Graphics Reference
In-Depth Information
Section 2.4
2.4.1.
Prove that the inverse of the shear
xx
y xy
¢ =
¢ =+
is a shear and find its equations.
2.4.2.
Find the affine map that sends
A
(1,0),
B
(0,3), and
C
(4,2) to
A
¢(-1,2),
B
¢(0,1), and
C
¢(1,3),
respectively.
Section 2.4.1
2.4.1.1.
Let
X
be the plane defined by x - 2y + z = 1.
Define the orthogonal projection of
R
3
onto
X
.
(a)
Define the parallel projection of
R
3
onto
X
parallel to
v
= (1,0,2).
(b)
Section 2.5
2.5.1.
Fill in the missing details in the proof of Theorem 2.5.1.
2.5.2.
Using the definition, find the equation of the reflection S about the plane x - 2y +
2z = 1.
Section 2.5.1
2.5.1.1.
Show that the following motion is a rotation and find its axis and angle of rotation:
1
6
2
6
1
6
1
6
1
3
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
x
¢ =
x
+
+
y
+-
+
z
2
6
1
6
1
6
1
6
1
3
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
y
¢ =-
+
x
+
y
+
+
z
1
6
1
3
1
6
1
3
2
3
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
z
¢ =
+
x
+-
+
y
+
z
2.5.1.2.
Using translations and rotations about the coordinate axes, find the equation of
a rigid motion that sends the plane
X
defined by 2x - 3y + 2z = 1 to the x-y
plane.
2.5.1.3.
Given a unit cube with one corner at (0,0,0) and the opposite corner at (1,1,1), derive
the transformations necessary to rotate the cube by q degrees about the main diago-
nal (from (0,0,0) to (1,1,1)) in the counterclockwise direction when looking along the
diagonal toward the origin. Use rotations about the coordinate axes.
