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

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