Technical Background - Computer Animation: Algorithms and Techniques - page 59

The inverse of a quaternion ,[ s , v ] 1 , is obtained by negating its vector part and dividing both parts by
the magnitude squared (the sum of the squares of the four components), as shown in Equation 2.27 .

p (2.27)
Multiplication of a quaternion, q, by its inverse, q 1 , results in the multiplicative identity
[1, (0, 0, 0)]. A unit-length quaternion (also referred to here as a unit quaternion), q, is created by divid-
ing each of the four components by the square root of the sum of the squares of those components (Eq.
2.28) .

s q 1

2

¼ð 1 =jjqjjÞ ½s; v where jjqjj ¼

2

þ x

2

þ y

2

þ z

2

q ¼ q=ðjjqjjÞ

(2.28)

Representing rotations using quaternions
A rotation is represented in a quaternion form by encoding axis-angle information. Equation 2.29 shows a
unit quaternion representation of a rotation of an angle, u , about a unit axis of rotation ( x , y , z ).

QuatRotð y ; ðx; y; zÞÞ ½ cos ð y = 2 Þ; sin ð y = 2 Þðx; y; zÞ (2.29)

Notice that rotating some angle around an axis is the same as rotating the negative angle around the
negated axis. In quaternions, this is manifested by the fact that a quaternion, q ¼ [ s , v ], and its negation,

q ¼ v ], represent the same rotation. The two negatives in this case cancel each other out and

produce the same rotation. In Equation 2.30 , the quaternion q represents a rotation of u about a unit axis
of rotation ( x , y , z ), i.e.,

[ s ,

q ¼½ cos ð y = 2 Þ; sin ð y = 2 Þðx; y; zÞ

¼½ cos ð 180 ð y = 2 ÞÞ; sin ð y = 2 Þððx; y; zÞÞ

¼½ cos ðð 360 y Þ= 2 Þ; sin ð 180 y = 2 Þððx; y; zÞÞ

(2.30)

¼½ cos ðð 360 y Þ= 2 Þ; sin ð 360 y = 2 Þððx; y; zÞÞ

QuatRotð y ; ðx; y; zÞÞ
QuatRotð y ; ðx; y; zÞÞ

Negating q results in a negative rotation around the negative of the axis of rotation, which is the
same rotation represented by q ( Eq. 2.30 ) .

Rotating vectors using quaternions
To rotate a vector, v , using quaternion math, represent the vector as [0, v ] and represent the rotation by
a quaternion, q . The vector is rotated according to Equation 2.31 .

0

Rot q ðvÞqvq 1

¼ v (2.31)

A series of rotations can be concatenated into a single representation by quaternion multiplication.
Consider a rotation represented by a quaternion, p , followed by a rotation represented by a quaternion,
q , on a vector, v ( Eq. 2.32 ).

¼ðqpÞvðqpÞ 1 Rot q ðRot p ðvÞÞ ¼ qðpvp 1 Þq 1 ¼ Rot qp ðvÞ (2.32)

The inverse of a quaternion represents rotation about the same axis by the same amount but in the
reverse direction. Equation 2.33 shows that rotating a vector by a quaternion, q , followed by rotating
the result by the inverse of that same quaternion produces the original vector.

ðRot q ðvÞÞ ¼ q 1 ðqvp 1 Rot q 1 Þq ¼ v (2.33)