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- 1. QUATERNION NOTATIONS<br />SMG<br />
- 2. What is a Quaternion?<br />Quaternions are 4d numbers used for events, potentials and 4 momentum.<br />They can be added, subtracted, multiplied or divided.<br />Even Quaternions are all positive.<br />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).<br />
- 3. We usually denote quaternions as entities with the form:<br />a + i b + j c + k d<br />Where a,b,c and d are scalar values and i,j and k are 'imaginary operators' which define how the scalar values combine.<br /> As a 2x2 matrix whose elements are complex numbers<br />
- 4. A complex number may be expressed as the sum of a real and imaginary part as follows: <br />a + i b<br />A quaternion adds two additional and independent imaginary parts as follows: <br />a + i b + j c + k d<br /> <br />So this adds two extra dimensions which square to a negative number, giving a total of:<br />One dimension which squares to a positive number (real part)<br />
- 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.<br />We could denote it like this: (s,v) <br />where:<br />s = scalar<br />v = 3D vector <br />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).<br />As the equivalent of a unit radius sphere in 4 dimensions.<br />
- 6. a quaternion can be represented in terms of axis-angle, in the usual notation this is: <br />q = cos(a/2) + i ( x * sin(a/2)) + j (y * sin(a/2)) + k ( z * sin(a/2))<br />where:<br />a = angle that we are rotating through <br />x,y,z = unit vector representing axis <br />Converting this to scalar & vector form simplifies this as follows, <br />q = (s*cos(a/2), v *sin(a/2)) <br />If this represents a pure rotation <br />q = (cos(a/2), axis*sin(a/2))<br />where:<br />a = angle that we are rotating through <br />axis = unit length axis vector <br />As a spinor in 3 dimensions.<br />
- 7. There is an alternative way to think of quaternions, imagine a complex number:<br />n1 + i n2<br />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,<br />n1 = a + jc<br />n2 = b + jd<br />If we substitute them into the first complex number this gives,<br />(a + jc) + i (b + jd)<br />since i*j = k (see under multiplication) this can be rearranged to give the same form as above.<br />a + i b + jc + kd<br />
- 8. p2=q * p1<br />where:<br />p2 = is a vector representing a point after being rotated <br />q = is a quaternion representing a rotation. <br />p1= is a vector representing a point before being rotated<br />
- 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. <br />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.<br />
- 10. Gimbal Lock<br />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.<br />
- 11.
- 12. http://www.flipcode.com/documents/matrfaq.html#Q56<br />http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/notations/index.htm<br />http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/code/index.htm<br />http://en.wikipedia.org/wiki/Gimbal_lock<br />
- 13. http://www.youtube.com/results?search_query=gimbal+lock&aq=f<br />

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