Practical Algorithms for 3D Computer Graphics, 2nd Edition

2.18. Angular interpolation (quaternions) 61

Quaternions have their own algebra with rules for addition and multiplication.

Addition is straightforward: add the scalar components and the vector compo-

nents. Multiplication is more interesting. Given two quaternions q

1

and q

2

the

product is the quaternion:

q

1

q

2

= (w, v) = (w

1

w

2

− v

1

· v

2

, w

1

v

2

+ w

2

v

1

+ v

1

× v

2

)

A conjugate quaternion to q is defined as q = (w, −v). Thus, the squared

magnitude of q is:

qq = |q|

2

= w

2

+ |v|

2

Note that if q is of unit magnitude its inverse q

−1

equals its conjugate, q

−1

= q

and qq = 1.

2.18.2 Quaternions and r ot a t ions

We have seen in section 2.12.3 that the action of rotating a vector ...