TRANSFORMATIONS AND PROJECTIONS 63
- Show that the reflection about an arbitrary line ax + by + c = 0 is given by
ba ab
ab a b
ac bc
ab
22
22
22
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
- Consider two lines L1: y = c and L2: y = mx + c which intersect at point C on y-axis. The angle θ between
these lines can be found easily. A point P (x 1 ,y 1 ) is first reflected through L1 and subsequently through L2.
Show that this is equivalent to rotating the point P about the intersection point C by 2θ.
- A point P (x,y) has been transformed to P(x,y*) by a transformation M. Find the matrix M.
- MatrixM =
10
10
001
b
c
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
shears an object by factors c and b along the Ox and Oy axes respectively.
Determine the matrix that shears the object by the same factors, but along Ox 1 and Oy 1 axes inclined at an
angleθ to the original axes.
Figure 2.33 Oblique projections for ψψψψψ = 45° and shown shrink factor f
3
2
1
0
–1
–2
–3
f = 1
3
2.5
2
1.5
1
0.5
0
–0.5
–1
–1.5
–2
3
2.5
2
1.5
1
0.5
0
–0.5
–1
–1.5
–3 –2 –1 0 1 2 3 4 5 6
f =^3 / 4
–2 –1 0 1 2 3 4 5 6
f =^1 / 2
