TRANSFORMATIONS AND PROJECTIONS 352.3.2 Shear
Consider a matrix Shx =
10
010
001⎡ shx⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥which when applied to a point P (x,y, 1) results inx*
y*sh x
yxshy
yxx1=10
010
0011=+1⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥(2.20)which in effect shears the point along the x axis. Likewise, application of Shy =
100
10
001shy⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥onP yields
x*
y* shx
yx
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
PT T T
1
*2
*3
*4
*=100
1.5 1 0
001311
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 ̇( ) jHence
dy
dxdy du
dx duyu
xuy
x=(/)
(/) =()
()μ
μ̇
̇ (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
scalingxUniform
scalingy