1
CS250 Review Questions
Notes:
a.
Unless otherwise specified, assume two-dimensional points and vectors are described
in right-handed reference frame
[ ]
d
d
W
d
W
d
j
i
O
f
2
2
2
2
ˆ
,
ˆ
,
=
whose reference point is
( )
0
,
0
2
=
W
d
O
while vectors
( )
0
,
1
ˆ
2
=
d
i
and
( )
1
,
0
ˆ
2
=
d
j
represent an ortho-normal
basis. Basis vector
d
i
2
ˆ
represents
−
X
axis and is horizontally oriented; while
d
j
2
ˆ
represents
−
Y
axis and is vertically oriented.
b.
Unless
otherwise
specified,
assume
three-dimensional
points
and
vectors
are
described in right-handed reference frame
[ ]
k
j
i
O
f
W
W
ˆ
,
ˆ
,
ˆ
,
=
whose reference
point
is
( )
0
,
0
,
0
=
W
O
while
vectors
( )
0
,
0
,
1
ˆ
=
i
,
( )
0
,
1
,
0
ˆ
=
j
,
and
( )
1
,
0
,
0
ˆ
=
k
represent an ortho-normal basis. Basis vector
i
ˆ
represents
−
X
axis
and is oriented right;
j
ˆ
represents
−
Y
axis and is oriented into the paper; while
k
ˆ
represents
−
Z
axis and is oriented up.
c.
v
r
represents a vector with arbitrary length while
v
ˆ
represents a unit-length vector.
d.
As in class, points and vectors are represented using column matrices.
1)
In
W
d
f
2
, compute the
3
3
×
matrix that aligns
d
i
2
ˆ
to
d
j
2
ˆ
.
2)
In
W
d
f
2
, compute the
3
3
×
matrix that aligns
d
i
2
ˆ
along vector
( )
1
,
1
.
3)
In
W
d
f
2
, find the
3
3
×
matrix that aligns
( )
y
x
d
v
v
v
,
2
=
r
along
( )
y
x
d
w
w
w
,
2
=
r
.
4)
Reflection
transform generates the mirror image of an object. In two-dimensional
reflection, the mirror image is generated relative to an
axis of reflection
by rotating
a vector (or, an object)
o
180
about the reflection axis. In a two-dimensional
reference frame, the axis of reflection can be a vector in
−
Y
X
plane or a vector
orthogonal to
−
Y
X
plane. In
W
d
f
2
, find
3
3
×
matrices for the following common
reflections:
a.
Reflection about line
0
=
y
.
b.
Reflection about line
0
=
x
.
c.
Reflection about line perpendicular to
−
Y
X
plane and passing through origin
W
d
O
2
.
d.