Game Physics Engine Development, 2nd Edition

198 Chapter 9 The M athematics of Rotations

Object

Object

Object

Object

Rotation

Origin

Origin

Origin

Origin

A

B

FIGURE 9.6 A matrix basis is changed.

The solution is to use M

b

and M

−1

b

in a three-stage process:

1. We perform the transformation M

−1

b

,whichtakesusfromB

2

back into B

1

.

2. We then perform the original transform M

t

, since we are now in the basis B

1

,

where it was originally correc t.

3. We then need to get back into basis B

2

, so we apply transformation M

b

.

Soweendupwith

M



t

=M

b

M

t

M

−1

b

bearing in mind that multiplied matrices are equivalent to transformations carried

out in right-to-left order.

We will need to use this function whenever we have a matrix exp