TRANSFORMATIONS AND PROJECTIONS 41Eq. (2.30) uses the equivalence [xyzs]T≡ ⎡
⎣⎢⎤
⎦⎥1x
sy
sz
sT
since both vectors represent the samepoint 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
000112 13
21 23
31 32sh sh
sh sh
sh sh⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥(2.31)whose effect on point P is
x
y
zsh sh
sh sh
sh shx
y
zx sh y sh z
shxyshz
sh x sh y z*
*
*
1=10
10
10
00011=+ +
+ +
+ +
112 13
21 23
31 3212 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 Rfzx1000
01 0 0
00–10
00 0 1, =1000
0–100
0010
0001and =–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 sizeZ
X Y(a) (b) (c)