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# 22

Theory and Design

nalysis

identities for multiplying i, j, and k are

i 2 = j 2 = k 2 = i j k =

1

(3.8)

q1q2

=

2 2 (w1 + x1i + y1j + z1k)(w2 + x2i + y j + z k)

=

w1w2 +

12 +y z

12 z y )i +

12 xz

12 + z x )j +

12 xy

12 y x )k

12 (w x + x1w2

12 xx

12 (w y + y1w2

12 yy

12 (w z + z1w2

12 zz

and from these identities, the remaining products such as ij = k can be derived.

Using Equation 3.6 and Equation 3.8, the quaternion product of q1, q2 is

(3.9)

# The addition of quaternions q1, q2 is defined in the same manner as vectors, i.e.,

q1 + q2 = w1 + w2 + (x1 + x2)i + (y1 + y2)j + (z1 + z2)k.

(3.10)

Quaternion rotations can be constructed in terms of an axis and a rotation angle. nion q can be defined using a rotation around an axis (l, m, n) by an angle θ using

quater-

q = cos

θ

2

+ l s i n θ 2 i + m s i n θ 2 j + n s i n θ 2 k .

(3.11)

For Equation 3.11 to hold, the axis (l, m, n) must be normalised to length 1.

Similarly an axis and angle can be determined from the components of a quaternion q using

w = cos

θ

x2 + y2 + z2 = sin

2 θ

2

p

(3.12)

that was derived from the definition of a quaternion. This leads to an angle and axis of

θ = 2 cos

x l = 1

1 w2 w, ,

y m = 1

w2

,

z n = 1

w2

.

(3.13)

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