Quaternion Notations

1. QUATERNION NOTATIONS SMG

2. What is a Quaternion? Quaternions are 4d numbers used for events, potentials and 4 momentum. They can be added, subtracted, multiplied or divided. Even Quaternions are all positive. 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).

3. We usually denote quaternions as entities with the form: a + i b + j c + k d Where a,b,c and d are scalar values and i,j and k are 'imaginary operators' which define how the scalar values combine. As a 2x2 matrix whose elements are complex numbers

4. A complex number may be expressed as the sum of a real and imaginary part as follows: a + i b A quaternion adds two additional and independent imaginary parts as follows: a + i b + j c + k d So this adds two extra dimensions which square to a negative number, giving a total of: One dimension which squares to a positive number (real part)

5. We can think of quaternions as an element consisting of a scalar number together with a 3 dimensional vector. In other words we have combined the 3 imaginary values into a vector. We could denote it like this: (s,v) where: s = scalar v = 3D vector So the quaternion still has 4 degrees of freedom, its just that we group the 4 scalars as 1+3 scalars, the quaternion is still an element but the vector is a sub-element within it (if that's not a contradiction in terms). As the equivalent of a unit radius sphere in 4 dimensions.

6. a quaternion can be represented in terms of axis-angle, in the usual notation this is: q = cos(a/2) + i ( x * sin(a/2)) + j (y * sin(a/2)) + k ( z * sin(a/2)) where: a = angle that we are rotating through x,y,z = unit vector representing axis Converting this to scalar & vector form simplifies this as follows, q = (s*cos(a/2), v *sin(a/2)) If this represents a pure rotation q = (cos(a/2), axis*sin(a/2)) where: a = angle that we are rotating through axis = unit length axis vector As a spinor in 3 dimensions.

7. 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

8. p2=q * p1 where: p2 = is a vector representing a point after being rotated q = is a quaternion representing a rotation. p1= is a vector representing a point before being rotated

9. 'Euler Parameters' which are just quaternions but with a different notation. It is shown as four numbers separated by commas instead of the usual notation with the imaginary parts denoted with i, j and k. 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° instead of 360° to get back where it started.

10. Gimbal Lock Gimbal Lock – is the loss of one degree of freedom in a three-dimensional space that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate two-dimensional space.

12. http://www.flipcode.com/documents/matrfaq.html#Q56 http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/notations/index.htm http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/code/index.htm http://en.wikipedia.org/wiki/Gimbal_lock

13. http://www.youtube.com/results?search_query=gimbal+lock&aq=f