Graphics Reference
In-Depth Information
Linearly interpolated intermediate points
Projection of intermediate points onto circle
Equal intervals
Unequal intervals
FIGURE 3.23
Equally spaced linear interpolations of straight-line path between two points on a circle generate unequal spacing
of points after projecting onto a circle. Interpolating quaternion four-tuples exhibit the same problem.
quaternion vectors. This can be determined by using the four-dimensional dot product of the quater-
nions to compute the cosine of the angle between
q
1
and
q
2
(
Eq. 3.28
)
. If the cosine is positive, then the
path from
q
1
to
q
2
is shorter; otherwise the path from
q
1
to
cos
ðyÞ¼q
1
q
2
¼ s
1
s
2
þ v
1
v
2
(3.28)
The formula for spherical linear interpolation (
slerp
) between unit quaternions
q
1
and
q
2
with
cos(
y
). Notice that this does
q
2
θ
q
1
Center of
sphere
180
θ
q
2
FIGURE 3.24
The closer of the two representations of orientation, q
2
, is the better choice to use in interpolation. In this case
q
2
is closer to
q
2
, as measured from the center of the sphere, is computed by
taking the dot product of q
1
and q
2
. The same is done for q
2
.
q
1
than
q
2
. The angle between
q
1
and
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