Game Development Reference

In-Depth Information

2

2

12

−−

y

2

z

2

xy

−

2

wz

2

xz

+

2

wy

2

2

R

=

22

xy

+

z

122

−

x z

−

22

z

−

x

.

(4.60)

q

2

2

22

xz

−

y

22

z

+

x

12 2

−

x y

−

4.6.3 Spherical Linear Interpolation

Because quaternions are represented by vectors, they are well suited for interpo-

lation. When an object is being animated, interpolation is useful for generating

intermediate orientations that fall between precalculated key frames.

The simplest type of interpolation is
linear interpolation
. For two unit qua-

ternions
q
and

()

q
, the linearly interpolated quaternion

q

t

is given by

() (

)

q

t

=−

1

t

qq
.

+

t

(4.61)

1

2

()

The function

q
changes smoothly along the line segment connecting
q
and
q

as
t
varies from 0 to 1. As shown in Figure 4.7,

()

q

t

does not maintain the unit

length of
q
and

q
, but we can renormalize at each point by instead using the

function

(

)

1

1

−+

t

qq

t

()

1

2

q

t

=

qq
.

(4.62)

(

)

−+

t

t

1

2

q

1

()

q

t

()

()

q

q

t

t

q

2

Figure 4.7.
Linear interpolation of quaternions.

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