CS480/CS680 Problem Set 1
Due in class Tuesday, February 12 at the beginning of lecture.
Please prepare the answers to these questions, neatly written or typed, on separate paper.
1. (a) (3 points) Write a
4
×
4
homogeneous transform matrix
M
that when applied to a point
(
x,y,z,
1)
yields
(
x
0
,y
0
,z
0
,w
0
)
where
x
0
=

1
√
2
x

1
√
2
z
+
a
y
0
=

2
y
z
0
=
1
√
2
z

1
√
2
x
+
a
w
0
=
1
(b) (12 points)In words
, what four basic computer graphics transforms occur when we apply
M
to a 3D point? Give a homogeneous transform matrix for each, and show the order in
which they are multiplied.
2. (15 points) We have a unit cube centered at the point
c
= (1
.
5
,
0
.
5
,
2
.
5)
. Derive the ho
mogeneous transformation matrix that will rotate the cube by angle
θ
around a vector in the
direction
v
= (1,0,1). The pivot point for the rotation is the cube’s center
c
.
3. Use quaternions in your answers to the following.
•
(5 points) Prove that in general two 3D rotations about different rotation axes do not
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 Spring '08
 Sclaroff
 Computer Graphics, Euclidean geometry, Hearn

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