Graphics Programs Reference
In-Depth Information
Exercise 1.17:
Given the two parallel lines
y
=0and
y
=
c
, calculate the double
reflection of a point (
x, y
)aboutthem.
Exercise 1.18:
Consider the shearing transformation
T
a
of Equation (1.15), followed
by the 90
◦
rotation
T
b
. What is the combined transformation, and what kind of trans-
formation is it?
⎛
⎞
⎛
⎞
cos 90
◦
sin 90
◦
010
200
001
−
0
⎝
⎠
,
⎝
⎠
.
sin 90
◦
cos 90
◦
T
a
=
T
b
=
0
(1.15)
0
0
1
Exercise 1.19:
Given the two rotations
⎛
⎞
⎛
⎞
cos
θ
1
−
sin
θ
1
0
cos
θ
2
−
sin
θ
2
0
⎝
⎠
⎝
⎠
,
T
1
=
sin
θ
1
cos
θ
1
0
and
T
2
=
sin
θ
2
cos
θ
2
0
0
0
1
0
0
1
calculate the combined transformation
T
1
T
2
.
Is it identical to a rotation through
(
θ
1
+
θ
2
)?
Exercise 1.20:
Given the two shearing transformations
⎛
⎞
⎛
⎞
1
b
0
010
001
100
c
10
001
⎝
⎠
⎝
⎠
,
T
1
=
and
T
2
=
calculate the combined transformation
T
1
T
2
. Is it identical to a shearing by factors
b
and
c
?
Exercise 1.21:
Prove that three successive shearings about the
x
,
y
,and
x
axes is
equivalent to a rotation about the origin.
Exercise 1.22:
Matrix
a
0
0
d
scales an object by factors
a
and
d
along the
x
and
y
axes,
respectively. If we want to scale the object by the same factors, but in the
i
and
j
directions (see Figure 1.8, where
i
and
j
are perpendicular and form an angle
θ
with
the
x
and
y
axes, respectively), we need to (1) rotate the object
θ
degrees clockwise, (2)
scale along the
x
and
y
axes using matrix
a
0
0
d
, and (3) rotate back. Write the three
transformation matrices and their product. Discuss the case
a
=
d
(uniform scaling).
Exercise 1.23:
We can perform an exercise with shearing, similar to Exercise 1.22.
Matrix
1
b
c
1
shears an object by factors
c
and
b
along the
x
and
y
axes, respectively.
Calculate the matrix that shears the object by the same factors, but in the
i
and
j
directions (Figure 1.8).
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