Quaternion Numbers
Quaternion Numbers
Definition
A Quaternion number is an extension to complex numbers invented by Lord William Hamilton. A Quaternion number has got two extra imaginary units which are called j and k. We write such a number as q = a + b*i + c*j + d*k, or for short: (a, b, c, d). As long as c and d are equal to zero, q is nothing else then a complex number. These are the multiplication rules for i, j, k:
i*i = j*j = k*k = -1
i*j = k, j*i = -k
j*k = i, k*j = -i
k*i = j, i*k = -jOne can see these rules as if i, j, k are three orthogonal unit vectors, the multiplication acts like a right-handed cross product of them. These rules does not cause any real difficulty until we look at multiplication of quaternions. Later on this page we see a very important fact: multiplication/division of quaternions is not commutative.
Conjugate, Norm and Absolute value
q = (a, b, c, d)
conj(q) = (a, -b, -c, -d)
norm(q) = a*a + b*b + c*c + d*d
abs(q) = sqrt(norm(q))Addition and subtraction
q = (a, b, c, d), p = (x, y, z, w)
q + p = (a+x, b+y, c+z, d+w)
q - p = (a-x, b-y, c-z, d-w)Scaling
q = (a, b, c, d)
t = Real value
t*q = q*t = (t*a, t*b, t*c, t*d)Multiplication
q = (a, b, c, d), p = (x, y, z, w)
q * p = (a + b*i + c*j + d*k) * (x + y*i + z*j + w*k)
= a * (x + y*i + z*j + w*k)
+ b*i * (x + y*i + z*j + w*k)
+ c*j * (x + y*i + z*j + w*k)
+ d*k * (x + y*i + z*j + w*k)
= a*x + a*y*i + a*z*j + a*w*k
+ b*x*i - b*y + b*z*k - b*w*j
+ c*x*j - c*y*k - c*z + c*w*i
+ d*x*k + d*y*j - d*z*i - d*w
= (a*x - b*y - c*z - d*w,
a*y + b*x + c*w - d*z,
a*z - b*w + c*x + d*y,
a*w + b*z - c*y + d*x)One can rewrite this if we define two 3-dimensional vectors:
qV = (b, c, d) q = (a, qV)
pV = (y, z, w) p = (x, pV)
q * p = (a*x - dotprod(qV, pV),
a*pV + x*qV + crossprod(qV, pV))Now it is easy to see that multiplication of quaternion is not commutative, since the cross product of the vectors is not commutative. One must take this in consideration while working with quaternion algebra.
If we square a quaternion, the cross product is zero:
q * q = (a*a - dotprod(qV, qV), 2*a*qV)In fact, One do not have to consider the multiplication order at all when raising a quaternion to any positive integer power.
Fast multiplication
There are some schemes available that reduces the number of internal multiplications when doing quaternion multiplication. Here is one:
q = (a, b, c, d), p = (x, y, z, w)
tmp_00 = (d - c) * (z - w)
tmp_01 = (a + b) * (x + y)
tmp_02 = (a - b) * (z + w)
tmp_03 = (c + d) * (x - y)
tmp_04 = (d - b) * (y - z)
tmp_05 = (d + b) * (y + z)
tmp_06 = (a + c) * (x - w)
tmp_07 = (a - c) * (x + w)
tmp_08 = tmp_05 + tmp_06 + tmp_07
tmp_09 = 0.5 * (tmp_04 + tmp_08)
q * p = (tmp_00 + tmp_09 - tmp_05,
tmp_01 + tmp_09 - tmp_08,
tmp_02 + tmp_09 - tmp_07,
tmp_03 + tmp_09 - tmp_06)With this method You get 7 less multiplications, but 15 more additions/subtractions. Generally, this is still an improvement.
Inverse and division
inv(q) = conj(q)/norm(q)
q / p = inv(p)*q or q*inv(p)Here it can be difficult to know which one to choose. I do not use division very often, but when I have to I always use q/p = +inv(p)*q. If I use it all the time, there seems to be no problem. I am not that good at explaining this, if someone knows better please send me an email.
