (b) twice the original size,
(c) half the original size
Z
X Y
(a) (b) (c)