Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 443

B.3.4 Quaternions
Quaternions are discussed in Chapter 2.4 and Chapter 3.1 . The equations from those chapters, along
with additional equations, are collected here to facilitate the discussion.

Quaternion arithmetic
Quaternions are four-tuples and can be considered as a scalar combined with a vector ( Eq. B.4 3). Addi-
tion and multiplication are defined for quaternions by Equations B.44 and B.45 , respectively. Quater-
nion multiplication is associative ( Eq. B.46 ), but it is not commutative ( Eq. B.47 ). The magnitude of a
quaternion is computed as the square root of the sum of the squares of its four components ( Eq. B.48 ).
Quaternion multiplication has an identity ( Eq. B.49 ) and an inverse ( Eq. B.50 ) . The inverse distributes
over quaternion multiplication similarly to how the inverse distributes over matrix multiplication
( Eq. B.51 ). A quaternion is normalized by dividing it by its magnitude ( Eq. B.52 ) . Equation B.52a
is used to compute the quaternion that represents the rotation r such that q represents the half-way rota-
tion between p and r .

q ¼ s; x; y; z ½ ¼s; v ½ (B.43)

½ s 1 ; v 1 þs 2 ; v 2 ½ ¼s 1 þ s 2 ; v 1 þ v 2 ½ (B.44)

½ s 1 ; v 1 s 2 ; v 2 ½ ¼s 1 s 2 v 1 : v 2 ; s 1 v 2 þ s 2 v 1 þ v 1 v 2 ½ (B.45)

ð q 1 q 2 Þq 3 ¼ q 1 q 2 q 3 ð Þ (B.46)

q 1 q 2 6¼ q 2 q 1 (B.47)

p

s

2 2 2 2

kqk¼ þ x þ y þ z (B.48)

½ s; v ½ 1 ; ð 0 ; 0 ; 0 Þ ¼ s; v ½ (B.49)

2

q 1 ¼ ð 1 =kqk Þ ½ s; v

(B.50)

q 1 q ¼ qq 1

¼ ½ 1 ; ð 0 ; 0 ; 0 Þ

pðÞ

1

¼ q 1 p 1

(B.51)

qunit ¼ q= kqðÞ (B.52)

r ¼ 2 p q ð Þq p (B.52a)

Rotations by quaternions
A point in space is represented by a vector quantity in quaternion form by using a zero scalar value
( Eq. B.53 ). A quaternion can be used to rotate a vector using quaternion multiplication ( Eq. B.54 ) . Com-
pound rotations can be implemented by premultiplying the corresponding quaternions ( Eq. B.55 ), similar
to what is routinely done when rotation matrices are used. As should be expected, compounding a rotation
with its inverse produces the identity transformation for vectors ( Eq. B.56 ) . An axis-angle rotation is repre-
sented by a unit quaternion, as shown in Equation B.57 . Any scalar multiple of a quaternion represents the
same rotation. In particular, the negation of a quaternion (negating each of its four components, q ¼

[ -s , x , y , z ]) represents the same rotation that the original quaternion represents ( Eq. B.58 ) .

v ¼ ½ 0 ; x; y; z (B.53)

0

Rot ðÞ¼qvq 1 v ¼ (B.54)