There is an alternative way to think of quaternions, imagine a complex number:
n1 + i n2
but this time make n1 and n2 (the real and imaginary parts) to be themselves complex numbers (but with a different imaginary part at right angles to the first), so,
- n1 = a + jc
- n2 = b + jd
If we substitute them into the first complex number this gives,
(a + jc) + i (b + jd)
since i*j = k (see under multiplication) this can be rearranged to give the same form as above.
a + i b + jc + kd
There are a number of notations and ways to think about quaternions:
- As a superset of complex numbers with two additional imaginary values.
- As the product of two independent complex planes.
- As a set of four scalar values with defined rules for combining them, known as euler parameters (as opposed to euler angles)
When we are using quaternions to represent rotations in 3 dimensions, then we restrict the quaternions to unit length and only use the multiplication operator, in this case there are other notations and ways to think about quaternions:
- As a quantity similar to axis-angle except that real part is equal to cos(angle/2) and the complex part is made up of the axis vector times sin(angle/2).
- As a 2x2 matrix whose elements are complex numbers, generated by Pauli matrices.
- As the equivalent of a unit radius sphere in 4 dimensions.
Even when normalised, there is still some redundancy when used for 3D rotations, in that the quaternions a + i b + j c + k d represents the same rotation as -a - i b - j c - k d. At least it does in classical mechanics. However in quantum mechanics a + i b + j c + k d and -a - i b - j c - k d represent different spins for particles, so a particle has to rotate through 720 degrees instead of 360 degrees to get back where it started.