Quaternion Multiplication

[Quaternion rules]
i² = j² = k² = i*j*k = -1

i*j = k     j*i = -k
j*k = i     k*j = -i
k*i = j     i*k = -j

[Example quaternion "q" and point "p"]
q(x,y,z,w) = q.w + q.x*i + q.y*j + q.z*k
q'(x,y,z,w) = q.w - q.x*i - q.y*j - q.z*k
p(x,y,z)   = p.x*i + p.y*j + p.z*k


p' = q * p * q'

p' = (q.w + q.x*i + q.y*j + q.z*k) * (p.x*i + p.y*j + p.z*k) * (q.w - q.x*i - q.y*j - q.z*k)

// first and second bracket combined
p' = (
q.w*p.x*i + q.x*i*p.x*i + q.y*j*p.x*i + q.z*k*p.x*i +
q.w*p.y*j + q.x*i*p.y*j + q.y*j*p.y*j + q.z*k*p.y*j +
q.w*p.z*k + q.x*i*p.z*k + q.y*j*p.z*k + q.z*k*p.z*k
) * (q.w - q.x*i - q.y*j - q.z*k)

// all brackets combined
p' = (
q.w*p.x*i*q.w + q.x*i*p.x*i*q.w + q.y*j*p.x*i*q.w + q.z*k*p.x*i*q.w +
q.w*p.y*j*q.w + q.x*i*p.y*j*q.w + q.y*j*p.y*j*q.w + q.z*k*p.y*j*q.w +
q.w*p.z*k*q.w + q.x*i*p.z*k*q.w + q.y*j*p.z*k*q.w + q.z*k*p.z*k*q.w +

q.w*p.x*i*(-q.x*i) + q.x*i*p.x*i*(-q.x*i) + q.y*j*p.x*i*(-q.x*i) + q.z*k*p.x*i*(-q.x*i) +
q.w*p.y*j*(-q.x*i) + q.x*i*p.y*j*(-q.x*i) + q.y*j*p.y*j*(-q.x*i) + q.z*k*p.y*j*(-q.x*i) +
q.w*p.z*k*(-q.x*i) + q.x*i*p.z*k*(-q.x*i) + q.y*j*p.z*k*(-q.x*i) + q.z*k*p.z*k*(-q.x*i) +

q.w*p.x*i*(-q.y*j) + q.x*i*p.x*i*(-q.y*j) + q.y*j*p.x*i*(-q.y*j) + q.z*k*p.x*i*(-q.y*j) +
q.w*p.y*j*(-q.y*j) + q.x*i*p.y*j*(-q.y*j) + q.y*j*p.y*j*(-q.y*j) + q.z*k*p.y*j*(-q.y*j) +
q.w*p.z*k*(-q.y*j) + q.x*i*p.z*k*(-q.y*j) + q.y*j*p.z*k*(-q.y*j) + q.z*k*p.z*k*(-q.y*j) +

q.w*p.x*i*(-q.z*k) + q.x*i*p.x*i*(-q.z*k) + q.y*j*p.x*i*(-q.z*k) + q.z*k*p.x*i*(-q.z*k) +
q.w*p.y*j*(-q.z*k) + q.x*i*p.y*j*(-q.z*k) + q.y*j*p.y*j*(-q.z*k) + q.z*k*p.y*j*(-q.z*k) +
q.w*p.z*k*(-q.z*k) + q.x*i*p.z*k*(-q.z*k) + q.y*j*p.z*k*(-q.z*k) + q.z*k*p.z*k*(-q.z*k)
)


// applying multiplication rules from above

p' = (
 q.w*p.x*q.w*i - q.x*p.x*q.w   - q.y*p.x*q.w*k + q.z*p.x*q.w*j
+q.w*p.y*q.w*j + q.x*p.y*q.w*k - q.y*p.y*q.w   - q.z*p.y*q.w*i
+q.w*p.z*q.w*k - q.x*p.z*q.w*j + q.y*p.z*q.w*i - q.z*p.z*q.w

+q.w*p.x*q.x   + q.x*p.x*q.x*i + q.y*p.x*q.x*j + q.z*p.x*q.x*k
+q.w*p.y*q.x*k - q.x*p.y*q.x*j + q.y*p.y*q.x*i - q.z*p.y*q.x
-q.w*p.z*q.x*j - q.x*p.z*q.x*k + q.y*p.z*q.x   + q.z*p.z*q.x*i

-q.w*p.x*q.y*k + q.x*p.x*q.y*j - q.y*p.x*q.y*i + q.z*p.x*q.y
+q.w*p.y*q.y   + q.x*p.y*q.y*i + q.y*p.y*q.y*j + q.z*p.y*q.y*k
+q.w*p.z*q.y*i - q.x*p.z*q.y   - q.y*p.z*q.y*k + q.z*p.z*q.y*j

+q.w*p.x*q.z*j + q.x*p.x*q.z*k - q.y*p.x*q.z   - q.z*p.x*q.z*i
-q.w*p.y*q.z*i + q.x*p.y*q.z   + q.y*p.y*q.z*k - q.z*p.y*q.z*j
+q.w*p.z*q.z   + q.x*p.z*q.z*i + q.y*p.z*q.z*j + q.z*p.z*q.z*k
)

// regroup according to imaginary units:

p' = (
- q.x*p.x*q.w   - q.y*p.y*q.w   - q.z*p.z*q.w   + q.w*p.x*q.x   - q.z*p.y*q.x   + q.y*p.z*q.x   + q.z*p.x*q.y   + q.w*p.y*q.y   - q.x*p.z*q.y   - q.y*p.x*q.z   + q.x*p.y*q.z   + q.w*p.z*q.z
+ q.w*p.x*q.w*i - q.z*p.y*q.w*i + q.y*p.z*q.w*i + q.x*p.x*q.x*i + q.y*p.y*q.x*i + q.z*p.z*q.x*i - q.y*p.x*q.y*i + q.x*p.y*q.y*i + q.w*p.z*q.y*i - q.z*p.x*q.z*i - q.w*p.y*q.z*i + q.x*p.z*q.z*i
+ q.z*p.x*q.w*j + q.w*p.y*q.w*j - q.x*p.z*q.w*j + q.y*p.x*q.x*j - q.x*p.y*q.x*j - q.w*p.z*q.x*j + q.x*p.x*q.y*j + q.y*p.y*q.y*j + q.z*p.z*q.y*j + q.w*p.x*q.z*j - q.z*p.y*q.z*j + q.y*p.z*q.z*j
- q.y*p.x*q.w*k + q.x*p.y*q.w*k + q.w*p.z*q.w*k + q.z*p.x*q.x*k + q.w*p.y*q.x*k - q.x*p.z*q.x*k - q.w*p.x*q.y*k + q.z*p.y*q.y*k - q.y*p.z*q.y*k + q.x*p.x*q.z*k + q.y*p.y*q.z*k + q.z*p.z*q.z*k
)

// factor out the imaginary units

p' = (
(- q.x*p.x*q.w + q.w*p.x*q.x - q.y*p.y*q.w + q.w*p.y*q.y - q.z*p.z*q.w + q.w*p.z*q.z - q.z*p.y*q.x+ q.x*p.y*q.z + q.y*p.z*q.x - q.x*p.z*q.y + q.z*p.x*q.y - q.y*p.x*q.z )+
(+ q.w*p.x*q.w - q.z*p.y*q.w + q.y*p.z*q.w + q.x*p.x*q.x + q.y*p.y*q.x + q.z*p.z*q.x - q.y*p.x*q.y + q.x*p.y*q.y + q.w*p.z*q.y - q.z*p.x*q.z - q.w*p.y*q.z + q.x*p.z*q.z)*i+
(+ q.z*p.x*q.w + q.w*p.y*q.w - q.x*p.z*q.w + q.y*p.x*q.x - q.x*p.y*q.x - q.w*p.z*q.x + q.x*p.x*q.y + q.y*p.y*q.y + q.z*p.z*q.y + q.w*p.x*q.z - q.z*p.y*q.z + q.y*p.z*q.z)*j+
(- q.y*p.x*q.w + q.x*p.y*q.w + q.w*p.z*q.w + q.z*p.x*q.x + q.w*p.y*q.x - q.x*p.z*q.x - q.w*p.x*q.y + q.z*p.y*q.y - q.y*p.z*q.y + q.x*p.x*q.z + q.y*p.y*q.z + q.z*p.z*q.z)*k
)

// eliminate real part and group terms after p.x, p.y and p.z

p' = (
(0 )+
(+ q.w*q.w*p.x + q.x*q.x*p.x - q.y*q.y*p.x - q.z*q.z*p.x - q.z*q.w*p.y + q.x*q.y*p.y + q.x*q.y*p.y - q.z*q.w*p.y + q.y*q.w*p.z + q.x*q.z*p.z + q.y*q.w*p.z + q.x*q.z*p.z)*i+
(+ q.z*q.w*p.x + q.y*q.x*p.x + q.w*q.z*p.x + q.x*q.y*p.x + q.w*q.w*p.y - q.x*q.x*p.y - q.z*q.z*p.y + q.y*q.y*p.y + q.z*q.y*p.z + q.y*q.z*p.z - q.x*q.w*p.z - q.w*q.x*p.z)*j+
(+ q.z*q.x*p.x - q.y*q.w*p.x + q.x*q.z*p.x - q.w*q.y*p.x + q.x*q.w*p.y + q.y*q.z*p.y + q.w*q.x*p.y + q.z*q.y*p.y - q.y*q.y*p.z + q.z*q.z*p.z + q.w*q.w*p.z - q.x*q.x*p.z)*k
)


// factor out p.x,p.y and p.z

p' = (
(0 )+
(+ (q.w*q.w + q.x*q.x - q.y*q.y - q.z*q.z)*p.x + (q.x*q.y + q.x*q.y - q.z*q.w - q.z*q.w)*p.y + (q.y*q.w + q.y*q.w + q.x*q.z + q.x*q.z)*p.z)*i+
(+ (q.z*q.w + q.y*q.x + q.w*q.z + q.x*q.y)*p.x + (q.w*q.w - q.x*q.x - q.z*q.z + q.y*q.y)*p.y + (q.z*q.y + q.y*q.z - q.x*q.w - q.w*q.x)*p.z)*j+
(+ (q.z*q.x - q.y*q.w + q.x*q.z - q.w*q.y)*p.x + (q.x*q.w + q.y*q.z + q.w*q.x + q.z*q.y)*p.y - (q.y*q.y + q.z*q.z + q.w*q.w - q.x*q.x)*p.z)*k
)


// simplify

p' = (
(0 )+
(+ (q.w*q.w+q.x*q.x-q.y*q.y-q.z*q.z)*p.x + (2*q.x*q.y - 2* q.z*q.w )        *p.y + (2*q.y*q.w + 2*q.x*q.z)          *p.z)*i+
(+ (2*q.z*q.w +2*q.x*q.y)           *p.x + (q.w*q.w-q.x*q.x-q.z*q.z+q.y*q.y)*p.y + (2*q.z*q.y - 2*q.x*q.w)          *p.z)*j+
(+ (2*q.z*q.x - 2*q.y*q.w)          *p.x + (2*q.x*q.w + 2*q.y*q.z)          *p.y - (q.y*q.y+q.z*q.z+q.w*q.w-q.x*q.x)*p.z)*k
)