TRANSFORMATIONS AND PROJECTIONS 35
2.3.2 Shear
Consider a matrix Shx =
10
010
001
⎡ shx
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
which when applied to a point P (x,y, 1) results in
x*
y*
sh x
y
xshy
y
xx
1
=
10
010
0011
=
+
1
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
(2.20)
which in effect shears the point along the x axis. Likewise, application of Shy =
100
10
001
shy
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
on
P yields
x*
y* sh
x
y
x
yysh x y
1
=
100
10
0011
= +
1
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
(2.21)
that is, the new point gets sheared along the y direction.
Example 2.6.For a rectangle with coordinates (3, 1), (3, 4), (8, 4) and (8, 1), respectively, applying
shear along the y direction (Figure 2.12) with a factor shy = 1.5 yields the new points as
P
P
P
P
T T T
1
*
2
*
3
*
4
*
=
100
1.5 1 0
001
311
341
841
811
=
3 5.5 1
3 8.5 1
8161
8131
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
Consider a curve, for instance, defined by r(u) = x(u)i + y(u)j, where parameter u varies in the
interval [0, 1]. The curve after scaling becomes r(u) = x(u)i + y(u)j=μxx(u)i+μyy(u)j and the
tangent to any point on this curve is obtained by
differentiating r(u) with respect to u, that is,
̇r*( ) = uxu yuμμxy ̇( ) + i ̇( ) j
Hence
dy
dx
dy du
dx du
yu
xu
y
x
=
(/)
(/) =
()
()
μ
μ
̇
̇ (2.19)
Thus, non-uniform scaling changes the tangent
vector proportionally while the slope remains
unaltered in uniform scaling for μx = μy. Figure 2.11 Uniform and non-uniform scaling
Non-uniform
scaling
x
Uniform
scaling
y