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Quaternion Numbers
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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 = -j
One can see these rules as if i, j, k are three orthognal 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.
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Last update by Henrik Engström 1994-Oct-27
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