Graphics Reference
In-Depth Information
for some real numbers r and s. Taking the dot product of both sides of this equation
with
a
and
b
leads to equations
q=
()
∑=+ ∑
cos t
ca
t
r
s
ba
(4.29)
and
(
)
()
∑= ∑+
cos
1 -
t
q
=
cbab
t
r
s
.
(4.30)
Solving equations (4.29) and (4.30) for r and s and using some standard trigonomet-
ric identities leads to the stated result.
4.14.2 Example.
Let
L
0
and
L
1
be the positively directed x- and y-axis, respectively.
Let R
0
and R
1
be the rotations about the directed lines
L
0
and
L
1
, respectively, through
angle p/3 and let M
0
and M
1
be the matrices that represent them. If M
t
, t Π[0,1], is
the 1-parameter family of in-betweening matrices in
SO
(3) between M
0
and M
1
, then
what is M
1/2
?
Solution.
The unit direction vectors for
L
0
and
L
1
are
n
0
= (1,0,0) and
n
1
= (0,1,0),
respectively. Therefore, by equation (4.25)
p
p
3
2
1
2
(
)
q
=
cos
+
sin
n
=
+
100
, ,
0
0
6
6
and
p
p
3
2
1
2
(
)
q
=
cos
+
sin
n
=
+
100
, ,
1
1
6
6
are the unit quaternions corresponding to rotation R
0
and R
1
. The angle q between
q
0
and
q
1
is defined by
3
4
cos
q=
q
01
∑
=
.
It follows that
7
4
q
=
-
1
cos
q
1
22
2
sin
q
=-
1
cos
q
=
and
sin
=
.
2
2
Using equation (4.28), let
2
14
3
14
1
14
1
14
02
3
14
1
7
1
2
1
2
+
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
q
=
(
q
+
q
)
=
2
,
,
=
+
,
,
0
.
12
0
1
Finally, equation (4.27) implies that
