Graphics Reference
In-Depth Information
The linear interpolation of the angle is controlled by the parameter , where
=
and
=
1
−
so that
ˆ
=
ˆ
+
ˆ
w
a
u
b
v
From Fig. 2.38 we get
=
−
and
sin
=
sin
Therefore, using the sine rule:
OW
sin
=
1
sin
=
a
sin
=
b
sin
sin
sin
=
sin1
sin
−
a
=
and
sin
sin
=
sin
sin
b
=
Finally,
sin1
sin
−
sin
sin
ˆ
w
ˆ
=
u
ˆ
+
v
(2.28)
Equation (2.28) is called a
slerp
, because in its 3D application the unit vectors are spherically
interpolated across a unit-radius sphere. This interpolant can also be used for unit quaternions.
Let's test Eq. (2.28) with a simple example.
Given
90
.
Now let's find the interpolated vector when
u
ˆ
=
i
and
v
ˆ
=
j
, then
=
=
05.
Y
v
w
45
°
45
°
u
X
Figure 2.39.