Game Development Reference
In-Depth Information
()
Figure 4.9(b) shows the same procedure used to find the length
b t , which is
given by
sin
sin
θt
()
b t
=
.
(4.66)
θ
()
We can now define the spherical linear interpolation function
q
t
as
follows.
(
)
sin
θ t
1
sin
θt
q
()
t
=
q
+
q
(4.67)
1
2
sin
θ
sin
θ
The angle θ is given by
1
(
qq ,
)
(4.68)
θ
=
cos
1
2
and thus, sin θ can be replaced by
(
)
2
sin
θ
=− ⋅
1
qq
(4.69)
1
2
if desired. Since the quaternions q and
q represent the same rotation, the signs
of the quaternions q and
q are usually chosen such that 1
q
q
0
. This also
2
ensures that the interpolation takes place over the shortest path.
Chapter 4 Summary
Orthogonal Matrices
An invertible nn
1
T
M M . A ma-
trix whose columns form an orthonormal set of vectors is orthogonal. Orthogonal
matrices preserve lengths and angles, and thus perform only rotations and
reflections.
×
matrix M is called orthogonal if and only if
=
Scaling Transforms
A scaling operation in three dimensions is performed using the transformation
matrix
a
00
0
b
0 .
00
c