Graphics Reference
In-Depth Information
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
−
1
y
22
+cos
30
◦
sin
30
◦
4
−
232
445
111
44
+sin
30
◦
4
+cos
30
◦
.
(10.77)
1
1
1
Both approaches are reasonably easy to work with.
There's a third approach—a variation of the second—in which we specify
where we want to send a point and two vectors, rather than three points. In this
case, we might say that we want the point
P
to remain fixed, and the vectors
e
1
and
e
2
to go to
x
⎡
⎤
⎡
⎤
y
cos
30
◦
sin
30
◦
0
sin
30
◦
cos
30
◦
0
−
⎣
⎦
and
⎣
⎦
,
(10.78)
respectively. In this case, instead of finding matrices that send the vectors
e
1
,
e
2
,
and
e
3
to the desired three points, before and after, we find matrices that send those
vectors to the desired point and two vectors, before and after. These matrices are
⎡
x
⎤
⎡
⎤
2
cos
30
◦
−
sin
30
◦
210
401
100
⎣
⎦
and
⎣
⎦
,
4
sin
30
◦
cos
30
◦
(10.79)
y
1
0
0
so the overall matrix is
⎡
⎤
⎡
⎤
−
1
2
cos
30
◦
−
sin
30
◦
210
401
100
x
⎣
⎦
⎣
⎦
4
sin
30
◦
cos
30
◦
.
(10.80)
1
0
0
These general techniques can be applied to create any linear-plus-translation
transformation of the
w
=
1 plane, but there are some specific ones that are good
to know. Rotation in the
xy
-plane, by an amount
Figure 10.15: The house after
translating
(
2, 4
)
to the origin,
after rotating by
30
◦
, and after
translating
θ
(rotating the positive
x
-axis
toward the positive
y
-axis) is given by
⎡
⎤
the
origin
back
to
cos
θ −
sin
θ
0
(
2, 4
)
.
⎣
⎦
.
R
xy
(
θ
)=
sin
θ
cos
θ
0
(10.81)
0
0
1
In some topics and software packages, this is called
rotation around
z
;
we prefer
the term “rotation in the
xy
-plane” because it also indicates the direction of rotation
(from
x
,toward
y
). The other two standard rotations are
⎡
⎣
⎤
1
0
0
⎦
0
cos
θ −
sin
θ
R
yz
(
θ
)=
(10.82)
0
sin
θ
cos
θ
and
⎡
⎤
cos
θ
θ
01 0
0
sin
⎣
⎦
;
R
zx
(
θ
)=
(10.83)
−
sin
θ
0
cos
θ
note that the last expression rotates
z
toward
x
, and
not
the opposite. Using this
naming convention helps keep the pattern of plusses and minuses symmetric.
